UNIVERSIDADE FEDERAL DO RIO GRANDE DO NORTE

CENTRO DE CIENCIAS EXATAS E DA TERRA

DEPTO. DE INFORMATICA E MATEMATICA APLICADA

POS-GRADUACAO EM SISTEMAS E COMPUTACAO

On Extension of Fuzzy Connectives

Eduardo Silva Palmeira

Natal/RN

february 2013

Eduardo Silva Palmeira

On Extension of FuzzyConnectives

Thesis presented to the Graduate Program in

Systems and Computation of Department of In-

formatics and Applied Mathematics of Federal

University of Rio Grande do Norte in a fulfill-

ment of the Requirements for the degree of Phi-

losophy Doctor .

Advisor:Prof. Dr. Benjamın Rene Callejas Bedregal

Natal/RN

2013

I dedicate this thesis to my family and specially to my wife and my

daughter who always encouraged and accompanied my studies

giving me all needed support.

Acknowledgements

I have a lot of thanks to do, because a thesis is not built in days or

weeks, it is the result of a lengthy effort and a lot of persistence and

perseverance, where we deal every day with frustrations and achieve-

ments in the struggle for the consolidation of ideas and conjectures

that constitute this work today.

First of all I thank God who gave me the gift of life and that, in good

and bad times, was always by my side watching and illuminating my

path. For the infinite blessings He give me every day, praise and glory

be given in His name.

I thank my parents Pedro and Telma, who directly or indirectly have

always believed in me, giving me full support and that, even in the

face of life’s difficulties, never failed to give me everything they could

so that I tread my path to greatness in life. Also to my brothers

Cristina and Danillo and my nephews Hendrick and Enzo for all the

warmth.

A special thank to the woman who gave me a family, my wife Stephanie

throughout companionship and dedication shown throughout these

four years of study doing me dreams her dreams. Thanks for always

being willing to listen to my bullshits, my complaints, for believing

and supporting my projects, for supporting my boredom when I was

exhausted. Anyway, for being an angel in my life and always be watch-

ing with great affection and love for me and our family. Maybe there

is no a word deep enough to express all my gratitude and what I’m

feeling right now writing these words. So I just say: I love you, all

this here is not mine, it is ours!

I could not forget to thank my daughter Lunna, my sweet, my smile.

Thanks for having me often waited at the door to give me a hug

overpowering covered with the purest feeling of love, able to renew

me and make me forget everything bad that might exist in the world.

For my master, professor Benjamın Bedregal, for believing in me from

the beginning and for putting all possible credibility in my work. Cer-

tainly this work would not be the same, if it not had the orientation

of a person so qualified and competent, able to demonstrate a humil-

ity so great as to always treat their students with equal professional.

Moreover, I would like to thank you for your friendship and partner-

ship, and for the beers and casual conversations. I leave this course

happy for having won much more than a research partner for doing a

great friend.

Also, for GIARA research group from Pamplona-Spain, in the figure

of the professor Humberto Bustince, thank you very much for giving

me the opportunity to be researching with them for six months and

the experience that they provided me with a personal and professional

enrichment of great value.

Finally, thank you Natal, this beautiful city where I could enjoy won-

derful days and made friends that I leave here, but that I will take

forever in my heart and my memory. In particular, I would like to

thank Isaac (and Lisa), Aluisio (and Bruninha), Fagner (and Fabi-

ana), Marcelo (and Lene), Anchieta, Guga, Samir, Rogerio, Macilon,

Liliane, among others, perchance, I may have forgotten, for all friend-

ship demonstrated. I could not also forget to thank the teachers and

other employees of the LoLITA group of UFRN for the support gen-

erated during this period. Also, I would like to thank for UESC and

CAPES for the financial support.

.

Stay Hungry, Stay Foolish

(Steve Jobs)

Abstract

Let M be a sublattice of lattice L and K be a fuzzy operator (eg. a

t-norm, a t-conorm, a fuzzy fuzzy negation, etc.) on M . So, how can

we extend this operator from M to L preserving the most possible

number of its properties?

This is a very known and interesting problem and a suitable solu-

tion for this is not so trivial. In this framework we present in this

thesis two different methods for extending t-norms, t-conorms, fuzzy

negations and implications from a sublattice to a lattice consider-

ing a generalized notion of sublattices, namely (r, s)-sublattice. In

one method, named extension method via retraction, we extend fuzzy

connectives trying to obtain the smallest possible extension while the

other method aims to make an extension able to preserve the largest

number of properties of extended connective using a special extension

operator (e-operator, for short). Also, we present some results involv-

ing extension, automorphisms, De Morgan triples and t-subnorms.

Moreover, leaving a bit the scope of fuzzy logic, we investigate the

behavior of extension methods proposed in this thesis for some func-

tions related to study of images. We do the extension of Fodor and

Roubens’ equivalence functions as well as the extension of a particu-

lar case of them named restricted equivalence functions. Also in this

framework, we extend restricted dissimilarity functions and normal

Ee,N -functions.

List of Figures

2.1 Hasse diagrams of lattices L and M . . . . . . . . . . . . . . . . . 13

2.2 The relaxed idea of sublattice . . . . . . . . . . . . . . . . . . . . 16

2.3 Hasse diagrams of lattices M , L1, L2 and L3 . . . . . . . . . . . . 18

2.4 Hasse diagrams of lattices M and L . . . . . . . . . . . . . . . . . 19

2.5 Hasse diagrams of lattices M and L . . . . . . . . . . . . . . . . . 26

4.1 Diagram of interval constructor . . . . . . . . . . . . . . . . . . . 81

4.2 Diagram of intuitive idea . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Hasse diagrams of lattices M and L . . . . . . . . . . . . . . . . . 83

vii

LIST OF FIGURES

viii

List of Tables

2.1 Tables of retractions r1, r2 and r3. . . . . . . . . . . . . . . . . . . 18

2.2 Tables of pseudo-inverses s1, s2 and s3. . . . . . . . . . . . . . . . 18

2.3 A function I on L . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Implication on M . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Table of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Properties preserved by TE and SE . . . . . . . . . . . . . . . . . 77

3.4 Properties preserved by IE . . . . . . . . . . . . . . . . . . . . . . 78

3.5 Properties preserved by NE . . . . . . . . . . . . . . . . . . . . . 78

4.1 Table of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2 Properties preserved by TE� and SE� . . . . . . . . . . . . . . . . . 104

4.3 Properties preserved by NE� . . . . . . . . . . . . . . . . . . . . . 104

5.1 Restricted equivalence function on lattice M . . . . . . . . . . . . 108

5.2 Restricted equivalence function on lattice L . . . . . . . . . . . . 113

5.3 The function IREF on lattice M . . . . . . . . . . . . . . . . . . . 115

6.1 Table of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2 Comparing results of TE and TE� . . . . . . . . . . . . . . . . . . 139

6.3 Comparing results of SE and SE� . . . . . . . . . . . . . . . . . . . 139

6.4 Comparing results of IE and IE� . . . . . . . . . . . . . . . . . . . 139

6.5 Comparing results of NE and NE� . . . . . . . . . . . . . . . . . . 139

ix

LIST OF TABLES

x

Contents

List of Figures vii

List of Tables ix

Contents xi

1 Introduction 1

1.1 On Extension Problem . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Restricted Equivalence Functions . . . . . . . . . . . . . . . . . . 5

1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 9

2.1 Lattice Theory: Definitions and Constructions . . . . . . . . . . . 9

2.1.1 Bounded Lattices and Homomorphisms . . . . . . . . . . . 10

2.1.2 Automorphisms and Conjugates . . . . . . . . . . . . . . . 13

2.1.3 Retractions and Sublattices . . . . . . . . . . . . . . . . . 15

2.1.4 Pseudo-quasi-metrics and Continuity . . . . . . . . . . . . 21

2.2 Fuzzy Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 T-norms and T-conorms . . . . . . . . . . . . . . . . . . . 23

2.2.2 Fuzzy Negations . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.3 Fuzzy Implications . . . . . . . . . . . . . . . . . . . . . . 31

2.2.4 T-subnorms . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.5 De Morgan Triples . . . . . . . . . . . . . . . . . . . . . . 37

xi

CONTENTS

3 Extension Method via Retractions 41

3.1 T-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 T-conorms and Fuzzy Negations . . . . . . . . . . . . . . . . . . . 46

3.2.1 Negations Obtained from Fuzzy Connectives . . . . . . . . 51

3.3 Fuzzy Implications . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 (S,N)-implications . . . . . . . . . . . . . . . . . . . . . . 62

3.3.2 R-implications . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 De Morgan Triples . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Extension and Automorphisms . . . . . . . . . . . . . . . . . . . . 69

3.6 T-subnorms and T-subconorms . . . . . . . . . . . . . . . . . . . 74

3.7 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Extension Method via e-operators 79

4.1 Toward e-operators . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 T-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 T-conorms and Fuzzy Negations . . . . . . . . . . . . . . . . . . . 89

4.4 De Morgan Triples . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5 Extension and Automorphisms . . . . . . . . . . . . . . . . . . . . 92

4.6 On Extension of n-dimensional T-norms . . . . . . . . . . . . . . 97

4.7 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 On Restricted Equivalence Functions 105

5.1 Restricted Equivalence Functions on L . . . . . . . . . . . . . . . 106

5.1.1 Restricted Equivalence Functions and Negations . . . . . . 109

5.2 Characterization Theorem for L-REF . . . . . . . . . . . . . . . . 111

5.3 Restricted Dissimilarity Functions . . . . . . . . . . . . . . . . . . 116

5.4 Normal Ee,N -functions on L . . . . . . . . . . . . . . . . . . . . . 119

5.5 REF on L([0, 1]): Definition and characterization . . . . . . . . . 121

5.6 Extension of REF via Retractions . . . . . . . . . . . . . . . . . . 124

5.7 Extension of REF via e-operators . . . . . . . . . . . . . . . . . . 127

5.7.1 Extension of Natural Negation of REF . . . . . . . . . . . 129

5.8 Extension of Restricted Dissimilarity Function . . . . . . . . . . . 130

5.9 Extension of Normal Ee,N -functions . . . . . . . . . . . . . . . . . 133

xii

CONTENTS

6 Remarks and Further Works 137

References 143

Index 151

xiii

CONTENTS

xiv

Chapter 1

Introduction

This thesis is devoted to discuss about the problem of extending fuzzy connec-

tives from sublattices to a greater one in such way to provide a suitable extension

of them able to preserve the largest number of their properties.

Besides being a naturally interesting and challenging mathematical problem

(Hestenes [1941]; Uspenskii [1966]; Whitney [1934]), the problem of extending

functions or operators arises in many situations in various branches of computer

science such as image processing, mathematical morphology, computational se-

mantics, object-oriented programming, etc. For instance, there is a very known

operator in object-oriented programming named inheritance and behavioral sub-

typing that works as an extension operator through which one can extend a

subclass to a superclass, leveraging their behaviors (methods) and variables (at-

tributes). In mathematical morphology, erosion and dilation operators play a

similar role switching images between lattices.

In fuzzy mathematics (particularly in fuzzy logic) most of operators are func-

tions in usual sense and hence it can be considered the extension problem for

them. Within this framework, we present here two extension methods based in

a generalized notion of sublattice defined using retractions which have different

behavior in preserving properties of fuzzy operators as one can see in Chapters

3 and 4. Also, for having an applied point of view we lead with extension of

restricted equivalence functions defined on bounded lattices.

1

1. INTRODUCTION

1.1 On Extension Problem

The problem of extending functions is very know and studied by many re-

searchers in many branches of knowledge. Particularly, in exact sciences, the

problem of extending functions is widely studied in some areas such as mathe-

matics, physics, computer science, etc (see Hans-Peter and Shapiro [1997]; Hori-

uchi and Murakami [1993]; Murota and Shioura [2000]). In a general way, we can

describe this problem as follows: how to extend a given function from a subset to

entire domain preserving its main properties, that is, if L is an arbitrary set and

supposing f is a function defined and possessing a property P on a nonempty

subset M of L how can f be extended to L in order to preserve property P for

the elements of L\M? In other words, which is the best choice to define f(x) for

elements x ∈ L\M?

The answer for this is: It depends! This is very simple if we want only to

construct a new function that has L as its domain. In this case, it is enough

to define f(x) = a for a suitable and fixed a belonging to L (i.e., define f as

a constant function for the elements belonging to L\M). However, this task

becomes more complex if we want that extension of f preserve its properties. For

instance, an important theorem of analysis states that a continuous function f

defined on a bounded closed set M can be extended to the whole space preserving

its continuity Apostol [1974]; Bartle and Sherbert [1982]; Lima [1982].

1.2 Fuzzy Logic

Emerging from the important work “Fuzzy Sets Theory” proposed by Zadeh

[1965], fuzzy logic is a generalization of classical logic which typically considers

for membership degrees values in the unit interval [0,1] instead of only {0, 1}, but

in modern fuzzy logic, lattices are used to range these degrees Goguen [1967];

Gratzer [2011]; Trillas [1979].

Nowadays, fuzzy logic has awakened the interest and curiosity of several re-

searchers in a variety of scientific areas due to its broad scope and the fact that

it provides a good framework for constructing models which better approach to

reality Bedregal and Figueira [2008]; Buckley [2005]; Chalco-Cano et al. [2011];

2

Pardo and de la Fuente [2010]; Takagi [2009]; Zadrozny and Kacprzyk [2009]. In

literature, one may find a substantial amount of works that introduce a fuzzy

version of classic concepts as well as some others that deal with the improvement

of techniques and tools utilizing fuzzy mathematics Bona [2006]; Gratzer [2009];

Hajek [1998]; Jacas and Recasens [1994]; Liu [2011]; Mitra and Pal [2005]; Siler

and Buckley [2005]; Takano [2002].

Talking about applications, fuzzy logic has been widely used to generalize and

define important theories and tools in control theory, artificial intelligence, etc.

According to Bojadziev and Bojadziev [1996]

“Fuzzy sets and fuzzy logic are powerful mathematical tools for mod-

eling and controlling uncertain systems in industry, humanity, and

nature; they are facilitators for approximate reasoning in decision

making in the absence of complete and precise information. Their

role is significant when applied to complex phenomena not easily de-

scribed by traditional mathematics.”

As for every logic, an appropriate definition of connectives and rules is essen-

tial. In fuzzy logic, conjunctions are interpreted by triangular norms (t-norms)

and disjunctions are generalized by triangular conorms (t-conorms). Such as in

classical case, a t-conorm S can be constructed from a given t-norm T , and a

fuzzy negation N , by

N(S(x, y)) = T (N(x), N(y)) (1.1)

In this case, t-conorm S is said to be N -dual of T (by dually principe).

Another concept which plays an important role in fuzzy logic are De Morgan

triples (see Calvo [1992]; da Costa et al. [2011]; Garcıa and Valverde [1989];

Klement et al. [2000]), a fuzzy version of De Morgan’s law. More specifically, a

triple 〈T, S,N〉 where T is a t-norm, S is a t-conorm and N is a negation, may

be called a De Morgan triple if 〈T, S,N〉 satisfies both the equality (1.1) and

N(T (x, y)) = S(N(x), N(y)) (1.2)

In other words, given a t-norm T , 〈T, S,N〉 is a De Morgan triple if and only if

N is a strong negation and S is N -dual to T .

3

1. INTRODUCTION

In particular, notice that t-norms, t-conorms, fuzzy negations and implications

are functions as well as most fuzzy other operators so, the problem of extending

functions can be considered for them. In this context, one of the main works is

due to Saminger-Platz et al. [2008], in which the authors propose a way to extend

a given t-norm T from a complete sublattice (in the sense of the usual concept of

sublattice) M to a bounded lattice L. However, their extension method is very

drastic due to the action of function x∗ = supM{y | y 6L x, y ∈ M ∪ {0L, 1L}}(since it works as a “collapsing function”, see (3.1)). This is motivated by the

fact that they wish to propose the least possible extension of a t-norm T .

In this sense, seeking to have a more general way to extend fuzzy connectives

we would like to propose a method which could be more flexible than one provided

by Saminger-Platz et al. [2008]. To do so, we consider a generalized version of

the concept of sublattice in which M is not necessarily a subset of L, but can be

seen as a copy embedded in L.

Specifically speaking, we propose the following concept of sublattice: let M

and L be two bounded lattices. It is said that M is a (r, s)-sublattice of L if

there is a retraction r : L −→ M with pseudo inverse s : M −→ L such that

r ◦ s = idM ; i.e. M is a (r, s)-sublattice of L if M is a retract of L (see Definition

2.10). A natural example of sublattice in this sense can be given by taking a

bounded lattice K as M and the interval lattice K constructed from it as L (see

Example 4.1).

Based on notion of (r, s)-sublattice, our early research has led us to provide a

way to extend t-norms, t-conorms and fuzzy negations (named extension method

via retractions, see Palmeira and Bedregal [2012]), in which we try to achieve two

goals: (1) generalize the extension of t-norms presented in Saminger-Platz et al.

[2008]; (2) in order to preserve the largest possible number of properties of these

fuzzy connectives which are invariant by retractions. However, despite the first

goal has been achieved, the extension proposed in Palmeira and Bedregal [2012]

does not completely fulfill (2) since some important properties related to t-norms

and t-conorms are not preserved such as continuity, for example. For extension

of fuzzy negation this method does not preserve involution property and hence it

does not preserve strong negations (see Remark 3.2).

Thus, in order to find a way to solve problems presented by extension method

4

via retractions proposed in Palmeira and Bedregal [2012]; Palmeira et al. [2009] we

have turned our attention to investigate extensions that are more effective in pre-

serving properties of fuzzy connectives even this new extensions do not fulfill goal

(1). The inspiration for this comes from the following important fact: the interval

constructor (see Bedregal and Takahashi [2006]; Bedregal et al. [2006b]) provides

a natural way to identify a lattice F with F via retractions. So, we developed

in Palmeira et al. [2012b] a new method to extend t-norms (t-conorms and fuzzy

negations) using a special mapping named e-operator as in Definition 4.1 (this

method is named extension method via e-operator). To verify that this method is

more efficient than extension method via retractions in preserving properties we

investigated in Palmeira et al. [2012b] same issues as in Palmeira and Bedregal

[2012] and results were quite satisfactory. Every problem presented by extension

method via retractions were overcome. Another advantage of extension method

via e-operator becomes evident when we are extending t-norms (and t-conorms)

since some constraints in hypotheses of Theorem 3.1 are not necessary anymore,

namely M could not be a lower (r, s)-sublattice of L (see Theorem 4.1).

1.3 Restricted Equivalence Functions

Basically our studies are theoretical, however determine a wide range of pos-

sible applications. One of our main interest is applying our extension methods

for operators used in several branches of knowledge that lead with images. Our

motivation for choosing images comes from the fact that many works discuss

about images defined on lattices nowadays, and it is natural to generalize con-

cepts related to image processing for lattices in order to obtain a much more

general framework than [0,1]. In particular bounded lattices are of interest since

intensities in images can be considered as taking values in such lattices.

In this framework, after six months working together with GIARA group

(spanish acronym Research Group of Artificial Intelligence and Approximate

Reasoning) under supervision of professor H. Bustince and collaboration with

other group’s members we have worked on restricted equivalence functions, a

very known and powerful tool that provide a good measure for making a global

comparison of images, in which we discuss about the following issues:

5

1. INTRODUCTION

• To provide a formalization of the concept of restricted equivalence functions

on bounded lattices;

• To present a characterization theorem of these functions via implications;

• Apply our two extension methods for extending restricted equivalence func-

tions and test which one has a better behavior.

Results of this research period were submitted for specialized journals on fuzzy

sets (see, Palmeira et al. [submitted 2013a,s]) where we made a formalization of

definition of restricted equivalence functions on bounded lattices and its exten-

sions.

1.4 Objectives

The main objectives of this thesis is developing a consistent formalization of

methods for extending lattice-valued fuzzy connectives and other fuzzy operators

which preserve the largest possible number of their properties. Moreover, we

discuss about the efficiency of these methods for extending restricted equivalence

functions.

Hence, seeking to achieve expected results we turn our attention to investigate

the following specific goals:

• Propose a suitable definition of the generalized notion of sublattices ((r, s)-

sublattices) relaxing some classical constraints of this concept using retrac-

tions;

• Define a way to extend fuzzy connectives (t-norms, t-conorms, fuzzy nega-

tions and implications) from a (r, s)-sublattice to a lattice which is able to

preserves their properties. In this case, we develop two different methods:

(1) extension method via retraction seeking to generalize the method pro-

posed by Saminger-Platz et al. [2008] and (2) a method efficient in preserv-

ing properties of extended connectives (extension method via e-operator);

• Apply both extension method for other fuzzy operators namely t-subnorms,

De Morgan triples, etc;

6

• Establish the relation between extension and conjugate of a fuzzy connective

and other operators;

• Define restricted equivalence functions on bounded lattices and study some

related properties and operators. Also, apply our extension methods for

extending it;

• Develop a study about restricted dissimilarity functions and normal Ee,N -

functions defined on bounded lattices and its extensions.

1.5 Structure of Thesis

This thesis is composed of six chapters which are divided in sections in which

we discuss about, lattices, extension, fuzzy connectives, restricted equivalence

functions and other operators. We begin describing the research problem of this

thesis and making a discussion about state of art on our research area in Chapter

1 (Introduction). Other chapters are organized as follows:

• In Chapter 2 we make a specific formalization of main concepts used along

this work such as lattices, automorphisms, retractions, sublattices, continu-

ity and fuzzy connectives which constitute our theoretical bases;

• Chapter 3 is devoted to present our extension method via retractions. We

have chosen t-norms, t-conorms, fuzzy negations, fuzzy implications, De

Morgan triples and t-subnorms for applying this method and to test it

efficiency in preserving properties. For a better organization, we divided

this chapter in six section in order to study these issues separately. We also

discuss about the relation between extension and conjugate in Section 3.5;

• Also within the framework of extension, we propose in Chapter 4 the ex-

tension method via e-operators. In section 4.1 it is introduced a notion of

e-operator and some results are proved. The next four sections are dedi-

cated to discuss about similar issues like in Chapter 3. Section 4.6 presents

a study on extension of n-dimensional t-norm;

7

1. INTRODUCTION

• Restricted equivalence functions are addressed in Chapter 5. In section

5.1, we present the definition of L-REF and then we characterize them by

implications in Section 5.2. Section 5.3 and 5.4 are dedicated to introduce

the notion of restricted dissimilarity functions and normal Ee,N -functions

on bounded lattices and to prove some results. Section 5.5 discusses about

restricted equivalence functions on L([0, 1]) and presents a generalization

of the REF characterization theorem given in Bustince et al. [2006]. In

Sections 5.6 and 5.7 we apply our extension method for L-REF;

• Finally, in Chapter 6 we write about our main contributions and publica-

tions beyond to present ideas and conjectures for further works.

8

Chapter 2

Preliminaries

This chapter is devoted to present and discuss about main concepts and results

we are leading in this work which constitute the framework of our studies.

We start in Section 2.1 making a formalization of lattices and its morphisms.

In which follows we introduce the key-concept of (r, s)-sublattices, a generalized

way to define the notion of sublattice, some useful examples and related defini-

tions. Continuity for lattices is considered at the end of this section.

In Section 2.2 we turn our attention to recall the usual notions of lattice-

valued fuzzy connectives such as t-norms, t-conorms, fuzzy negations and fuzzy

implications. Also in this framework we discuss about t-subnorms and De Morgan

triples.

It is important to point out here that most of definitions and results present

is this chapter are from the literature, but some of them were proposed by us.

So, to highlight this fact definitions and results that came from literature we cite

its respective authors.

2.1 Lattice Theory: Definitions and Construc-

tions

As a basis for our developments a clever and consistent formalization on lattice

theory and related concepts is necessary since all of operators we are considering

in this work are lattice-valued. It is important to say that throughout this section

9

2. PRELIMINARIES

L represents a bounded lattice.

Some elementary concepts will not be shown here, however for a further read-

ing about such concepts we recommend Bedregal et al. [2007]; Birkhoff [1973];

Burris and Sankappanavar [2005]; Chen and Pham [2001]; de Cooman and Kerre

[1994]; Klement and Mesiar [2005]; Klement et al. [2000]; Klir and Yuan [1995];

Lowen [1996].

2.1.1 Bounded Lattices and Homomorphisms

It is very known that the concept of lattices has two approaches, namely: an

algebraic and an order-theoretical approaches.

Definition 2.1 Birkhoff [1973] Let L be a nonempty set and 6L be a partial

order on it. We define 〈L,6L〉 as an ord-lattice if for all a, b ∈ L the set {a, b}has a supremum and an infimum. If there are two elements 0L and 1L in L such

that 0L 6L x (bottom) and x 6L 1L (top) for each x ∈ L, then 〈L,6L, 0L, 1L〉 is

called a bounded lattice.

Definition 2.2 Birkhoff [1973] Let L be a nonempty set. If ∧L and ∨L are two

binary operations on L, then 〈L,∧L,∨L〉 is an alg-lattice provided that for each

x, y, z ∈ L, the following properties stand:

1. x ∧L y = y ∧L x and x ∨L y = y ∨L x (commutativity);

2. (x∧L y)∧L z = x∧L (y∧L z) and (x∨L y)∨L z = x∨L (y∧L z) (associativity);

3. x ∧L (x ∨L y) = x and x ∨L (x ∧L y) = x (absorption law).

If in L there are elements 0L and 1L such that, for all x ∈ L, x∨L0L = x (bottom)

and x ∧L 1L = x (top), then 〈L,∧L,∨L, 0L, 1L〉 is a bounded lattice .

A very interesting fact is that Definitions 2.1 and 2.2 are equivalent. Indeed,

it is known from the literature that given an alg-lattice L, the relation

x 6L y if and only if x ∧L y = x (2.1)

10

defines a partial order on L and hence L can be seen as an ord-lattice and vice-

versa. This allows us to use both definitions indiscriminately. We consider in

this whole work the algebraic notion of this concept (Definition 2.2) taking into

account that from this structure we can always define a partial order on L what

is very important for us to compare elements. The reason for choosing it rises up

from the lattice homomorphism as we discuss in rest of this subsection.

Remark 2.1 From now on when we say that L is a bounded lattice it means that

L has a structure as in Definition 2.2. Otherwise, an appropriate distinction will

be made.

Definition 2.3 Birkhoff [1973] A lattice L is called a complete lattice if every

subset of it has a supremum and an infimum element. Notice that every complete

lattice is bounded.

Example 2.1 The set [0, 1] endowed with the operations defined by x ∧ y =

min{x, y} and x∨ y = max{x, y} for all x, y ∈ [0, 1] is a complete bounded lattice

in the sense of Definitions 2.2 and 2.3 which has 0 as the bottom and 1 as the

top element.

Example 2.2 For all x, y ∈ [0, 1] it is possible to define the interval set L([0, 1]) =

{[x, y] ; 0 6 x 6 y 6 1} . This set equipped with the operations

[x, y] ∧L [w, z] = [x ∧ w, y ∧ z] and [x, y] ∨L [w, z] = [x ∨ w, y ∨ z].

with a ∧ b = min(a, b) and a ∨ b = max(a, b), is a complete bounded lattice (in

the sense of Definition 2.2) which has [0, 0] and [1, 1] as a bottom and a top

respectively. It is easy to see that this lattice is also obtained by considered in

L([0, 1]) the partial order [a, b] 62 [c, d] if and only if a 6 c and b 6 d.

Remark 2.2 When 6L is a partial order on L and there are two elements x

and y belonging to L such that neither x 6L y nor y 6L x, these elements are

said to be incomparable and we denote this by x ‖ y . Otherwise we say they are

comparable (notation: x ¨ y) .

11

2. PRELIMINARIES

Definition 2.4 Davey and Priestley [2002] Let (L,6L, 0L, 1L) and (M,6M , 0M , 1M)

be bounded lattices. A mapping f : L −→M is said to be a lattice ord-homomorphism

if, for all x, y ∈ L, it follows that

1. If x 6L y then f(x) 6M f(y);

2. f(0L) = 0M and f(1L) = 1M .

Definition 2.5 Davey and Priestley [2002] Let (L,∧L,∨L, 0L, 1L) and (M,∧M ,∨M , 0M , 1M) be bounded lattices. A mapping f : L −→ M is said to be a lattice

alg-homomorphism if, for all x, y ∈ L, we have

1. f(x ∧L y) = f(x) ∧M f(y);

2. f(x ∨L y) = f(x) ∨M f(y);

3. f(0L) = 0M and f(1L) = 1M .

Definition 2.6 Hungerford [2000] A given lattice homomorphism f on L is

called:

1. A monomorphism if it is injective;

2. An epimorphism if f is surjective;

3. An isomorphism when f is bijective. An automorphism is an isomorphism

from a lattice to itself.

Proposition 2.1 Every alg-homomorphism is an ord-homomorphism.

Proof: Let f : L −→ M be an alg-homomorphism. Since x 6L y if only if

x∧L y = x, therefore f(x) = f(x∧L y) = f(x)∧M f(y) and hence f(x) 6M f(y).

�

12

However, in general, the reciprocal of Proposition 2.1 does not hold. If f :

L −→ M is an ord-homomorphism, since x ∧L y 6L x and x ∧L y 6L y, so

f(x∧L y) 6M f(x) and f(x∧L y) 6M f(y). Thus, f(x∧L y) 6M f(x)∧M f(y) =

inf{f(x), f(y)}, however it is possible for f(x ∧L y) 6= inf{f(x), f(y)} to occur.

For example, consider the lattices L and M , as depicted in Hasse diagram shown

in Figure 2.1. Nevertheless, the map f : L −→ M defined by f(0L) = 0M ,

f(1L) = 1M , f(x) = u and f(y) = v, preserves infimum and supremum elements

and, hence, is an ord-homomorphism, though it is not an alg-homomorphism as

∧ operation is not preserved.

L◦

◦ ◦

◦

1L

x y

0L@@@

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M◦

◦ ◦

◦

◦

1M

u v

w

0M

@@@

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Figure 2.1: Hasse diagrams of lattices L and M

In other words the Proposition 2.1 says that every alg-homomorphism is order-

preserving. Due to this fact we have chosen to use the algebraic approach to

lattice homomorphisms (to see the other way to define lattice homomorphisms

we recommend Birkhoff [1973]; Davey and Priestley [2002]). From now on, alg-

homomorphisms will be called just homomorphisms for simplicity.

2.1.2 Automorphisms and Conjugates

Proposition 2.2 Let L be a bounded lattice. Then a function f : L −→ L is an

automorphism if and only if

1. f is bijective and

2. x 6L y if and only if f(x) 6L f(y).

13

2. PRELIMINARIES

Proof: If f is an automorphism, from Proposition 2.1 we just need to prove that

x 6L y when f(x) 6L f(y). Suppose that f(x) 6L f(y), then f(x)∧Lf(y) = f(x)

and so, as f is a lattice homomorphism, f(x ∧L y) = f(x). Thus, since f is

bijective, x ∧L y = x and therefore x 6L y.

Conversely, suppose that either 1. or 2. is not satisfied. If 1. is violated,

i.e., if f is not bijective, then it is not an automorphism by definition. If 2. is

violated, then we have two possible cases:

(i) either there are x, y ∈ L such that x 6L y but not f(x) 6L f(y). However,

then f(x)∧L f(y) <L f(x) = f(x∧L y), and thus f is not a homomorphism, and

hence neither an automorphism;

(ii) or there are x, y ∈ L such that f(x) 6L f(y) but not x 6L y. Put z = x∧L y.

Then z <L x, and f(x ∧L y) = f(z) 6= f(x) = f(x) ∧L f(y) due to the fact that

f is bijective; again this mean that f is not a homomorphism, and hence not an

automorphism.

�

Remark 2.3 We denote the set of all automorphisms on a bounded lattice L by

Aut(L) . This set endowed with the composition operation is a group that has

as neutral element the identity function idL. In algebra, an important tool is

the action of the groups on sets Burris and Sankappanavar [2005]; Hungerford

[2000]. In our case, the action of automorphism group transforms lattice functions

in other lattice functions.

Definition 2.7 Bustince et al. [2003] Let L be a bounded lattice and Ln = L ×· · · × L be a cartesian product of L n-times. Given a function f : Ln → L, the

action of an automorphism ρ over f results in the function fρ : Ln → L defined

as

fρ(x1, . . . , xn) = ρ−1(f(ρ(x1), . . . , ρ(xn))) (2.2)

In this case, fρ is called a conjugate of f .

Notice that if f : Ln → L is a conjugate of g : Ln → L and g is a conjugate

of h : Ln → L then f is a conjugate of h (automorphisms are closed under

14

composition) and if f is a conjugate of g, then g is also a conjugate of f since the

inverse of an automorphism is also an automorphism. Thus, an automorphism

action on set of n-ary functions on L (LLn) determines an equivalence relation on

LLn.

Let f : Ln → L be a conjugate of g : Ln → L. If f(x1, . . . , xn) 6Lg(x1, . . . , xn) for every (x1, . . . , xn) ∈ Ln then we denote it by f 6 g.

2.1.3 Retractions and Sublattices

Recall that the classical notion of sublattice is given as following.

Definition 2.8 Birkhoff [1973] An ordinary sublattice of a lattice L is a subset

M of L such that x, y ∈M imply x ∧L y ∈M and x ∨L y ∈M .

Nevertheless, we would like to work in a more flexible framework of sublattice

where M does not need to be a subset of L. Focusing on this idea, we have the

interest to define a generalized notion of sublattice using retractions.

Definition 2.9 Burris and Sankappanavar [2005] A homomorphism r of a lattice

L onto a lattice M is said to be a retraction if there exists a homomorphism s of

M into L which satisfies r ◦ s = idM . A lattice M is called a retract of a lattice

L if there is a retraction r, of L onto M .

Remark 2.4 In the literature, homomorphism s presented in Definition 2.9 has

no specific name. But here, this function play an important hole in our studies

and by this reason we give to it a special name, viz. a pseudo-inverse of retraction

r.

Notice that, if a lattice M is a retract of lattice L, then we have an identifica-

tion of M with a subset K = s(M) of L in which it is carried on some properties

of M to K, including its lattice structure via retraction r (see Figure 2.2). In

this case, K works as a algebraic copy of M (i.e. they are isomorphic) embedded

into L since r is an isomorphism when restricted to K.

Definition 2.10 Let L and M be arbitrary bounded lattices. We say that M is

a (r, s)-sublattice of L if M is a retract of L (i.e. M is a sublattice of L up to

15

2. PRELIMINARIES

Figure 2.2: The relaxed idea of sublattice

isomorphisms). In other words, M is a (r, s)-sublattice of L if there is a retraction

r of L onto M with pseudo-inverse s : M → L.

Proposition 2.3 The following statements hold:

1. M is a (r, s)-sublattice of L if and only if for all x, y ∈ M we have that

r(s(x) ∧L s(y)) ∈M and r(s(x) ∨L s(y)) ∈M ;

2. If M is an ordinary sublattice of L (in the sense of Definition 2.8) then

M is a (r, s)-sublattice of L where s is the identity function and r(x) =

sup{z ∈ L | z 6 x};

3. If M is a (r, s)-sublattice of L then s(M) equipped with restriction of oper-

ation of L is an ordinary sublattice of L.

Proof: Straight.

�

Based on the above proposition we can list at least four main advantages in

working with (r, s)-sublattice notion instead of the classical one:

• Generalize the ordinary notion: Every sublattice in the classical sense

(Definition 2.8) is also a (r, s)-sublattice (Definition 2.10). Indeed, it is

enough to consider s as the inclusion of M into L and as r the mapping

which keeps unchanged M , sends x to supM if the maximum of x and

supM is equal to x, sends x to inf M if the minimum of x and inf M is x,

and send x to x in any other case;

16

• Algebraic invariance: Since M is isomorphic with a subset s(M) of L

every property invariant under homomorphisms that holds in M works on

s(M) as well;

• Flexible identification: In Definition 2.10, a pseudo-inverse s of a re-

traction r can not be unique. This means that if there exist more than

one pseudo-inverse for the same retraction it is possible to identify M with

a subset of L in different ways1 what give us the possibility to chose the

best one for our proposes. But it must be clear that when we say that M

is a (r, s)-sublattice of L we are considering the existence of at least one

pseudo-inverse s and fixing it. However, no matter which pseudo-inverse is

taken, every result presented here remains working;

• Subclasses of sublattices: Considering (r, s)-sublattice notion allows us

to define some subclasses of this concept (see Definition 2.11) what it is not

possible for ordinary sublattices.

Remark 2.5 Throughout this thesis, it is used the concept of (r, s)-sublattice as

in Definition 2.10 instead of retract. Whenever the usual definition of sublattice

is used and it is not clear from the context, this sublattice will be called ordinary

sublattice.

Definition 2.11 Every retraction r : L −→ M (with pseudo-inverse s) which

satisfies s ◦ r 6 idL2 (idL 6 s ◦ r) is called a lower (an upper) retraction . In this

case, M is a lower (an upper) retract of L.

Example 2.3 One can easily see in Figure 2.3 that M is a lower retract (but it

is not an upper retract) of L1, M is an upper retract (but is not a lower retract)

of L2 and M is a retract (but is neither an upper nor a lower retract) of L3. In

fact, the unique possible retractions are ri : Li −→ M with i ∈ {1, 2, 3} defined

as in the Table 2.1. Their pseudo-inverses si : M → Li with i ∈ {1, 2, 3} are

respectively given as in the Table 2.2.

1For each pair (r, s) we have a different identification between M and its image by s.2If f and g are functions on a lattice L it is said that f 6 g if and only if f(x) 6L g(x) for

all x ∈ L.

17

2. PRELIMINARIES

M

◦

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◦

d

b c

a@@@

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L1

◦

◦ ◦

◦

◦

5

3 4

2

1

@@@

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L2

◦

◦

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◦

6

5

3 4

2@

@@

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L3

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6

5

3 4

2

1

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Figure 2.3: Hasse diagrams of lattices M , L1, L2 and L3

x r1

1 a2 a3 b4 c5 d

x r2

2 a3 b4 c5 d6 d

x r3

1 a2 a3 b4 c5 d6 d

Table 2.1: Tables of retractions r1, r2 and r3.

Example 2.4 Let M and L be bounded lattices as shown in Figure 2.4. A map-

ping r : L −→ M given by r(x) = sup{z ∈ M | s(z) 6L x} is a lower retraction

whose pseudo-inverse is a mapping s : M −→ L defined by s(1M) = 1L, s(a) = v,

s(b) = x, s(c) = y, s(d) = z and s(0M) = 0L. Therefore, it follows that M is a

(r, s)-sublattice of L in the sense of Definition 2.10.

Remark 2.6 Note that, given a lower retraction it is possible sometimes to define

an upper retraction with the same pseudo-inverse. For instance, let L and M be

x s1 s2 s3

a 1 2 1b 3 3 3c 4 4 4d 5 6 6

Table 2.2: Tables of pseudo-inverses s1, s2 and s3.

18

M◦

◦

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1

a

b c

d

0

@@@

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L◦

◦

1

t

u◦◦

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◦

◦

v

x y

z

0

@@@

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@@@

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Figure 2.4: Hasse diagrams of lattices M and L

lattices as shown in Figure 2.4. If r is a lower retraction with pseudo-inverse

s as defined in the Example 2.4, then the function r′ given by r′(x) = inf{z ∈M | s(z) >L x} is an upper retraction since idL 6 s ◦ r′. It is easy to check that

s is also a pseudo-inverse of r′.

It is worth noting that if M is a (r, s)-sublattice of L then there is a retraction

r from L onto M , but it is not required to r to be a lower or an upper retrac-

tion. Nevertheless, as shown in the above remark, there may be more than one

retraction from L onto M with the same pseudo-inverse. This is a very useful

particularity of Definition 2.10 and we would like to highlight it in a definition.

Definition 2.12 Let M be a (r1, s)-sublattice of L. We say that

1. M is a lower (r1, s)-sublattice of L if r1 is a lower retraction. Notation:

M < L with respect to (r1, s);

2. M is an upper (r1, s)-sublattice of L whenever r1 is an upper retraction.

Notation: M > L with respect to (r1, s);

3. If r1 is a lower retraction and there is an upper retraction r2 : L −→M such

that its pseudo-inverse is also s, then M is called a full (r1, r2, s)-sublattice

of L. Notation: M E L with respect to (r1, r2, s).

Remark 2.7 Let L be a complete lattice. We denote the case when M is a

complete and lower (respectively upper) (r, s)-sublattice of L by M lL (by M mL) with respect to (r, s).

19

2. PRELIMINARIES

An immediate consequence of the definition of lower (upper) retraction is that,

if M E L then it follows that s ◦ r1 6 idL 6 s ◦ r2.

Proposition 2.4 Let K, M and L be bounded lattices. If KEMEL then KEL.

Proof: We shall prove that there exists a lower and an upper retraction r and

r′ from L onto K, respectively, both with a pseudo-inverse s such that s ◦ r 6idL 6 s ◦ r′.

Supposing K E M with respect to (r1, r2, s1) and M E L with respect to

(r3, r4, s2), it follows that

r1 ◦ s1 = r2 ◦ s1 = idK and s1 ◦ r1 6 idM 6 s1 ◦ r2

and

r3 ◦ s2 = r4 ◦ s2 = idM and s2 ◦ r3 6 idL 6 s2 ◦ r4

Thus, letting r = r1 ◦ r3, r′ = r2 ◦ r4 and s = s2 ◦ s1 then, for all x ∈ K, we have

r ◦ s(x) = r1(r3(s2(s1(x)))) = r1(s1(x)) = x = idK(x)

r′ ◦ s(x) = r2(r4(s2(s1(x)))) = r2(s1(x)) = x = idK(x)

and hence

s ◦ r = s2 ◦ (s1 ◦ r1) ◦ r3 = s2 ◦ r3 6 idL 6 s2 ◦ r4 6 s2 ◦ (s1 ◦ r2) ◦ r4 = s ◦ r′

Since r, r′ and s are homomorphisms (composition of homomorphisms is also a

homomorphism) then K E L with respect to (r, r′, s).

�

Proposition 2.5 Let M lL with respect to (r, s). For every nonempty set A ⊆M it holds that s(supA) = sup s(A). In other words, a pseudo-inverse of a lower

retract preserves supremum element.

Proof: Recall that s(A) = {s(t) ∈ L | t ∈ A}. Putting a = supA and

a′ = sup s(A) we shall prove that s(a) = a′.

20

On one hand, for all t ∈ s(A) there is a k ∈ A such that t = s(k). Since

k 6M a for each k ∈ A, then t = s(k) 6L s(a) by monotonicity of s, i.e. s(a) is

an upper bound of s(A) and hence a′ 6L s(a) since a′ = sup s(A).

On the other hand, if we take an arbitrary element k ∈ A it follows that

s(k) 6L a′ that implies k = r(s(k)) 6M r(a′) for all k ∈ A which means that

r(a′) is an upper bound of A. Thus by properties of supremum we have that

a 6M r(a′) and hence s(a) 6L s(r(a′)) 6L a′ since r is a lower retraction.

�

Analogously, we can prove the following.

Proposition 2.6 Let M mL with respect to (r, s). For every nonempty set A ⊆M it holds that s(inf A) = inf s(A), i.e. a pseudo-inverse of an upper retract

preserves infimum element.

Remark 2.8 It is important to point out that in Proposition 2.5 (Proposition

2.6) the hypothesis that r should be a lower retraction (an upper retraction) is

just used to prove inequality s(supA) 6L sup s(A) (inf s(A) 6L s(inf A)). It

means that sup s(A) 6L s(supA) (s(inf A) 6L inf s(A)) always holds no matter

which kind of retraction r is (i.e., lower, upper or neither).

2.1.4 Pseudo-quasi-metrics and Continuity

Intuitively, a distance among two values is a non-negative real value and so a

distance function on a set X is a function d : X × X −→ R+, where R+ is the

set of non-negative real numbers. In mathematics, several conditions have been

considered for distance functions, but the more used ones are those for metric

distances. However, in order to consider a reasonable general notion of distance,

we will consider the notion of pseudo-quasi-metric distance.

Definition 2.13 Kim [1968] Let X be a set. A function dX : X ×X −→ R+, is

a pseudo-quasi-metrics 1 if for each x, y ∈ X it satisfies the following conditions:

1In other papers, such as in Kasahara [1968], pseudo-quasi-metrics are called pre-metrics .But, the name of pre-metric have been used for a different notion of distance such as in Yaacovet al. [2011].

21

2. PRELIMINARIES

1. dX(x, x) = 0 and

2. dX(x, z) 6 dX(x, y) + dX(y, z)

In mathematics, an important requirement for functions is its continuity,

which intuitively means that small changes in the input result in small changes

in the output of function. Therefore, it is necessary to consider some way of

measuring changes, which can be made by using some distance notion.

Definition 2.14 Smyth [1992] Let dX and dY pseudo-quasi-metrics on sets X

and Y , respectively. A function f : X −→ Y is continuous w.r.t. dX and dY , or

just (dX , dY )-continuous , if for each x ∈ X and ε > 0 there exists δ > 0 such

that for any y ∈ X, if dX(x, y) 6 δ then dY (f(x), f(y)) 6 ε.

Proposition 2.7 Let X be a set and d be a pseudo-quasi-metric on X. Then

d2X : X ×X −→ R+ defined by d2X((x1, x2), (y1, y2)) =√dX(x1, y1)2 + dX(x2, y2)2

is also a pseudo-quasi-metric.

Proof: Straightforward.

�

Proposition 2.8 Let dX and dY be pseudo-quasi-metrics on the sets X and Y ,

respectively. If f : X −→ Y is (dX , dY )-continuous then f 2 : X −→ Y ×Y defined

by f 2(x) = (f(x), f(x)) is (dX , d2Y )-continuous.

Proof: Straightforward.

�

Proposition 2.9 Let dX , dY and dZ be pseudo-quasi-metrics on X, Y and Z, re-

spectively. If g : X −→ Y and f : Y −→ Z are (dX , dY ) and (dY , dZ)-continuous

then f ◦ g is (dX , dZ)-continuous.

Proof: Straightforward.

�

22

2.2 Fuzzy Connectives

Typically fuzzy logic considers for membership degrees values the unit interval

[0,1]. This logic is very rich due to its connectives (t-norm, t-conorm, fuzzy

negation and implication) can be defined in various different ways and have many

different properties since the unique consensus about how to define them is that

truth functions of these connectives have to behave classically on extremal values

0 and 1. For instance, there are several paper in the literature presenting different

definitions for fuzzy implications and variances of it.

Modern fuzzy logic consider lattices to range its degrees of truth for having

a much more general framework. In this section, we discuss about these fuzzy

connectives on bounded lattices taking into account definitions given in Baczynski

and Jayaram [2008]; Bedregal et al. [2006b, 2013]. Moreover, we discuss about

De Morgan triples and t-subnorms.

2.2.1 T-norms and T-conorms

It presented here a short formalization for the notion of t-norm and t-conorm on

bounded lattices. Moreover, some results are demonstrated as well.

Definition 2.15 Bedregal et al. [2006b] Let L be a bounded lattice. A binary

operation T : L× L −→ L is a t-norm if, for all x, y, z ∈ L, it satisfies:

(T1) T (x, y) = T (y, x) (commutativity);

(T2) T (x, T (y, z)) = T (T (x, y), z) (associativity);

(T3) If x 6L y then T (x, z) 6L T (y, z), ∀ z ∈ L (monotonicity);

(T4) T (x, 1L) = x (boundary condition).

Definition 2.16 Klement and Mesiar [2005] Let L be a bounded lattice and dL

a pseudo-quasi-metric on L. A t-norm T on L is called

1. Archimedean if for any x, y ∈ L\{0L, 1L}, there exists n ∈ N such that

T n(x) 6L y, where for any m ∈ N

T 0(x) = 1L and Tm+1(x) = T (Tm(x), x); (2.3)

23

2. PRELIMINARIES

2. dL-nilpotent if it is (d2L, dL)-continuous and each x ∈ L\{0L, 1L} is a nilpo-

tent element of T , i.e. there exists n ∈ N such that T n(x) = 0L;

3. idempotent if for each x ∈ L, T (x, x) = x; and

4. positive if T (x, y) = 0L if and only if either x = 0L or y = 0L;

5. An element a ∈ L is called a zero divisor if there exists a b ∈ L such that

T (a, b) = 0L.

Example 2.5 Let L be a bounded lattice. Thus, the function T : L × L → L

defined by T (x, y) = x ∧L y is a t-norm that generalize classical fuzzy t-norm of

minimum, i.e. TM : [0, 1] × [0, 1] → [0, 1] such that TM(x, y) = min{x, y} for all

x, y ∈ [0, 1].

Dually, it is possible to define the concept of t-conorms.

Definition 2.17 Bedregal et al. [2006b] Let L be a bounded lattice. A binary

operation S : L× L −→ L is said be a t-conorm if, for all x, y, z ∈ L, we have:

(S1) S(x, y) = S(y, x) (commutativity);

(S2) S(x, S(y, z)) = S(S(x, y), z) (associativity);

(S3) If x 6 y then S(x, z) 6 S(y, z), ∀ z ∈ L (monotonicity);

(S4) S(x, 0L) = x (boundary condition).

Remark 2.9 Notice that T (x, y) 6L x (or T (x, y) 6L y) and x 6L S(x, y) (or

y 6L S(x, y)) for all x, y ∈ L. In fact, T (x, y) 6L x∧y 6L x and x 6L x∨L y 6LS(x, y).

Example 2.6 Given an arbitrary bounded lattice L, the function S given by

S(x, y) = x ∨L y for all x, y ∈ L is a t-conorm on L that generalize the clas-

sical fuzzy t-conorm of maximum, i.e. SM(x, y) = max{x, y} for all x, y ∈ [0, 1].

Proposition 2.10 Let ρ be an automorphism on L. A t-conorm S : L×L −→ L

satisfies

S(x, y) = 1L if and only if x = 1L or y = 1L (2.4)

if and only if Sρ satisfies also it. A t-conorm satisfying (2.4) is called positive.

24

Proof: We have that Sρ(x, y) = 1L if and only if ρ−1(S(ρ(x), ρ(y))) = 1L if

and only if S(ρ(x), ρ(y)) = 1L if and only if ρ(x) = 1L or ρ(y) = 1L (by (2.4)) if

and only if x = 1L or y = 1L.

�

Similarly, it can be proved the following.

Proposition 2.11 Let ρ be an automorphism on L. A t-norm T : L× L −→ L

is positive if and only if T ρ satisfies also it.

2.2.2 Fuzzy Negations

A natural extension of notion of fuzzy negations can be done by considering

arbitrary bounded lattices as possible sets of truth values.

Definition 2.18 Bedregal et al. [2013] A mapping N : L→ L is a fuzzy negation

on L if the following properties are satisfied for each x, y ∈ L:

(N1) N(0L) = 1L and N(1L) = 0L and

(N2) If x 6L y then N(y) 6L N(x).

Moreover, a fuzzy negation N is strong if it also satisfies the involution property,

i.e.

(N3) N(N(x)) = x for each x ∈ L.

N is strict if satisfies the property:

(N4) N(x) <L N(y) whenever y <L x.

and it is called frontier if it satisfies property:

(N5) N(x) ∈ {0L, 1L} if and only if x = 0L or x = 1L.

Example 2.7 Bedregal et al. [2013] If L is an arbitrary bounded lattice, then the

functions N⊥, N> : L→ L defined by

N⊥(x) =

{1L, if x = 0L;

0L, otherwise.

and

N>(x) =

{0L, if x = 1L;

1L, otherwise.

25

2. PRELIMINARIES

for each x ∈ L are fuzzy negations on L.

Remark 2.10 An element e ∈ L is an equilibrium point of a fuzzy negation N

on L if N(e) = e. But differently from usual case and interval-valued case (see

Bedregal [2010]), strong fuzzy negations on lattices can not have an equilibrium

point.

Example 2.8 Let L and M be bounded lattices as shown in the Figure 2.5. The

function N1 : M → M defined by N1(0M) = 1M , N1(x) = y, N1(y) = x and

N1(1M) = 0M is a strong M-negation. Nevertheless, N1 has no equilibrium point.

Now, consider a function N2 : L→ L given by N2(0L) = 1L, N2(a) = e, N2(e) =

a, N2(1L) = 0L and N2(u) = u for each u ∈ {b, c, d}. In this case, N2 is a strong

fuzzy negation with three equilibrium points, namely b, c and d.

M

◦

◦ ◦

◦

1M

x y

0M@@@

���

���

@@@

L◦

◦

◦ ◦ ◦

◦

◦

1L

e

b c d

a

0L

@@@

���

���

@@@

Figure 2.5: Hasse diagrams of lattices M and L

Proposition 2.12 Let N be a strong fuzzy negation on L. Then

1. N is strict;

2. If N(x) 6L N(y) then y 6L x; and

3. N is bijective.

Proof: If y <L x then by (N2), N(x) 6L N(y). Supposing N(x) = N(y) then

N(N(x)) = N(N(y)) and so x = y, which is a contradiction with the premise.

Therefore, N(x) <L N(y).

If N(x) 6L N(y) then, by (N2), N(N(y)) 6L N(N(x)) and so y 6L x.

26

Since N is strict then N is trivially injective. Moreover, for any y ∈ L we

have N(N(y)) = y and hence N also is surjective. Thus, N is bijective.

�

From this proposition it follows that for each strong fuzzy negation we have

that x ‖ y if and only if N(x) ‖ N(y). Notice that although every lattice admits

a negation N , it is not true that every lattice admits an involutive negation.

Example 2.9 Let L be any lattice such that there exists x0 ∈ L with x0 6= 0L, 1L.

Then the mapping:

N(x) =

0L if x = 1L;

1L if x = 0L;

x0 otherwise.

is a frontier negation. Notice that this example proves that for every lattice L

with at least three elements it is possible to define a frontier negation.

Example 2.10 Consider lattice L0 obtained from lattice L in Figure 2.5 by omit-

ting the point a. Then, it does not exist a strong negation for this lattice. Indeed, if

N is such negation, then we should have that 0L < N(e) < N(b), N(c), N(d) < 1L

which is not possible due to the injectivity of a strong negation. It is important

to point out that the fact we can not define a strong negation over a lattices in

general, as shown in this example, is not due to the partial order of the lattice.

It is easy to see that it is not possible to define a strong negation on the linear

ordered lattice L = {0, 1}∪ [2, 3] as well, considering the usual linear order of real

numbers 6 on L.

Proposition 2.13 Let N : L → L be a function, ρ be an automorphism on L

and i ∈ {1, 2, 3, 4, 5}. N satisfies (Ni) if and only if Nρ satisfy (Ni). Moreover,

e is an equilibrium point of N if and only if ρ−1(e) is an equilibrium point of Nρ.

Proof: Suppose N satisfies (Ni) with i ∈ {1, 2, 3, 4}, then

(N1) Nρ(0L) = ρ−1(N(ρ(0L))) = ρ−1(N(0L)) = ρ−1(1L) = 1L. Analogously it

can be proved that Nρ(1L) = 0L;

(N2) If x 6L y then ρ(x) 6L ρ(y) and hence N(ρ(y)) 6L N(ρ(x)). Therefore, by

27

2. PRELIMINARIES

isotony of ρ−1, ρ−1(N(ρ(y)) 6L ρ−1(N(ρ(x)));

(N3) Nρ(Nρ(x)) = ρ−1(N(ρ(ρ−1(N(ρ(x)))))) = ρ−1(N(N(ρ(x)))) = ρ−1(ρ(x)) =

x;

(N4) Straight;

(N5) If Nρ(x) ∈ {0L, 1L} then, by Equation (2.2) and considering that ρ−1(0L) =

0L and ρ−1(1L) = 1L we have N(ρ(x)) ∈ {0L, 1L}. Thus, by (N5), ρ(x) ∈ {0L, 1L}which implies that x = 0L or x = 1L;

If N(e) = e then N(ρ(ρ−1(e))) = e and hence Nρ(ρ−1(e)) = ρ−1(N(ρ(ρ−1(e)))) =

ρ−1(e).

Reciprocal is straightforward from the previous item and the fact that for any

function f : L→ L, (fρ)ρ−1

= f .

�

Corollary 2.1 Let N : L → L be a function and ρ be an automorphism on L.

N is a (strong, frontier) fuzzy negation if and only if Nρ is a (strong, frontier)

fuzzy negation.

Proof: Straightforward from Proposition 2.13.

�

Klement et al. [2000]; Klir and Yuan [1995] observed that it is possible to

obtain, in a canonical way, a fuzzy negation NT from a t-norm T . This negation

is called natural negation of T or negation induced by T . In the most general

case, where we consider a t-norm on a bounded lattice L, it is not always possible

to obtain a natural negation, because the construction of NT is based in the

supremum of, possibly, infinite sets (this concept was generalized for lattices by

Bedregal et al. [2012a]).

Proposition 2.14 Let L be a complete lattice and T be a t-norm on L. Then

the function NT : L→ L defined by

NT (x) = sup{z ∈ L | T (x, z) = 0L} (2.5)

is an fuzzy negation .

28

Proof: According to Definition 2.18 we shall prove that NT satisfies (N1) and

(N2). Hence

(N1) NT (1L) = sup{z ∈ L | T (1L, z) = 0L} = sup{0L} = 0L and NT (0L) =

sup{z ∈ L | T (0L, z) = 1L} = supL = 1L;

(N2) If x 6L y then for any z ∈ L, T (x, z) 6L T (y, z) and therefore, if T (y, z) =

0L then T (x, z) = 0L. So, {z ∈ L | T (y, z) = 0L} ⊆ {z ∈ L | T (x, z) = 0L}.Hence, NT (y) = sup{z ∈ L | T (y, z) = 0L} 6L sup{z ∈ L | T (x, z) = 0L} =

NT (x).

�

Theorem 2.1 Let T be a t-norm on L. If T is positive then NT = N⊥.

Proof: If x 6= 0L and z ∈ L then, by (T4), T (x, z) = 0L if and only if z = 0L.

So, by Equation (2.5), NT (x) = sup{0L} = 0L. Therefore, NT = N⊥.

�

Theorem 2.2 Let T be a t-norm on L. If NT is a frontier negation then each

x ∈ L\{0L} is a zero divisor of T .

Proof: If x 6= 1L, then, as NT is frontier, NT (x) 6= 0L and so sup{z ∈L | T (x, z) = 0L} 6= 0L, that is, {z ∈ L | T (x, z) = 0L} 6= {0L}. Thus, since

T (x, 0L) = 0L, {0L} ⊂ {z ∈ L | T (x, z) = 0L}. Therefore, there exists z ∈ L\{0L}such that T (x, z) = 0L. Hence, x is a zero divisor of T .

�

Theorem 2.3 Let T be a t-norm on L and ρ be an automorphism on L. Then

NρT = NT ρ.

Proof: Let x ∈ L, then

NρT (x) = ρ−1(NT (ρ(x)))

= ρ−1(sup{z∈L | T (ρ(x), z)=0L})= ρ−1(sup{z∈L | T ρ(x, ρ−1(z))=0L})= sup{ρ−1(z) ∈ L | T ρ(x, ρ−1(z))=0L} (∗)= sup{z∈L | T ρ(x, z) = 0L}= NT ρ(x)

29

2. PRELIMINARIES

(∗) Notice that every monotone homomorphism preserves supremum element.

�

It is also possible to define this kind of negation from a given t-conorm S on

a complete bounded lattice L, as one can see in the following proposition.

Proposition 2.15 Let L be a complete lattice and S be a t-conorm on L. Then

the function NS : L→ L defined by

NS(x) = inf{z ∈ L | S(x, z) = 1L} (2.6)

is an L-negation called natural negation of S .

Proof:

(N1)

NS(1L) = inf{z ∈ L | S(1L, z) = 1L} = inf L = 0L and NS(0L) = inf{z ∈L | S(0L, z) = 1L} = inf{1L} = 1L.

(N2)

If x 6L y then for any z ∈ L we have that S(x, z) 6L S(y, z) and hence

if S(x, z) = 1L then S(y, z) = 1L. Thus, NS(y) = inf{z ∈ L | S(y, z) =

1L} 6L inf{z ∈ L | S(x, z) = 1L} = NS(x).

�

Proposition 2.16 Bedregal et al. [2013] Let S be a t-conorm on a complete

bounded lattice L. If S is positive then NS = NL>

Proposition 2.17 Bedregal et al. [2013] Let S be a t-conorm on L and ρ be an

automorphism on L. Then (NS)ρ = NSρ.

30

2.2.3 Fuzzy Implications

In the literature several notions for fuzzy implications have been considered

(see for example Baczynski [2004]; Bustince et al. [2003]; Fodor and Roubens

[1994]; Mas et al. [2007]; Yager [1983]). But, the unique consensus is that a fuzzy

implication should have the same behavior than classical implication when the

crisp case is considered Fodor and Roubens [1994], i.e. for values 0 and 1. Here

we consider the notion given by Bedregal et al. [2013].

Definition 2.19 Bedregal et al. [2013] Let L be a bounded lattice. A function

I : L2 → L is called a fuzzy implication if it satisfies the following properties:

(FPA) I(y, z) 6L I(x, z) whenever x 6L y (First place antitonicity);

(SPI) I(x, y) 6L I(x, z) whenever y 6L z (Second place isotonicity);

(CC1) I(0L, 0L) = 1L (Corner condition 1);

(CC2) I(1L, 1L) = 1L(Corner condition 2);

(CC3) I(1L, 0L) = 0L (Corner condition 3)

The set of implications on L will be denoted by IL.

Example 2.11 Let L be a bounded lattice. Thus, functions I⊥, I> : L × L → L

given by

I⊥(x, y) =

{1L, if x = 0L or y = 1L;

0L, otherwise.

and

I>(x, y) =

{0L, if x = 1L and y = 0L;

1L, otherwise.

for all x, y ∈ L are fuzzy implications.

For all x, y, z ∈ L, we define the following properties of I:

(WFPA) if x 6L y, x ¨ z and y ¨ z, then I(y, z) 6L I(x, z) (weak first place

antitonicity);

(WSPI) if y 6L z, x ¨ y and x ¨ z, then I(x, y) 6L I(x, z) (weak second place

isotonicity);

(RB) I(x, 1L) = 1L (right boundary conditions);

31

2. PRELIMINARIES

(LB) I(0L, y) = 1L (left boundary condition);

(CC4) I(0L, 1L) = 1L(corner condition 4);

(NP) I(1L, y) = y for each y ∈ L (left neutrality principle);

(EP) I(x, I(y, z)) = I(y, I(x, z)) for all x, y, z ∈ L (exchange principle);

(IP) I(x, x) = 1L for each x ∈ L (identity principle);

(OP) I(x, y) = 1L if and only if x 6L y (ordering property);

(LOP) if x 6L y then I(x, y) = 1L (left ordering property);

(WLOP) if x 6L y or x ‖L y then I(x, y) = 1L (weak left ordering property);

(IBL) I(x, I(x, y)) = I(x, y) for all x, y, z ∈ L (iterative Boolean law);

(CP) I(x, y) = I(N(y), N(x)) being N a strong L-negation (contrapositivity

property);

(L-CP) I(N(x), y) = I(N(y), x) (left contraposition law);

(R-CP) I(x,N(y)) = I(y,N(x)) (right contraposition law);

(P) I(x, y) = 0L if and only if x = 1L and y = 0L (positivety) .

It is easy to verify that (SPI) and (CC1) imply (LB). Moreover, (FPA) and

(CC2) imply (LB) and consequently (CC4).

Remark 2.11 Notice that if L is a totally ordered set then (WFPA), (WSPI)

and (WLOP) are equivalent to (FPA), (SPI) and (LOP), respectively.

Example 2.12 Let L be a bounded lattice (see Figure 2.5) and N2 the strong

fuzzy negation on L in Example 2.8. The function I : L2 → L given by

I(x, y) =

1L, if x 6L y;

N2(x), if y = 0L and x 6= 0L;

y, if x = 1L;

e, otherwise.

(2.7)

satisfies the properties (FPA), (OP ), (CP ) (with respect to N2) and (P ). It is a

easy to see that it holds by Table 2.3.

Lemma 2.1 If a function I : L2 → L satisfy (FPA) and (CP) for some strong

fuzzy negation N , then I also satisfies (SPI).

32

I 0L a b c d e 1L

0L 1L 1L 1L 1L 1L 1L 1La e 1L 1L 1L 1L 1L 1Lb b e 1L e e 1L 1Lc c e e 1L e 1L 1Ld d e e e 1L 1L 1Le a e e e e 1L 1L

1L 0L a b c d e 1L

Table 2.3: A function I on L

Proof: If y 6L z then N(z) 6L N(y) and so by (FPA), I(N(y), N(x)) 6LI(N(z), N(x)) for each x ∈ L. Therefore, by (CP), I(N(N(x)), N(N(y))) 6LI(N(N(x)), N(N(z))). Hence, I(x, y) 6L I(x, z) whereas N is strong.

�

Proposition 2.18 Let ρ be an automorphism on L, I : L×L→ L be a function

and P ∈ {(FPA), (SPI), (CC1), (CC2), (CC4), (LB), (RB)}. I satisfies P if

and only if Iρ also satisfies P .

Proof: See Proposition 10 in Bedregal et al. [2013].

�

A special type of fuzzy implication that we would like to study is the (S,N)-

implication, that is, an implication defined from a t-conorm S and a fuzzy nega-

tion N .

Definition 2.20 Baczynski and Jayaram [2008] Let S be a t-conorm on L and

N be a fuzzy negation on L. The function IS,N : L× L −→ L given by

IS,N(x, y) = S(N(x), y) (2.8)

for all x, y ∈ L is called a (S,N)-implication. If N is strong then I is called

a strong implication or S-implication. In this case, S and N are said to be

generators of I.

33

2. PRELIMINARIES

Proposition 2.19 Bedregal et al. [2013] Let I : L× L→ L be a function and ρ

an automorphism on L. Thus, I is an (S,N)-implication on L generated from S

and N if and only if Iρ is an (S,N)-implication on L generated from Sρ and Nρ.

In other words, (IS,N)ρ = ISρ,Nρ.

Another special type of implication we would like to consider in this work is

R-implication. Taking into account that there exists an isomorphism between

classical two-valued logic and classical set theory, it is possible to see that if K

and G are subsets of a set X then the identity

Kc ∪G = (K\G)c =⋃{P ⊆ X | K ∩ P ⊆ G}

holds, where Kc is the complement of set K (see Baczynski and Jayaram [2008]).

The R-implications (residual implications) are generalizations of the this iden-

tity in fuzzy logic.

Definition 2.21 Baczynski and Jayaram [2008] Let L be a complete bounded

lattice. A function I : L × L → L is called an R-implication if there exists a

t-norm T such that for all x, y ∈ L we have

I(x, y) = sup{t ∈ L | T (x, t) 6L y} (2.9)

We denote this implication generated from a t-norm T by IT .

To finish this section we present the notion of negations defined from implica-

tions. There exists a natural way to define this particular class of fuzzy negations

on [0, 1] based on the fact that a propositional formula p is logically equivalent

(in classic logic) to p →⊥ where ⊥ denotes the absurd (see Lemma 1.4.14, pg

18 in Baczynski [2004]). Bedregal et al. [2013] have generalized this concept for

bounded lattices as in the following proposition.

Proposition 2.20 Let L be a bounded lattice. If a function I : L × L → L

satisfies (FPA), (CC1) and (CC3) then the function NI : L→ L defined for each

x ∈ L by

NI(x) = I(x, 0L) (2.10)

is a fuzzy negation on L called the natural negation of I .

34

Next lemma provides necessary conditions for NI to be a fuzzy negation and

also establishes some properties.

Lemma 2.2 Bedregal et al. [2013] Let L be a bounded lattice. If a function

I : L× L→ L satisfies (EP) and (OP) then

1. NI is a fuzzy negation;

2. x 6L NI(NI(x)) for each x ∈ L;

3. NI ◦NI ◦NI = NI .

2.2.4 T-subnorms

As a generalization of the concept of t-norms, it was presented by Klement

et al. [2000] the notion of triangular subnorms as follows.

Definition 2.22 A function F : [0, 1]2 −→ [0, 1] that satisfies, for all x, y, z ∈[0, 1], the properties

1. F (x, y) = F (y, x)

2. F (x, F (y, z)) = F (F (x, y), z)

3. F (x, z) 6 F (y, z) whenever x 6 y

4. F (x, y) 6 min(x, y)

is called a t-subnorm.

It is clear that every t-norm is a t-subnorm. However, the reciprocal of this

affirmation is not true, in general. For instance, the function F : [0, 1]2 −→ [0, 1],

defined by F (x, y) = 0, is a t-subnorm, but not a t-norm. However, it is always

possible to construct a t-norm from a t-subnorm as is shown in the following

proposition.

35

2. PRELIMINARIES

Proposition 2.21 If F is a t-subnorm then the function T : [0, 1]2 −→ [0, 1],

given by

T (x, y) =

{F (x, y), if(x, y) ∈ [0, 1[2

min(x, y), otherwise

is a t-norm.

Proof: See Klement et al. [2000].

�

Naturally, the concept of t-subnorm may be generalized for bounded lattices.

Definition 2.23 Let L be a bounded lattice. A function F : L×L −→ L is called

a t-subnorm on L if it satisfies the following properties:

1. F (x, y) = F (y, x)

2. F (x, F (y, z)) = F (F (x, y), z)

3. F (x, z) 6L F (y, z) whenever x 6L y

4. F (x, y) 6L x ∧L y

Proposition 2.22 If F is a t-subnorm on bounded lattice L, then T defined by

T (x, y) =

{F (x, y), if(x, y) ∈ (L\{1L})2

x ∧L y, otherwise

is a t-norm on L.

Proof: Since F is a t-subnorm and x∧L y is a t-norm, then T is commutative,

associative and monotone. Thus, we shall only prove that T satisfies the boundary

condition, i.e., T (x, 1L) = x for each x ∈ L.

But, for each x ∈ L we have x = x ∧L 1L = T (x, 1L) since x 6 1L for all

x ∈ L. Therefore, T is a t-norm on L.

36

�

Note that it is possible to define a t-subconorm R dually from the definition of

t-subnorm just by replacing the property F (x, y) 6L x∧L y for x∨L y 6L R(x, y).

Precisely, we have

Definition 2.24 Let L be a bounded lattice. A function R : L×L −→ L is called

a t-subconorm on L if it satisfies the following properties

1. R(x, y) = R(y, x)

2. R(x,R(y, z)) = R(R(x, y), z)

3. R(x, z) 6L R(y, z) whenever x 6L y

4. x ∨L y 6L R(x, y)

Of course, a dual proof of Proposition 2.22 can be given to prove the following

proposition.

Proposition 2.23 If R is a t-subconorm on bounded lattice L, then S defined by

S(x, y) =

{R(x, y), if(x, y) ∈ (L\{0L})2

x ∨L y, otherwise

is a t-conorm on L.

2.2.5 De Morgan Triples

De Morgan’s laws represent, among other things, a way to relate disjunctive

and conjunctive operators via negations. Both in set theory and formal logics

these laws are very important tools to simplify equations, formal proofs and

other tasks. Specifically, these laws are given by

α ∧ β ≡ ¬(¬α ∨ ¬β) and α ∨ β ≡ ¬(¬α ∧ ¬β) (2.11)

37

2. PRELIMINARIES

or

¬(α ∧ β) ≡ ¬α ∨ ¬β and ¬(α ∨ β) ≡ ¬α ∧ ¬β (2.12)

In fuzzy logic, De Morgan’s laws can be naturally generalized using t-norms, t-

conorms and negations as operators. Nevertheless, due to the several possibilities

that exist to define these operators in fuzzy logic, De Morgan’s laws can be

generalized in different ways as seen in Calvo [1992]; da Costa et al. [2011]; Garcıa

and Valverde [1989]; Kolesarosa and Mesiar [2010]; Palmeira and Bedregal [2012].

Here, we choose a version that we believe is the most general one because it does

not impose any constraints for operators involved and it is more fateful with

Equations (2.11) and (2.12).

Definition 2.25 Calvo [1992] Let T be a t-norm, S a t-conorm and N a fuzzy

negation, all defined on the same bounded lattice L. We say that 〈T, S,N〉 is a

De Morgan triple if, for all x, y ∈ L, we have

1. N(T (x, y)) = S(N(x), N(y));

2. N(S(x, y)) = T (N(x), N(y)).

Remark 2.12 Naturally, every time we are talking about De Morgan triple 〈T, S,N〉its operators are considered defined on the same lattice. In order to highlight the

lattice concerned, we shall simply say that 〈T, S,N〉 is a De Morgan triple on L

when T , S and N are a t-norm, a t-conorm and a fuzzy negation all defined on

L.

It is important to point out that there are some fuzzy negations which are not

involutive and hence for some t-norms, t-conorms and fuzzy negations only one

of the items of Definition 2.25 holds true, as can be seen in the example below.

Example 2.13 Let L be the bounded lattice shown in Figure 2.4. It is easy to

check that the function N : L −→ L, given by setting N(0L) = 1L, N(z) = v,

N(x) = x, N(y) = N(u) = y, N(v) = N(t) = z and N(1L) = 0L, is a fuzzy

38

negation. Thus, considering the t-norm T and the t-conorm S defined by

T 0 z x y v u t 1

0 0 0 0 0 0 0 0 0

z 0 z z z z z z z

x 0 z x z x z x x

y 0 z z y y y y y

v 0 z x y v u v v

u 0 z z y u y y u

t 0 z x y v y v t

1 0 z x y v u t 1

S 0 z x y v u t 1

0 0 z x y v u t 1

z z z x y v 1 1 1

x x x x v v 1 1 1

y y y v y v 1 1 1

v v v v v v 1 1 1

u u 1 1 1 1 1 1 1

t t 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

then it follows that

N ◦ T 0 z x y v u t 1

0 1 1 1 1 1 1 1 1

z 1 v v v v v v v

x 1 v x v x v x x

y 1 v v y y y y y

v 1 v x y z y z z

u 1 v v y y y y y

t 1 v x y z y z z

1 1 v x y z y z 0

S ◦ (N ×N) 0L z x y v u t 1L

0 1 1 1 1 1 1 1 1

z 1 v v v v v v v

x 1 v x v x v x x

y 1 v v y y y y y

v 1 v x y z y z z

u 1 v v y y y y y

t 1 v x y z y z z

1 1 v x y z y z 0

Therefore N(T (a, b)) = S(N(a), N(b)) for all a, b ∈ L. However, on the other

hand, we see that N ◦ S 6= T ◦ (N ×N) as evidenced on the following tables:

N ◦ S 0 z x y v u t 1

0 1 v x y z y z 0

z v v x y z 0 0 0

x x x x z z 0 0 0

y y y z y z 0 0 0

v z z z z z 0 0 0

u y 0 0 0 0 0 0 0

t z 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0

T ◦ (N ×N) 0 z x y v u t 1

0 1 v x y z y z 0

z v v x y z y z 0

x x x x z z z z 0

y y y z y z y z 0

v z z z z z z z 0

u y y z z z y z 0

t z z z z z z z 0

1 0 0 0 0 0 0 0 0

39

2. PRELIMINARIES

Thus, taking into account such a possibility, we define a relaxed notion of De

Morgan triples.

Definition 2.26 Let T be a t-norm, S be a t-conorm and N be a fuzzy negation,

all of which are defined on bounded lattice L. A triple 〈T, S,N〉 is a De Morgan

T -semitriple (De Morgan S-semitriple) if, for all x, y ∈ L we have N(T (x, y)) =

S(N(x), N(y)) (N(S(x, y)) = T (N(x), N(y))).

It is worth noting that, in the case that N is a strong negation, the notions

of T -semitriple and S-semitriple coincide.

Notice also that it is possible to define De Morgan triples for t-subnorms,

t-subconorms and fuzzy negations as follows.

Definition 2.27 Let F be a t-subnorm on L, R be a t-subconorm on L and N

be a fuzzy negation. A triple 〈F,R,N〉 is called a De Morgan triple if, for all

x, y ∈ L, it satisfies

1. N(F (x, y)) = R(N(x), N(y));

2. N(R(x, y)) = F (N(x), N(y)).

Definition 2.28 Let F be a t-subnorm on L, R be a t-subconorm on L and

N be a fuzzy negation. A triple 〈F,R,N〉 is called a De Morgan F -semitriple

(R-semitriple) if, for all x, y ∈ L, it holds that N(F (x, y)) = R(N(x), N(y))

(N(R(x, y)) = F (N(x), N(y))).

40

Chapter 3

Extension Method via

Retractions

As we have seen in Chapter 1, the extension problem is a well investigated

subject in several different branches of knowledge. Here we are interested in

developing a general method to solve the problem of extending fuzzy operator

from a sublattice in order to preserve the largest number of its properties.

Our main goal in this chapter is presenting an extension method for fuzzy

connectives. Without loss of generality let’s introduce the extension problem for

t-norms. Consider an ordinary sublattice M of a bounded lattice L (i.e. M ⊆ L)

and T a t-norm on M . Since a t-norm is particularly a function it is natural to

think to extend T from M to L in order to obtain a new t-norm TE on L in

such way that this extension should preserve some properties of T by means TE

should have same properties of T . This problem is not trivial even when M is an

ordinary sublattice of L, as one can see in Example 3.1 below.

Example 3.1 Let M be an ordinary sublattice of L and TM a t-norm on M .

Define T : L× L −→ L such that

T (x, y) =

TM(x, y), if (x, y) ∈M ×M ;

x, if y = 1L;

y, if x = 1L;

0L, otherwise.

41

3. EXTENSION METHOD VIA RETRACTIONS

Thus, we have T |M×M = TM , i.e., T is an extension of TM . However, T is

not a t-norm as the order-preserving property of T fails (item (4) of Definition

2.15). This is due to the fact that we set T (x, y) = 0 for elements such that

(x, y) /∈M ×M , x 6= 1L and y 6= 1L.

Recent investigations points to a solution to this problem. In Saminger-Platz

et al. [2008] is shown a way to make this kind of extension of t-norms using the

ordinary definition of sublattice. Precisely,

WLTM (x, y) =

{x ∧L y, if 1L ∈ {x, y};

TM∪{0L,1L}(x∗, y∗), otherwise.(3.1)

where M is a (complete) ordinary sublattice of L, TM is a t-norm on M , x∗ =

supM{z | z 6L x, z ∈M ∪ {0L, 1L}} and

TM∪{0L,1L}(x, y) =

x ∧L y, if 1L ∈ {x, y};

0L, if 0L ∈ {x, y};TM(x, y), otherwise.

(3.2)

Notice that operator x∗ acts as an collapsing function flattening elements

belonging to L\M on M .

However, to have a more flexible environment where the collapsing function

can be defined in different ways, we would like to extend t-norms considering

our generalized notion of sublattice (see Definition 2.10). In fact, if M is a

(r, s)-sublattice of L (in this case, there are a retraction r : L −→ M and a

pseudo-inverse s : M −→ L such that r ◦ s = idM) and T a t-norm on M , how

can we define a function TE on L that works on s(M) as T works on M and that

is able to “carry on” to TE the largest possible number of properties that T has.

For instance, if T is Archimedean (see Definition 2.16) how to define a t-norm

TE such way that it is also Archimedean?

In this chapter it is presented a extension method via retractions that gener-

alize the Saminger-Platz’s extension method and its efficiency in preserving prop-

erties is demonstrated. Throughout this chapter L always represents a bounded

lattice.

42

3.1 T-norms

Notice that the idea behind the extension of t-norms proposed by Saminger-

Platz is to take the minimum values of those elements that do not belong to M .

We approach this problem in a similar way seeking to consider a more general

environment provided by the concept of (r, s)-sublattice.

Theorem 3.1 Let M < L with respect to (r, s). If T is a t-norm on M then,

the function TE : L× L −→ L defined by

TE(x, y) =

{x ∧L y, if 1L ∈ {x, y}

s(T (r(x), r(y))), otherwise.(3.3)

is a t-norm that extends T from M to L.

Proof: Firstly note that, if 0L ∈ {x, y} then TE(x, y) = 0L, so TE satisfies

the Equation (3.2). Moreover, it is clear that TE extends T from M to L, is

commutative and has neutral element 1L. Thus, it remains to prove that TE is

monotone and associative.

• Monotonicity

Let x, y, z ∈ L where x 6L y. In this case, supposing 1L /∈ {x, y, z},we have that r(x) 6M r(y) and hence TE(x, z) = s(T (r(x), r(z))) 6Ls(T (r(y), r(z))) = TE(y, z). Moreover, we have the following property:

(i) If x = 1L then y = 1L then TE(x, z) = TE(y, z).

(ii) Consider y = 1L and 1L /∈ {x, z}. Due to T (x, y) 6L y for all x, y ∈ Lthen

TE(x, z) = s(T (r(x), r(z))) 6L s(r(z)) 6L z = 1L ∧ z = TE(y, z)

(iii) Suppose z = 1L. Thus, TE(x, z) = x∧Lz = x 6L y = y∧Lz = TE(y, z).

• Associativity

43

3. EXTENSION METHOD VIA RETRACTIONS

First, suppose that 1L /∈ {x, y, z}. Then:

TE(TE(x, y), z) = TE(s(T (r(x), r(y))), z)

= s(T (r(s(T (r(x), r(y)))), r(z)))

= s(T (T (r(x), r(y)), r(z)))

= s(T (r(x), T (r(y), r(z))))

= s(T (r(x), r(s(T (r(y), r(z))))))

= TE(x, s(T (r(y), r(z))))

= TE(x, TE(y, z))

(i) If x = 1L then:

TE(TE(x, y), z) = TE(TE(1L, y), z) = TE(y, z) = TE(x, TE(y, z))

(ii) If y = 1L then:

TE(TE(x, y), z) = TE(TE(x, 1L), z) = TE(x, z) = TE(x, TE(y, z))

(iii) If z = 1L then:

TE(TE(x, y), z) = TE(TE(x, y), 1L) = TE(x, y) = TE(x, TE(y, z))

�

Notice that TE is a t-norm on L satisfying the identity r(TE(x, y)) = T (r(x), r(y)),

for all x, y ∈ L. In other words, TE is defined in such way that the following di-

agram commutes:

L× L r×r //

TE

��

M ×M

T

��L

r //M

Remark 3.1 If M is a complete sublattice of L in the sense of Definition 2.8

(i.e. M is a subset of L), r(x) = sup{z | s(z) 6L x} for each x ∈ L and s

44

is an inclusion map, then TE = WLTM . In other words, the extension proposed

by Saminger-Platz as shown in Equation (3.1) is an intense of our extension.

Another interesting fact is that T can be extended in several ways since pseudo-

inverse s is not unique. For each possible pseudo-inverse, TE is a different ex-

tension of T from M to L.

Corollary 3.1 Let M < L with respect to (r, s) and T be a t-norm on M . If

r : L −→M is such that r(x) = 0M if and only if x = 0L and TE is the extension

of T from M to L, as defined in Equation (3.3), then

1. if T is Archimedean then TE is Archimedean;

2. if T has a nilpotent element then TE has a nilpotent element;

3. if T has an idempotent element then TE has an idempotent element.

4. if a is a zero divisor of T then s(a) is a zero divisor of TE.

Proof:

1. Let (x, y) ∈ (L\{0L, 1L})2. Since r(x) = 0M if and only if x = 0L then

(r(x), r(y)) ∈ (M\{0M , 1M})2. Thus, by hypothesis T is Archimedean, and

hence, there is an n ∈ N with T (r(x), . . . , r(x))︸ ︷︷ ︸n−times

6L r(y). Therefore,

TE(x, . . . , x) = s(T (r(x), . . . , r(x))) 6L s(r(y)) 6L y

2. Let x ∈M\{0M , 1M} be a nilpotent element of T . Then s(x) ∈ L\{0L, 1L}is a nilpotent element of TE. In fact, let k be an integer such that

T (x, . . . , T (x, x))︸ ︷︷ ︸k−times

= 0M

Hence, since r ◦ s = IdM , it follows that

TE(s(x), . . . , TE(s(x), s(x)) · · ·) = s(T (r(s(x)), . . . , r(s(T (r(s(x)), r(s(x)))))))

= s(T (x, . . . , T (x, x))) = s(0M) = 0L

45

3. EXTENSION METHOD VIA RETRACTIONS

3. Suppose that a ∈ M is an idempotent element of T , i.e., T (a, a) = a.

Thus, since TE(s(a), s(a)) = s(a) ∧L s(a) = s(a) when s(a) = 1L and,

TE(s(a), s(a)) = s(T (r(s(a)), r(s(a)))) = s(T (a, a)) = s(a) otherwise, then

s(a) is an idempotent element of TE.

4. If a ∈ M\{0M , 1M} is a zero divisor of T then there is a b ∈ M\{0M , 1M}such that T (a, b) = 0M . Thus

TE(s(a), s(b)) = s(T (r(s(a)), r(s(b)))) = s(T (a, b)) = s(0M) = 0L

�

Example 3.2 However, extension method via retractions presented in Theorem

3.1 fails in preserving some properties of a given t-norm T . Indeed, if M < L

with respect to (r, s) and T is a t-norm on M , except for the trivial cases (i.e.

when r is not an isomorphism), even T satisfies cancellation law

T (x, y) = T (x, z) ⇒ x = 0M or y = z (3.4)

then its extension TE does not satisfies it. Notice that, if a, b, c ∈ L\{1L}without loss of generality and supposing TE(a, b) = TE(a, c) then we have that

s(T (r(a), r(b))) = s(T (r(a), r(c))) implies T (r(a), r(b)) = T (r(a), r(c)) since s is

injective and hence r(a) = 0M and r(b) = r(c). But, we can not conclude that

a = 0L or b = c.

3.2 T-conorms and Fuzzy Negations

It is also possible to apply the method of extending t-norms for t-conorms and

fuzzy negations under similar conditions as one can see in Propositions 3.1 and

3.3 below.

Proposition 3.1 Let M > L with respect to (r, s). If S is a t-conorm on M

46

then SE : L× L −→ L defined by

SE(x, y) =

{x ∨L y, if 0L ∈ {x, y}

s(S(r(x), r(y))), otherwise.(3.5)

is a t-conorm which extends S from M to L.

Proof: As in the prove of Theorem 3.1 it is easy to see that SE is commutative

and monotone. We shall prove properties (S2) and (S3) of Definition 2.17.

• Associativity

Let x1, x2, x3 ∈ L. The case x1 = x2 = x3 = 0L is straightforward from

definition.

Suppose that 0L /∈ {x1, x2, x3}. Thus,

SE(SE(x1, x2), x3) = SE(s(S(r(x1), r(x2))), x3)

= s(S(r(s(S(r(x1), r(x2)))), r(x3)))

= s(S(S(r(x1), r(x2)), r(x3)))

= s(S(r(x1), S(r(x2), r(x3))))

= s(S(r(x1), r(s(S(r(x2), r(x3))))))

= SE(x1, s(S(r(x2), r(x3))))

= SE(x1, SE(x2, x3))

If x1 = 0L and 0L /∈ {x2, x3} we have

SE(SE(x1, x2), x3) = SE(x2, x3) = SE(x1, SE(x2, x3))

If x2 = 0L and 0L /∈ {x1, x3} then

SE(SE(x1, x2), x3) = SE(x1, x3) = SE(x1, SE(x2, x3))

If x3 = 0L and 0L /∈ {x1, x2} we have

SE(SE(x1, x2), x3) = SE(x1, x2) = SE(x1, SE(x2, x3))

47

3. EXTENSION METHOD VIA RETRACTIONS

Finally, if only xi 6= 0L, where i ∈ {1, 2, 3}, then

SE(SE(x1, x2), x3) = xi = SE(x1, SE(x2, x3))

• Monotonicity

Let x1, x2 and x3 be elements of L such that x1 6L x2. Thus, we have the

following cases:

(i) Firstly, suppose that 0L /∈ {x1, x2, x3}. As such,

SE(x1, x3) = s(S(r(x1), r(x3))) 6L s(S(r(x2), r(x3))) = SE(x2, x3)

(ii) Now, if x1 = 0L and 0L /∈ {x2, x3}, then SE(x1, x3) = x3 6L s(r(x3)) 6Ls(S(r(x2), r(x3))) = SE(x2, x3);

(iii) In the case that x3 = 0L and 0L /∈ {x1, x2} we have: SE(x1, x3) =

x1 ∨L 0L = x1 6L x2 = x2 ∨L 0L = SE(x2, x3);

(iv) If x1 = x2 = x3 = 0L then SE(x1, x3) = 0L = SE(x2, x3).

�

Proposition 3.2 Let M > L with respect to (r, s). Suppose that S is a t-conorm

on M and r is such that r(x) = 1M (r(x) = 0M) if and only if x = 1L (x = 0L).

Thus

1. if S has a nilpotent element then SE has a nilpotent element;

2. if S has an idempotent element then SE has an idempotent element;

3. if a is a zero divisor of S then s(a) is a zero divisor of SE;

4. if S is positive then SE is positive.

Proof:

48

1. Let x ∈M\{0M , 1M} be a nilpotent element of S. Then s(x) ∈ L\{0L, 1L}is a nilpotent element of SE. In fact, let k be an integer such that

S (x, . . . , S(x, x))︸ ︷︷ ︸k−times

= 0M

Hence, since r ◦ s = IdM , it follows that

SE(s(x), . . . , s(x)) = SE(s(x), . . . , SE(s(x), s(x)) · · ·)= s(S(r(s(x)), . . . , r(s(S(r(s(x)), r(s(x)))))))

= s(S(x, . . . , S(x, x))) = s(0M) = 0L

2. Suppose that a ∈M is an idempotent element of S, i.e., S(a, a) = a. Thus,

since SE(s(a), s(a)) = s(a) ∨L s(a) = s(a) when s(a) = 0L and

SE(s(a), s(a)) = s(S(r(s(a)), r(s(a)))) = s(S(a, a)) = s(a)

otherwise. Then s(a) is an idempotent element of SE.

3. If a ∈ M\{0M , 1M} is a zero divisor of S then there is a b ∈ M\{0M , 1M}such that S(a, b) = 0M . Thus

SE(s(a), s(b)) = s(S(r(s(a)), r(s(b)))) = s(S(a, b)) = s(0M) = 0L

4. By (3.5) if SE(x, y) = 1L and y = 0L (or x = 0L) then 1L = SE(x, 0L) =

x ∨L 0L = x (or 1L = SE(0L, y) = 0L ∨L y = y), i.e. SE(x, y) = 1L implies

x = 1L or y = 1L. Now, suppose that SE(x, y) = 1L and 0L /∈ {x, y}. Thus

s(S(r(x), r(y))) = 1L and hence S(r(x), r(y)) = 1M by injetivety of s. By

hypotesis S is positive, then r(x) = 1M or r(y) = 1M what allow us to

conclude that x = 1L or y = 1L.

Reciprocally, if x = 1L then SE(1L, y) = 1L whereas y = 0L and

SE(1L, y) = s(S(r(1L), r(y))) = s(S(1M , r(y))) = s(1M) = 1L

if y 6= 0L. Analogously, one can prove that if y = 1L then SE(x, y) = 1L.

49

3. EXTENSION METHOD VIA RETRACTIONS

�

Proposition 3.3 Let M be a (r, s)-sublattice of L and N : M −→M be a fuzzy

negation. Then NE(x) = s(N(r(x))) for each x ∈ L is a fuzzy negation that

extends N from M to L.

Proof: Let x, y ∈ L such that x 6L y. As r is monotone, we have r(x) 6L r(y).

In this case, N(r(y)) 6M N(r(x)) since N is a fuzzy negation. Consequently, con-

sidering the monotonicity of s, we can conclude that s(N(r(y))) 6L s(N(r(x))),

that is, NE(y) 6L NE(x).

Moreover,

NE(0L) = s(N(r(0L))) = s(N(0M)) = s(1M) = 1L

NE(1L) = s(N(r(1L))) = s(N(1M)) = s(0M) = 0L

Therefore, according to Definition 2.18, it is clear that NE is a fuzzy negation on

L which extends N from M to L.

�

It is worth noting that the hypotheses of Proposition 3.3 require only that r

can be a retraction (it does not need to be neither a lower nor an upper retraction)

and hence if r is a lower retraction or an upper retraction the result remains valid.

This fact allows us to extend fuzzy negations in a more flexible way than t-norms

and t-conorms.

Remark 3.2 Note that the extension of fuzzy negation does not preserve invo-

lution property, i.e. in general NE is not a strong negation even if N is. For

instance, let M = [0, 1] and L = [0, 2]. Considering the retraction r : L → M

such that r(x) = x if x ∈ [0, 1] and else r(x) = 1 which has as a pseudo-inverse

function s : M → L given by s(x) = x for all x ∈ [0, 1) and s(1) = 2, then if N is

an arbitrary strong negation on M we have that NE(NE(4\3)) = 2 which shows

that NE does not satisfies the involution property.

Proposition 3.4 Let M be a (r, s)-sublattice of L and N : M −→ M a strong

negation. Then e ∈ M is an equilibrium point of N if and only if s(e) is an

equilibrium point of NE.

50

Proof: Let e ∈ M be an equilibrium point of N , i.e. N(e) = e. Hence by

Proposition 3.3 we have that NE(s(e)) = s(N(r(s(e)))) = s(N(e)) = s(e) what

allow us to say that s(e) is a equilibrium point of NE.

Conversely, if s(e) is an equilibrium point ofNE thus sinceN(r(x)) = r(NE(x))

for each x ∈ L it follows that N(e) = N(r(s(e))) = r(NE(s(e))) = r(s(e)) = e.

�

3.2.1 Negations Obtained from Fuzzy Connectives

In this subsection we discuss about a particular class of fuzzy negations, those

that one can obtain from either a t-norm or a t-conorm or a fuzzy implication

as introduced in Section 2.2.2. Also, for such negations we establish the relation

between its extensions and the negation obtained from extension of each fuzzy

connective (for instance, see Palmeira et al. [2012a]).

Theorem 3.2 Let MlL with respect to (r, s). Moreover, suppose that r(x) = 0M

if and only if x = 0L. If T is a t-norm on M , then (NT )E 6 NTE .

Proof: By Theorem 3.3 and Proposition 2.14 it follows that

NTE(x) = sup{y ∈ L | TE(x, y) = 0L} (3.6)

and

(NT )E(x) = s(NT (r(x))) = s(sup{z ∈M | T (r(x), z) = 0M}) (3.7)

Take an arbitrary x ∈ L and let be A = {y ∈ L | TE(x, y) = 0L}. We will

prove that (NT )E(x) ∈ A and hence it is possible to conclude that (NT )E(x) 6LNTE(x) since NTE(x) = supA.

Thus, we shall prove that TE(x, (NT )E(x)) = 0L. In fact, if x = 1L it fol-

lows that (NT )E(x) = 0L then by Theorem 3.1 we have TE(x, (NT )E(x)) = 0L.

On the other hand, if (NT )E(x) = 1L then 1L = s(NT (r(x))) = s(sup{z ∈M | T (r(x), z) = 0M}) and hence sup{z ∈ M | T (r(x), z) = 0M} = 1M since s

is injective. Thus, {z ∈ M | T (r(x), z) = 0M} = M that implies r(x) = 0M , i.e.

x = 0L. Then it can be concluded that TE(x, (NT )E(x)) = 0L.

51

3. EXTENSION METHOD VIA RETRACTIONS

Otherwise, when x 6= 1L and (NT )E(x) 6= 1L we have that

TE(x, (NT )E(x)) = s(T (r(x), r((NT )E(x))))

= s(T (r(x), r(s(NT (r(x))))))

= s(T (r(x), NT (r(x))))

Note thatNT (r(x)) = sup{z ∈M | T (r(x), z) = 0M} then T (r(x), NT (r(x))) =

0M . Thus s(T (r(x), NT (r(x)))) = 0L and hence TE(x, (NT )E(x)) = 0L by (5).

Therefore, (NT )E(x) 6 NTE(x) for all x ∈ L, i.e. (NT )E 6 NTE .

�

Proposition 3.5 Let M mL with respect to (r, s). If S is a t-conorm on M then

(NS)E(x) = s(NS(r(x))) = s(inf{z ∈M | S(r(x), z) = 1M}) is a negation on L.

Proof: Straightforward from Proposition 3.3.

�

Theorem 3.3 Let M m L with respect to (r, s). If S is a t-conorm on M then

NSE 6 (NS)E.

Proof: Firstly, take x ∈ L such that x 6= 0L. In this case we have

(NS)E(x) = s(NS(r(x))) = s(inf{z ∈M | S(r(x), z) = 1M}) (3.8)

and(NSE)(x) = inf{t ∈ L | SE(x, t) = 1L}

= inf{t ∈ L | s(S(r(x), r(t))) = 1L}= inf{t ∈ L | S(r(x), r(t)) = 1M}

(3.9)

It is important to say here that in Equation (3.9) above we can hide the case

where SE(x, t) = x ∨L t = 1L without loss of generality because if t = 0L in this

case we have that SE(x, t) = 1L if and only if x = 1L and it is easy to see that

(NS)E(1L) = (NSE)(1L).

Thus, being

A = {z ∈M | S(r(x), z) = 1M} and B = {t ∈ L | S(r(x), r(t)) = 1M}

52

thus for all z ∈ A, by (3.8), we have that S(r(x), r(s(z))) = S(r(x), z) = 1M which

means that s(z) ∈ B for each z ∈ A, i.e. s(A) ⊆ B and hence inf B 6 inf s(A)1. But s(inf A) = inf s(A) by Proposition 2.6 what allow us to conclude that

inf B 6 s(inf A). Therefore, it follows that NSE 6 (SN)E.

Finally, to complete the proof we shall consider the case when x = 0L. Thus

(NS)E(0L) = s(NS(r(1L)))

= s(NS(1M))

= s(inf{z ∈M | S(1M , z) = 1M})= s(inf{1M}) = s(1M) = 1L

and

(NSE)(0L) = inf{t ∈ L | TE(0L, t) = 1L} = inf{1L} = 1L

�

Note that, considering the proof of Theorem 3.3 above, for each t ∈ B we

have that S(r(x), r(t)) = 1M which means that r(t) ∈ A and hence s(r(t)) ∈ s(A)

what does not allows us to conclude that t ∈ s(A) since it holds only t 6L s(r(t)).

However, if we suppose that for each x ∈ L the set Ax = {z ∈ M | S(r(x), z) =

1M} is an ideal2 of M for all t ∈ L such that s(r(t)) ∈ s(Ax) then we have that

t ∈ s(Ax) since t 6L s(r(t)). Therefore for all x ∈ L it follows that B ⊆ s(Ax)

and hence s(inf Ax) = inf s(Ax) 6 B which implies (NS)E 6 NSE . In this case,

considering Theorem 3.3 we have the following corollary.

Corollary 3.2 Let M m L with respect to (r, s) and S be a t-conorm on M . If

for each x ∈ L the set Ax = {z ∈ M | S(r(x), z) = 1M} is an ideal of M then

NSE = (NS)E.

Proposition 3.6 If M > L with respect to (r, s) then SL> 6 (SM> )E.

Proof: By definition we have that

SL>(x, y) =

{x ∨L y, if x = 0L or y = 0L;

1L, otherwise.

1note that if C ⊆ D then inf D 6 inf C, see Lima [1982]2A nonempty subset K of a bounded lattice L is called an ideal if x ∈ K and y 6L x then

y ∈ K.

53

3. EXTENSION METHOD VIA RETRACTIONS

and

(SM> )E(x, y) =

{x ∨L y, if x = 0L or y = 0L;

s(SM> (r(x), r(y))), otherwise.

Note that

0M /∈ {r(x), r(y)} ⇒ s(SM> (r(x), r(y))) = s(1M) = 1L (3.10)

On the other hand

r(x) = 0M or r(y) = 0M ⇒ SM> (r(x), r(y)) = r(x) ∨M r(y) (3.11)

and hence

x ∨L y 6 s(r(x)) ∨M s(r(y)) = s(r(x) ∨M r(y)) = s(SM> (r(x), r(y))) (3.12)

Therefore, by Equations (3.10), (3.11) and (3.12) we can conclude that SL> 6

(SM> )E.

�

Proposition 3.7 Let M m L with respect to (r, s). Suppose r(x) = 1M if and

only if x = 1L, then it follows that NL> = (NM

> )E. Moreover, if S is a positive

t-conorm on M then NSE = (NM> )E.

Proof: Note that

NL>(x) =

{0L, if x = 1L;

1L, otherwise.

and (NM> )E(x) = s(NM

> (r(x))) where

NM> (r(x)) =

{0M , if r(x) = 1M ;

1M , otherwise.

But, by hipothesis r(x) = 1M if and only x = 1L and hence

NM> (r(x)) =

{0M , if x = 1L;

1M , otherwise.

54

Therefore, (NM> )E(x) = 0L if x = 1L and (NM

> )E(x) = 1L else, i.e. (NM> )E = NL

>.

Moreover, if S is a positive t-conorm on M then by Proposition 3.2 SE is a

positive t-conorm on L which implies that NSE = NL> = (NM

> )E by Proposition

2.16 .

�

3.3 Fuzzy Implications

In the following theorem it is presented a way to extend fuzzy implications

as an application of the method of extending fuzzy operators as introduced in

Theorem 3.1.

Now, we apply extension method via retractions for fuzzy implications and

discuss about related results. Also within the framework of implications two

special classes of this fuzzy connective are considered, namely (S,N)-implications

and R-implications.

Theorem 3.4 Let M be a (r, s)-sublattice of L. If I is an implication on M then

function IE : L× L −→ L given by

IE(x, y) = s(I(r(x), r(y))) (3.13)

for all x, y ∈ L, is an implication on L. In this case, IE is called the extension

of I from M to L.

Proof: Let x, y, z ∈ L. We shall prove that IE satisfies the axioms of Definition

2.19. Thus,

1. (FPA) Suppose x 6L y. Thus, by monotonicity of r, we have r(x) 6L r(y)

and hence by (FPA) and the isotonicity of s it follows that

IE(y, z) = s(I(r(y), r(z)))

6L s(I(r(x), r(z)))

= IE(x, z)

55

3. EXTENSION METHOD VIA RETRACTIONS

2. (SPI) Now, let y 6L z. Again, by monotonicity of r we have r(y) 6L r(z).

ThenIE(x, y) = s(I(r(x), r(y)))

6L s(I(r(x), r(z)))

= IE(x, z)

3. Moreover, we have that

– (CC1) IE(0L, 0L) = s(I(r(0L), r(0L))) = s(I(0M , 0M)) = s(1M) = 1L;

– (CC2) IE(1L, 1L) = s(I(r(1L), r(1L))) = s(I(1M , 1M)) = s(1M) = 1L;

– (CC3) IE(1L, 0L) = s(I(r(1L), r(0L))) = s(I(1M , 0M)) = s(0M) = 0L;

Therefore, by (1), (2) and (3) it can be concluded that IE is an implication on L.

�

It is worth noting that as for fuzzy negations we do not need to impose an

additional property for the retract r to extend implications. Note that for t-

conorms, r must be an upper retraction (for t-norms r must be a lower retraction

(see Palmeira and Bedregal [2012])).

Proposition 3.8 Under the same conditions as in Theorem 3.4, if I is an im-

plication on M satisfying some of properties (LB), (RB), (CC4), (EP), (IP),

(IBL), (CP), (L-CP) and (R-CP) then IE is an implication on L which satisfies

the same properties.

Proof: If I is an implication onM then, by Theorem 3.4, IE(x, y) = s(I(r(x), r(y))

for all x, y ∈ L. Thus:

(LB)

By hypothesis I(0M , x) = 1M for all x ∈M . Then IE(0L, y) = s(I(r(0L), r(y))) =

s(I(0M , r(y))) = s(1M) = 1L for all y ∈ L.

(RB)

Now, considering that I(x, 0M) = 1M for all x ∈ M , it follows that IE(a, 0L) =

s(I(r(a), r(0L))) = s(I(r(a), 0M)) = s(1M) = 1L for all a ∈ L.

(CC4)

IE(0L, 1L) = s(I(r(0L), r(1L))) = s(I(0M , 1M)) = s(1M) = 1L.

56

(EP)

IE(x, IE(y, z)) = s(I(r(x), r(IE(y, z)))) by eq. (3.13)

= s(I(r(x), r(s(I(r(y), r(z)))))) by eq. (3.13)

= s(I(r(x), I(r(y), r(z))) by def. 2.9

= s(I(r(y), I(r(x), r(z))) by (EP )

= s(I(r(y), r(s(I(r(x), r(z)))))) by def. 2.9

= IE(y, IE(x, z))

(IP)

Supposing that I(x, x) = 1M for each x ∈ M then IE(y, y) = s(I(r(y), r(y))) =

s(1M) = 1L for all y ∈ L.

(IBL)

For all x, y ∈ L, supposing I satisfies (IBL) we have that

IE(x, IE(x, y)) = s(I(r(x), r(IE(x, y)))) by eq. (3.13)

= s(I(r(x), r(s(I(r(x), r(y)))))) by eq. (3.13)

= s(I(r(x), I(r(x), r(y)))) by def. 2.9

= s(I(r(x), r(y))) by (IBL)

= IE(x, y)

(CP)

Let N be a fuzzy negation on M such that I(x, y) = I(N(y), N(x)). Thus, by

Proposition 3.3, NE is a fuzzy negation on L. Hence

IE(NE(y), NE(x)) = s(I(r(NE(y)), r(NE(x)))) by eq. (3.13)

= s(I(r(s(N(r(y)))), r(s(N(r(x)))))) by prop. 3.3

= s(I(N(r(y)), N(r(x)))) by def. 2.9

= s(I(r(x), r(y))) by (CP )

= IE(x, y)

(L-CP)

57

3. EXTENSION METHOD VIA RETRACTIONS

Now, being N a fuzzy negation for which I satisfies (L-CP), then

IE(NE(x), y) = s(I(r(NE(x)), r(y))) by eq. (3.13)

= s(I(r(s(N(r(x)))), r(y))) by prop. 3.3

= s(I(N(r(x)), r(y))) by def. 2.9

= s(I(N(r(y)), r(x))) by (L− CP )

= s(I(r(s(N(r(y)))), r(x))) by def. 2.9

= IE(NE(y), x)

(R-CP) Analogously to previous item.

�

Corollary 3.3 Let M be a (r, s)-sublattice of L, ρ be an automorphism on M

and I : M ×M →M be a function. If I satisfies P ∈ {(FPA), (SPI), (CC1),

(CC2), (CC4), (LB), (RB)} then (Iρ)E also satisfies P .

Proof: Straightforward from Proposition 3.8 and Proposition 2.18.

�

Example 3.3 We show here that the method of extending implications fails in

preserving properties (NP) and (OP). Let M and L be the bounded lattices shown

in Figure 2.4. It is clear that the function I given by

I(x, y) =

{1M , if x 6M y;

y, otherwise.

is a fuzzy implication on M (this function works as a generalization of Godel

implication for an arbitrary bounded lattice). It can be easily seen in Table 3.1

that implication I satisfies the properties (NP) and (OP).

However, considering the lower retraction r(x) = sup{z ∈ M | s(z) 6L x}from L into M and its pseudo-inverse s defined by s(1M) = 1L, s(a) = v, s(b) = x,

s(c) = y, s(d) = z and s(0M) = 0L (see Example 2.4), the extension IE of fuzzy

implication I does not satisfy properties (NP) and (OP). Indeed, if we take the

pair (1L, t) ∈ L2 then we have

IE(1L, t) = s(I(r(1L), r(t)) = s(I(1M , a)) = s(a) = v 6= t

58

i.e. for IE the property (NP) does not hold. To see that IE also does not satisfy

(OP) it is enough to take the pair (u, v) ∈ L2 and consider that

IE(u, v) = s(I(r(u), r(v)) = s(I(c, a)) = s(1M) = 1L

while we have that u ‖ v. In other words, IE(m,n) = 1L does not imply that

m 6L n, for all m,n ∈ L.

I 0M d c b a 1M

0M 1M 1M 1M 1M 1M 1Md 0M 1M 1M 1M 1M 1Mc 0M d 1M b 1M 1Mb 0M d c 1M 1M 1Ma 0M d c b 1M 1M

1M 0M d c b a 1M

Table 3.1: Implication on M

However, if I is a fuzzy implication on M satisfying (NP) and (OP), the

following weak versions of these properties hold for its extension IE:

(W-NP): For each y ∈ L, if I satisfies (NP), we have that

IE(1L, y) = s(I(r(1L), r(y))) = s(I(1M , r(y))) = s(r(y)) 6L y

(L-OP): If x 6L y then IE(x, y) = 1L. Indeed, let x, y ∈ L such that x 6L y

and suppose that I satisfies (L-OP). In this case, r(x) 6M r(y) since r

is monotone. Thus, by (L-OP), I(r(x), r(y)) = 1M and hence IE(x, y) =

s(I(r(x), r(y))) = s(1M) = 1L.

Remark 3.3 It is worth noting also that property (P) is not preserved by this

method of extending fuzzy implications, in general. It is easy to see that if an

implication I satisfies (P) then we have that IE(1L, 0L) = 0L. But if IE(x, y) = 0L

it does not imply that x = 1L and y = 0L. For instance, let M and L1 be the

bounded lattices shown in Figure 2.3 and take the retraction r1 : L1 → M with

pseudo-inverse s1 : M → L1 as defined in Example 2.3. In this case the fuzzy

59

3. EXTENSION METHOD VIA RETRACTIONS

implication I> : M ×M → M given by I>(x, y) = a if x = d and y = a and

I>(x, y) = d else (see Example 2.11) satisfies (P) but for its extension (I>)E we

have that

(I>)E(5, 2) = s1(I>(r1(5), r1(2))) = s1(I>(d, a)) = s1(a) = 1

which means that there are x, y ∈ L1 (in this case x = 5 and y = 2. Notice that

0L1 = 1 and 1L1 = 5) such that (I>)E(x, y) = 0L1 but (x, y) 6= (1L1 , 0L1). The

following proposition shows a weak version of the sufficiency part of property (P).

Proposition 3.9 Let M be a (r, s)-sublattice of L and I be a fuzzy implication

on M which satisfies property (P). Then:

1. If M < L and IE(x, y) = 0L then x = 1L;

2. If M > L and IE(x, y) = 0L then y = 0L.

Proof: If M is a (r, s)-sublattice of L then there exists a retraction r : L→M

and a pseudo-inverse s : M → L such that r ◦ s = idM . Thus, supposing that

IE(x, y) = 0L then s(I(r(x), r(y)) = 0L = s(0M) for each x, y ∈ L, and hence, by

injectivity of s and (P), we have that

I(r(x), r(y)) = 0M implies r(x) = 1M and r(y) = 0M (3.14)

1. If M < L then we also have that s ◦ r 6 idL and hence 1L = s(1M) =

s(r(x)) 6L x by (3.14). Therefore, we must have x = 1L since 1L is the

supremum element of L.

2. In case that M > L it follows that idL 6 s ◦ r. Again, by (3.14) we have

y 6L s(r(y)) = s(0M) = 0L. It means that y = 0L considering that 0L is

the infimum of L.

�

In which follows we present a result about negation defined from fuzzy impli-

cations.

60

Proposition 3.10 Let M be a (r, s)-sublattice of L and I : M ×M → M be a

function satisfying (EP) and (OP). Then

1. NIE(x) := IE(x, 0L) for all x ∈ L is a fuzzy negation on L;

2. NIE = (NI)E;

3. If r is an upper retraction then x 6L NIE(NIE(x)) for each x ∈ L;

4. NIE ◦NIE ◦NIE = NIE .

Proof:

1. Let x, y ∈ L such that x 6L y. Since r is monotone then r(x) 6M r(y)

and hence I(r(y), 0M) 6M I(r(x), 0M) by (FPA) (properties (EP) and (OP)

imply (FPA) as one can see in Lemma 6 in Baczynski [2004] for 〈[0, 1],6〉.The proof for a generic lattice L is similar.). Thus

NIE(y) = s(I(r(y), r(0L))) = s(I(r(y), 0M)) 6M s(I(r(x), 0M)) = NIE(x)

Moreover,

NIE(0L) = s(I(r(0L), r(0L))) = s(I(0M , 0M)) = s(1M) = 1L

and

NIE(1L) = s(I(r(1L), r(0L))) = s(I(1M , 0M)) = s(0M) = 0L

Therefore, it can be concluded that NIE is a fuzzy negation on L.

2. For each x ∈ L we have

NIE(x) = s(I(r(x), r(0L))) = s(I(r(x), 0M)) = s(NI(r(x))) = (NI)E(x)

61

3. EXTENSION METHOD VIA RETRACTIONS

3. Note that for each x ∈ L

NIE(NIE(x)) = s(I(r(NIE(x)), 0M))

= s(I(r(s(I(r(x), 0M))), 0M))

= s(I(I(r(x), 0M), 0M))

= s(NI(NI(r(x))))

(3.15)

If r is an upper retraction then idL 6 s ◦ r. By item 2 of Lemma 2.2 we

have that r(x) 6L NI(NI(r(x))) for each x ∈ L and hence x 6L s(r(x)) 6Ls(NI(NI(r(x)))). Thus by (3.15) it follows that x 6L NIE(NIE(x)) for each

x ∈ L.

4. Recall that r ◦ s = idM and that NI ◦ NI ◦ NI = NI (by item 3, Lemma

2.2). Thus,

NIE(NIE(NIE(x))) = s(I(r(NIE(NIE(x))), 0M))

= s(I(r(s(I(r(s(I(r(x), 0M))), 0M))), 0M))

= s(I(I(I(r(x), 0M), 0M), 0M))

= s(NI(NI(NI(r(x)))))

= s(NI(r(x)))

= s(NI(r(x), 0M))

= NIE(x)

for each x ∈ L.

�

3.3.1 (S,N)-implications

Proposition 3.11 Let M > L with respect to (r, s). If S is a t-conorm on M

and N is a negation on M then a function ISE ,NE : L × L −→ L defined by

ISE ,NE(x, y) = SE(NE(x), y)) for all x, y ∈ L is an implication in the sense of

Definition 2.19.

Proof: Straightforward from Propositions 3.1 and 3.3.

�

62

Corollary 3.4 Under the same conditions as in Proposition 3.11 it follows that I

is an (S,N)-implication on M generated from S and N if and only if IE = (IS,N)E

is an (S,N)-implication on L.

It is important to point out here that in general if I is a S-implication then

IE is not a S-implication since the extension of a strong negation is not a strong

negation (see Remark 3.2).

Proposition 3.12 Let M > L with respect to (r, s). Considering S a t-conorm

and N a negation both defined on M then it follows that ISE ,NE 6 (IS,N)E.

Proof: By Proposition 3.11 and definition of ISE ,NE we have that

ISE ,NE(x, y) = SE(NE(x), y) =

{NE(x) ∨L y, 0L ∈ {NE(x), y}

s(S(r(NE(x)), r(y))), otherwise.

Since NE(x) = s(N(r(x))) and r ◦ s = idM then

ISE ,NE(x, y) =

{NE(x) ∨L y, 0L ∈ {NE(x), y}

s(S(N(r(x))), r(y))), otherwise.

On the other hand, (IS,N)E(x, y) = s(IS,N(r(x), r(y))) = s(S(N(r(x)), r(y)))

for all x, y ∈ L. Thus, it is clear that

ISE ,NE(x, y) = (IS,N)E(x, y) whenever 0L /∈ {NE(x), y}. (3.16)

Moreover, if y = 0L then

ISE ,NE(x, 0L) = NE(x) = s(N(r(x)))

= s(S(N(r(x)), 0M))

= s(IS,N(r(x), 0M))

= (IS,N)E(x, 0L)

(3.17)

Now, if NE(x) = 0L then s(N(r(x))) = 0L = s(0M) which implies N(r(x)) =

63

3. EXTENSION METHOD VIA RETRACTIONS

0M since s is injective. Hence

ISE ,NE(x, y) = y

6 s(r(y))

= s(S(0M , r(y)))

= s(S(N(r(x)), r(y)))

= s(IS,N(r(x), r(y)))

= (IS,N)E(x, y)

(3.18)

Therefore, by (3.16), (3.17) and (3.18) it can be concluded that ISE ,NE(x, y) 6L(IS,N)E(x, y) for all x, y ∈ L.

�

Corollary 3.5 Under the same conditions as in Proposition 3.12 it follows that

NISE,NE

6 N(IS,N )E .

Proof: Direct from Proposition 3.10 and Proposition 3.12.

�

3.3.2 R-implications

Proposition 3.13 Let M l L with respect to (r, s). If T is a t-norm on M and

N is a negation on M then function ITE : L × L −→ L defined by ITE(x, y) =

sup{t ∈ L | TE(x, t) 6L y} for all x, y ∈ L is an implication in the sense of

Definition 2.19.

Proof: Straightforward from Definition 2.21 and the fact that under these

hypotheses TE is a t-norm on L as proved in Theorem 3.1.

�

Theorem 3.5 Let M l L with respect to (r, s). If T is a t-norm on M then

(IT )E 6 ITE ;

64

Proof: Firstly, take x, y ∈ L such that 1L /∈ {x, y}. In this case we have

(IT )E(x, y) = s(IT (r(x), r(y))) = s(sup{z ∈M | T (r(x), z) 6M r(y)}) (3.19)

and(ITE)(x, y) = sup{t ∈ L | TE(x, t) 6L y}

= sup{t ∈ L | s(T (r(x), r(t))) 6L y}= sup{t ∈ L | T (r(x), r(t)) 6M r(y)}

(3.20)

Being

A = {z ∈M | T (r(x), z) 6M r(y)} and B = {t ∈ L | T (r(x), r(t)) 6M r(y)}

thus for all z ∈ A, by (3.19), we have that T (r(x), r(s(z))) = T (r(x), z) 6Mr(y) which means that s(z) ∈ B for each z ∈ A, i.e. s(A) ⊆ B and hence

sup s(A) 6 supB (note that if C ⊆ D then supC 6 supD, see Lima [1982]).

But s(supA) = sup s(A) by Proposition 2.5 what allows us to conclude that

s(supA) 6 supB. Therefore (IT )E 6 ITE .

Finally, to complete the proof we must consider the cases when 1L ∈ {x, y}.We have two possibilities:

(i) y = 1L:

(IT )E(x, 1L) = s(IT (r(x), r(1L)))

= s(IT (r(x), 1M))

= s(sup{z ∈M | T (r(x), z) 6M 1M})= s(supM) = s(1M) = 1L

and

(ITE)(x, 1L) = sup{t ∈ L | TE(x, t) 6L 1L} = supL = 1L

65

3. EXTENSION METHOD VIA RETRACTIONS

(ii) x = 1L:

(IT )E(1L, y) = s(IT (r(1L), r(y)))

= s(IT (1M , r(y)))

= s(sup{z ∈M | T (1M , z) 6M r(y)})= s(sup{z ∈M | z 6M r(y)})

and

(ITE)(1L, y) = sup{t ∈ L | TE(1L, t) 6L y} = sup{t ∈ L | t 6L y}

Note that if z 6M r(y) then s(z) 6L s(r(y)) 6L y and hence using the

same argumentation of the first part of the proof of this proposition we can

conclude that (IT )E(1L, y) 6L (ITE)(1L, y).

Therefore by (i) and (ii) we have that (IT )E(x, y) 6L (ITE)(x, y) whereas 1L ∈{x, y}.

�

Corollary 3.6 Let M l L with respect to (r, s). If T is a t-norm on M then

N(IT )E 6 N(ITE

). In other words, the natural negation of (IT )E is less than or

equal to the natural negation of ITE .

Proof: Straightforward from Theorem 3.5 and Definition 2.21.

�

3.4 De Morgan Triples

After describing how to extend t-norms, t-conorms and fuzzy negations a

natural question that arises is under which conditions a De Morgan triple can be

extended, i.e. if M is a (r, s)-sublattice of L and 〈T, S,N〉 is a De Morgan triple

on M is 〈TE, SE, NE〉 also a De Morgan triple in the sense of Definition 2.25?

Unfortunately using the extension method via retraction it does not hold for

no trivial cases. There are two main reasons for that:

66

1. For extending a t-norm T from M to L r should be a lower retraction

while for extending a t-conorm S r should be an upper retraction and

hence r should be an isomorphism (in this case M and L are equal up to

isomorphism) what is a trivial case.

2. On the other hand, suppose M < L with respect to (r1, s) and M > L with

respect to (r2, s). Then we can extend T using retraction r1 as shown in

Equation (3.3) and extend S using retraction r2. For extending the negation

N no matter which retraction is token since the extension of negations there

is no constraints for retraction. So, suppose we extend N using retraction

r1. Thus, under this conditions for x, y /∈ {0L, 1L} we have that

NE(TE(x, y)) = NE(s(T (r1(x), r1(y))))

= s(N(r1(s(T (r1(x), r1(y))))))

= s(N(T (r1(x), r1(y))))

= s(S(N(r1(x)), N(r1(y))))

= s(S(r2(s(N(r1(x)))), r2(s(N(r1(y))))))

= SE(s(N(r1(x))), s(N(r1(y))))

= SE(NE(x), NE(y))

but identityNE(SE(x, y)) = TE(NE(x), NE(y)) does not hold sinceNE(x) =

S(N(r1(x))) and

NE(SE(x, y)) = NE(s(S(r2(x), r2(y))))

= s(N(r1(s(S(r2(x), r2(y))))))

= s(N(S(r2(x), r2(y))))

= s(T (N(r2(x)), N(r2(y))))

= s(T (r1(s(N(r2(x)))), r1(s(N(r2(y))))))

In other words, under this conditions Identities (1) and (2) of Definition 2.25

can not be valid at the same time. Therefore, we can affirm that extension method

via retraction does not preserve De Morgan triples.

Nevertheless, we have seen in Example 2.13 there exists triple 〈T, S,N〉 that

just satisfies one of axioms (1) and (2), i.e. when it is a De Morgan T -semitriple

67

3. EXTENSION METHOD VIA RETRACTIONS

or a De Morgan S-semitriple (see Definition 2.26).

Proposition 3.14 Let M < L with respect to (r, s). If 〈T, S,N〉 is a De Morgan

T -semitriple on M then the triple 〈TE, SE, NE〉 is a De Morgan T -semitriple on

L.

Proof: Let r and r′ be lower and upper retractions, respectively, with the same

pseudo-inverse s, such that, for each x, y ∈ L\{0L, 1L}, TE(x, y) = s(T (r(x), r(y)))

and SE(x, y) = s(S(r′(x), r′(y))). In addition, suppose that NE(x) = s(N(r(x)))

for all x ∈ L.

Let x, y ∈ L. Suppose that 0L ∈ {x, y}. Without loss of generality, we can

consider x = 0L. Then

NE(TE(x, y)) = NE(TE(0L, y)) = NE(0L) = 1L = SE(NE(x), NE(y))

Analogously, the above identity may be verified when y = 0L.

Subsequently, consider 1L ∈ {x, y}. We prove only the case of x = 1L, as the

case where y = 1L is analogous. Thus,

NE(TE(x, y)) = NE(TE(1L, y)) = NE(y) = SE(0L, NE(y)) = SE(NE(x), NE(y))

Finally, suppose that 0L, 1L /∈ {x, y}. Recall that r◦s = idM and r′ ◦s = idM .

Thus, it follows that

NE(TE(x, y)) = NE(s(T (r(x), r(y))))

= s(N(r(s(T (r(x), r(y))))))

= s(N(T (r(x), r(y))))

= s(S(N(r(x)), N(r(y))))

= s(S(r′(s(N(r(x)))), r′(s(N(r(y))))))

= SE(s(N(r(x))), s(N(r(y))))

= SE(NE(x), NE(y))

Therefore, it can be concluded that NE(TE(x, y)) = SE(NE(x), NE(y)), for all

x, y ∈ L.

�

68

Proposition 3.15 Let M > L with respect to (r, s). If 〈T, S,N〉 is a De Morgan

S-semitriple on M then the triple 〈TE, SE, NE〉 is a De Morgan S-semitriple on

L.

Proof: Similar to the proof of Proposition 3.14, now takingNE(x) = s(N(r′(x)))

for all x ∈ L.

�

3.5 Extension and Automorphisms

Studying the action of an automorphism over algebraic structures is a very in-

teresting issue. In fuzzy logic it is used among other classifying fuzzy connectives

through the concept of conjugate (see Definition 2.7).

In this section we discuss about some results involving extension and conjugate

of t-norms, t-conorms, fuzzy negations and implications. We investigate their

relation also for De Morgan triples and t-subnorms.

The main issue in this framework is: For instance, let M be a (r, s)-sublattice

of L, T a t-norm on M and ρ : M →M an automorphism. Is it possible to define

an automorphism ψ on L from ρ in order to have (T ρ)E = (TE)ψ?

Of course, a natural candidate to solve this problem should be ρE, i.e. ρE(x) =

s(ρ(r(x))) for each x ∈ L. But ρE is not an automorphism in general. For

instance, let M and L1 be the bounded lattices shown in the Figure 2.3 and

r1 : L1 → M the lower retract defined in Example 2.3 which has as pseudo-

inverse the homomorphism s1 : L1 → M given by s1(a) = 1, s1(b) = 2, s1(c) = 3

and s1(d) = 5 (as in Example 2.3 also). Thus, if we define the automorphism

ρ : M → M by ρ(a) = a, ρ(b) = c, ρ(c) = b and ρ(d) = d then it is easy to see

that ρE is not an automorphism since ρE(1) = ρE(2) = 1, i.e. ρE is not injective.

However, it is possible to establish conditions under which the question raised

up above can be considered for fuzzy connectives.

Theorem 3.6 Let M < L with respect to (r, s), ρ be an automorphism on M

and T be a t-norm on M . Moreover, suppose ψ : L −→ L is an automorphism

on L such that r ◦ ψ = ρ ◦ r. Then, (T ρ)E 6 (TE)ψ.

69

3. EXTENSION METHOD VIA RETRACTIONS

Proof: By definition, we have

(T ρ)E(x, y) =

{x ∧L y, if 1L ∈ {x, y};

s(ρ−1(T (ρ(r(x)), ρ(r(y))))), otherwise.

On the other hand, considering that 1L ∈ {ψ(x), ψ(y)} if and only if 1L ∈ {x, y},then

(TE)ψ(x, y) =

{x ∧L y, if 1L ∈ {x, y};

ψ−1(s(T (r(ψ(x)), r(ψ(y))))), otherwise.

It is clear that

1L ∈ {x, y} ⇒ (T ρ)E(x, y) = (TE)ψ(x, y) (3.21)

Otherwise, considering that r ◦ ψ = ρ ◦ r then

(T ρ)E(x, y) =

{x ∧L y, if 1L ∈ {x, y};

s(ρ(T (r ◦ ψ−1 ◦ s(r(x)), r ◦ ψ−1 ◦ s(r(y))))), otherwise.

(3.22)

(TE)ψ(x, y) =

{x ∧L y, if 1L ∈ {x, y};

s ◦ ρ ◦ r(s(T (r(ψ−1(x)), r(ψ−1(y))))), otherwise.

(3.23)

As r is a lower retraction, we have r ◦ s = IdM and s ◦ r 6 IdL. Thus

s(ρ(T (r ◦ ψ−1 ◦ s(r(x)), r ◦ ψ−1 ◦ s(r(y))))) 6L s(ρ(T (r(ψ−1(x)), r(ψ−1(y)))))

(3.24)

and

s ◦ ρ ◦ r(s(T (r(ψ−1(x)), r(ψ−1(y))))) = s(ρ(T (r(ψ−1(x)), r(ψ−1(y))))) (3.25)

for all x 6= 1L and y 6= 1L. Thus, by Equations (3.22), (4.11), (4.12) and (3.25),

we can conclude that

(T ρ)E(x, y) 6L (TE)ψ(x, y) for x 6= 1L and y 6= 1L (3.26)

70

Therefore, by Equations (4.8) and (3.26), it follows that (T ρ)E 6 (TE)ψ.

�

It is worth noting that (T ρ)E = (TE)ψ, regardless of whether s ◦ r = IdL,

i.e., r is an isomorphism of bounded lattices or M is a complete sublattice of

L in ordinary sense. The first case is trivial while second one is presented in

Proposition 3.16.

Lemma 3.1 If M is a complete sublattice of bounded lattice L in ordinary sense

(i.e., M ⊆ L), ψ is an automorphism on L such that ψ|M is an automorphism

on M and r : L −→M is a function given by r(x) = supM{y ∈M | y 6L x}, then

r ◦ ψ−1 = (ψ−1)|M ◦ r.

Proof: Note that (ψ−1)|M(r(x)) = (ψ−1)|M(supM{y ∈ M | y 6L x}) and

r(ψ−1(x)) = supM{w ∈ M | w 6L ψ−1(x)}. Thus, if k1 = sup

M{y ∈ M | y 6L x}

and k2 = supM{w ∈M | w 6L ψ−1(x)}, then we shall show that (ψ−1)|M(k1) = k2.

(i) We uphold that k1 6 x. Since (ψ−1)|M preserves order, hence (ψ−1)|M(k1) 6

ψ−1(x). So, (ψ−1)|M(k1) 6 k2 (because k2 = supM{w ∈M | w 6L ψ−1(x)}).

(ii) Conversely, k2 6 ψ−1(x) and hence ψ(k2) 6 ψψ−1(x) = x (note that

ψ preserves order). Thus, ψ(k2) 6 k1 and then k2 = ψ−1ψ(k2) 6 ψ−1(k1) =

(ψ−1)|M(k1).

Therefore, by (i) and (ii), we have (ψ−1)|M(k1) = k2.

�

Proposition 3.16 Let M be a complete ordinary sublattice of L as in Definition

2.8. Consider ψ an automorphism on L and suppose that ρ = ψ|M is an auto-

morphism on M . If T is a t-norm on M then (TE)ψ = (T ρ)E. In other words,

the conjugate of the extension of T is equal to the extension of the conjugate of

T .

Proof: Note that, due to M ⊆ L by hypothesis, the pseudo-inverse s of

retraction r works as the identity function in the definition of TE, i.e., TE(x, y) =

71

3. EXTENSION METHOD VIA RETRACTIONS

T (r(x), r(y)) for all x 6= 1L and y 6= 1L. Thus, by definition we have

(T ρ)E(x, y) =

{x ∧L y, if 1L ∈ {x, y};

ρ−1(T (ρ(r(x)), ρ(r(y)))), otherwise.(3.27)

On the other hand, considering that 1L ∈ {ψ−1(x), ψ−1(y)} if and only if 1L ∈{x, y}, then

(TE)ψ(x, y) =

{x ∧L y, if 1L ∈ {x, y};

ψ−1(T (r(ψ(x)), r(ψ(y))), otherwise.(3.28)

It is clear that if 1L ∈ {x, y}, then (T ρ)E(x, y) = (TE)ψ(x, y). Otherwise,

Identities (3.27) and (3.28) are different only in arguments of T , i.e., in the

Identity (3.27) the arguments of T are ρ−1(r(x)) and ρ−1(r(y)) while in Identity

(3.28) the arguments are r(ψ−1(x)) and r(ψ−1(y)). However, in accordance with

Lemma 4.2, we have r ◦ ψ−1 = ρ−1 ◦ r. Hence, (T ρ)E = (TE)ψ.

�

Let M and L be bounded lattices. If CM and CL are the classes of all t-norms

on M and L, respectively, then proposition above establishes that the following

diagram is commutative:

CMTE //

ψ|M

��

CL

ψ

��CM

(Tψ|M )E // CL

Remark 3.4 It is clear that if M > L with respect to (r, s) and ρ and S are

respectively an automorphism and a t-conorm on M it is possible to prove that

(Sρ)E > (SE)ψ and (Sρ)E = (SE)ψ whenever M is a complete ordinary sublattice

of L.

Theorem 3.7 Let M be a (r, s)-sublattice of L, ρ be an automorphism on M .

If N and I are respectively a fuzzy negation and an implication on M , supposing

ψ : L→ L is an automorphism on L such that r ◦ψ = ρ◦ r and ψ−1 ◦ s = s◦ρ−1,then

72

1. (Iρ)E = (IE)ψ;

2. (Nρ)E = (NE)ψ.

Proof:

1. By definition, for all x, y ∈ L we have

(Iρ)E(x, y) = s(ρ−1(I(ρ(r(x)), ρ(r(y))))) (3.29)

and

(IE)ψ(x, y) = ψ−1(s(I(r(ψ(x)), r(ψ(y))))) (3.30)

Considering that ψ−1 ◦ s = s ◦ ρ−1 and r ◦ ψ = ρ ◦ r then by (3.29) and

(3.30) it follows that

(Iρ)E(x, y) = ψ−1(s(I(r(ψ(x)), r(ψ(y))))) = (IE)ψ(x, y)

2. Similar to item 1.

�

Considering what it was established in Theorems 3.14 and 3.7, another ques-

tion arises involving extensions and automorphisms: If M is a complete sublattice

of the bounded lattice L, 〈T, S,N〉 is a De Morgan T -semitriple on M and ρ an

automorphism on M , then, is 〈(T ρ)E, (Sρ)E, (Nρ)E〉 a De Morgan T -semitriple?

There is a solution to a similar question presented in Theorem 7, proved by

da Costa et al. [2011]. Specifically speaking, it is shown that, if 〈T, S,N〉 is a De

Morgan triple on [0, 1] and ρ is an automorphism on [0, 1], then 〈T ρ, Sρ, Nρ〉 is

also a De Morgan triple. This is naturally generalized to lattices and De Morgan

T -semitriples, as shown in the following proposition.

Proposition 3.17 If 〈T, S,N〉 is a De Morgan T -semitriple on M a complete

sublattice of a bounded lattice L and ρ is an automorphism on L, then we have

that 〈(TE)ρ, (SE)ρ, (NE)ρ〉 is a De Morgan T -semitriple on L.

Proof: Analogous to the proof of Theorem 7 in da Costa et al. [2011].

73

3. EXTENSION METHOD VIA RETRACTIONS

�

Observe that, according to Proposition 3.16, if M is a complete sublattice of

bounded lattice L in ordinary sense, ψ is an automorphism on L and ρ = ψ|M is

an automorphism on M , then 〈(T ρ)E, (Sρ)E, (Nρ)E〉 = 〈(TE)ψ, (SE)ψ, (NE)ψ〉.

3.6 T-subnorms and T-subconorms

Our primary purpose in this section is to prove that, if M is a complete

sublattice of bounded lattice L and F is a t-subnorm on M , then we may apply

our technique of extending t-norms to extend t-subnorms. This is what we present

in the proposition below.

Proposition 3.18 Let M < L with respect to (r, s). If F is a t-subnorm on M

then the function FE : L× L −→ L given by

FE(x, y) =

{x ∧L y, if1L ∈ {x, y}

s(F (r(x), r(y))), otherwise

is a t-subnorm on L.

Proof: Clearly, FE is commutative, associative and monotone (see the proof

of Theorem 3.1). It remains to prove that FE(x, y) 6 x ∧L y.

Suppose x, y ∈ L. If 1L ∈ {x, y}, then, by definition, FE(x, y) = x ∧L y.

Otherwise, if z = x ∧L y then

FE(x, y) = s(F (r(x), r(y)))

6L s(r(x) ∧L r(y))

= s(r(x ∧L y))

= s(r(z))

6L z = x ∧L y

�

Corollary 3.7 Let M < L with respect to (r, s). If F is a t-subnorm on M then

a t-norm T generated from the extension FE of F to L is equal to the extension

(T ′)E of the t-norm T ′ defined from F (as in Proposition 3.18).

74

Proof: In Propositions 2.22 and 3.18, we have

T ′(x, y) =

{F (x, y), if(x, y) ∈ (L\{1L})2

x ∧L y, otherwise

and

FE(x, y) =

{x ∧L y, if1L ∈ {x, y}

s(F (r(x), r(y))), otherwise

Thus, since T ′(x, y) = F (x, y) if (x, y) ∈ (L\{1L})2 and FE(x, y) = s(F (r(x), r(y)))

if 1L /∈ {x, y}, it follows that

(T ′)E(x, y) =

{x ∧L y, if1L ∈ {x, y}

s(F (r(x), r(y))), otherwise(3.31)

and

T (x, y) =

{s(F (r(x), r(y))), if(x, y) ∈ (L\{1L})2

x ∧L y, otherwise

Thus, rewriting the above identity more conveniently, we have

T (x, y) =

{s(F (r(x), r(y))), otherwise

x ∧L y, if 1L ∈ {x, y}(3.32)

Therefore, by Identities (3.31) and (3.32) it follows that (T ′)E(x, y) = T (x, y) for

all x, y ∈ L.

�

Naturally, the notion of a t-subconorm may be defined simply be replacing the

property F (x, y) 6L x ∧L y for R(x, y) >L x ∨L y in Definition 2.23, allowing us

to apply the concept of De Morgan semitriples using t-subnorms, t-subconorms

and fuzzy negations.

Definition 3.1 Let F be a t-subnorm on L, R be a t-subconorm on L and N

be a fuzzy negation. A triple 〈F,R,N〉 is called a De Morgan F -semitriple

(R-semitriple) if, for all x, y ∈ L, it holds that N(F (x, y)) = R(N(x), N(y))

(N(R(x, y)) = F (N(x), N(y))).

75

3. EXTENSION METHOD VIA RETRACTIONS

Proposition 3.19 Let M > L with respect to (r, s). If R is a t-subconorm on

M then the function RE : L× L −→ L given by

RE(x, y) =

{x ∨L y, if0L ∈ {x, y}

s(R(r(x), r(y))), otherwise

is a t-subconorm.

Proof: Analogous to the proof of Proposition 3.18.

�

Proposition 3.20 Let M < L with respect to (r, s). Suppose F is a t-subnorm

on M , R is a t-subconorm on M and N a fuzzy negation on M . If 〈F,R,N〉 is a

De Morgan F -semitriple, then 〈FE, RE, NE〉 is also a De Morgan F -semitriple.

Proof: Similar to the proof of Theorem 3.14.

�

It is worth noting that a dual version of Proposition 3.20 holds. Moreover,

assume that 〈F,R,N〉 is a De Morgan F -semitriple on M complete sublattice

of bounded lattice L. By Corollary 3.7 and Proposition 3.20, we may conclude

that the De Morgan T -semitriple 〈T, S,N〉 obtained from the extended De Mor-

gan F -semitriple 〈FE, RE, NE〉 is equal to the extended De Morgan T -semitriple

〈TEM , SEM , NE〉 obtained from the De Morgan T -semitriple 〈T ′, S ′, N〉, where T ′

(S ′) is a t-norm generated from F (R) by Proposition 2.22 (by a dual version of

Proposition 2.22).

3.7 Final Remarks

This section is devoted to present tables describing the main properties that

are preserved by extension method via retractions for each fuzzy connective stud-

ied in this thesis. We use the symbol (√

) to check those properties that are

preserved by extension method via retraction and the symbol (×) for those prop-

erties that are not preserved by this extension method. Symbol (?) means that we

76

have not discuss about it and (-) means that it does not make sense this property

for these operators. Moreover, properties (RB), (LB), (CC4), (NP), (EP), (IP),

(OP), (P), (IBL) and (CP) related to implications (see Table 3.4) are defined on

page 32. For simplicity, we used abbreviations for the following properties:

Abbreviation Property

AC Archimedean

NI Nilpotent

IDN Idempotent

ZD Zero divisor

CL Cancellation law

C Continuity

P Positive

EC Relation between extension and conjugate

RE Relation between NKE and (NK)E with K a fuzzy connective

Table 3.2: Table of Abbreviations

t-norm and t-conorm AC NI IDN ZD CL C P EC

TE√ √ √ √

× ×√

(T ρ)E 6 (TE)ψ

SE ×√ √ √

× ×√

(Sρ)E > (SE)ψ

Table 3.3: Properties preserved by TE and SE

77

3. EXTENSION METHOD VIA RETRACTIONS

fuzzy implication (RB) (LB) (CC4) (NP) (EP) (IP) (OP) (P) (IBL) (CP)

IE√ √ √

×√ √

× ×√ √

(IS,N )E ? ? ? ? ? ? ? ? ? ?

(IT )E ? ? ? ? ? ? ? ? ? ?

Table 3.4: Properties preserved by IE

fuzzy negation Strong Strict Eq. P. EC RE

NE × ×√

(Nρ)E = (NE)ψ –

(NT )E × × ? ? (NT )E 6 NTE

(NS)E × × ? ? (NS)E > NSE

(NI)E × × ? ? (NT )E = NTE

Table 3.5: Properties preserved by NE

78

Chapter 4

Extension Method via

e-operators

We have seen in Chapter 3 that extension method via retractions fulfills the

objective of extending fuzzy connectives and others related operator. Actually,

this method is very general and can be applied for extending many other kind

of fuzzy operators. However, a disadvantage of this extension method is the fact

that some properties of these operator are not preserved. For instance, it does not

preserve strong (strict) negations, De Morgan triples and other properties. (see

Sections 3.2 and 3.4). Taking this into account, we started the investigation of

another way to extend fuzzy connectives that could be more efficient in preserving

properties.

Seeking to find out this method we present in this chapter a new way of

extending t-norms, t-conorms and fuzzy negations inspired on the idea of interval

constructor (see Bedregal and Takahashi [2006]) considering the important fact

that every lattice L can be seen as a sublattice of L = {[x, y] | x, y ∈ L and x 6Ly} in the sense of Definition 2.10. To do so, we develop an essential mapping,

named e-operator, as discussed in Section 4.1. This operator plays in an essential

role for making extension.

Indeed, results have shown that the extension method via e-operator is more

robust allowing a better performance of this extension in preserving properties of

extended connectives or operators.

79

4. EXTENSION METHOD VIA E-OPERATORS

We study here similar issues as in the previous chapter in order to show

the efficiency of the extension method via e-operator in preserving properties

against extension method via retractions. Also, we discuss about extension of

n-dimensional t-norms

4.1 Toward e-operators

Let 〈L,∧L,∨L, 0L, 1L〉 be a bounded lattice. It is well known that it is possible

to construct an interval version of it. Specifically, 〈L,∧,∨, [0L, 0L], [1L, 1L]〉 is a

bounded lattice where

L = {[x, y] | x, y ∈ L and x 6L y} (4.1)

Then, considering r1([x, y]) = x, r2([x, y]) = y for each [x, y] ∈ L and s(x) = [x, x]

for all x ∈ L, we can conclude that L E L with respect to (r1, r2, s), i.e. L is a

sublattice of L in the sense of Definition 2.10.

Moreover, notice that if T is a t-norm on L, a function I[T ] : L × L −→ Lgiven by

I[T ](X, Y ) = [T (r1(x), r1(y)), T (r2(x), r2(y))] (4.2)

for all X = [x, x], Y = [y, y] ∈ L, is a t-norm on L. Operator I is called interval

constructor and it works as described in Figure 4.1 (see Bedregal and Takahashi

[2006]; Bedregal et al. [2006b]).

The interval constructor is a very suitable and useful tool for converting

lattice-valued fuzzy operator in interval ones with a power to carry on most

important properties of these operators. So, taking into account that L is a

(r1, r2, s)-sublattice of L this constructor inspire us to provide an similar opera-

tor that could be able to works as a bridge for extending fuzzy connectives from

a (r1, r2, s)-sublattice M of bounded lattice L.

To do so, considering MEL with respect to (r1, r2, s) and T a t-norm on M we

just need to define a function that plays the role of [·, ·] for interval constructor,

i.e. a function H : M ×M → L as in Figure 4.2.

Based on these ideas we developed a generic operator having the minimal

properties necessary to be a function as H.

80

L× L

r1×r1

��

r2×r2

��L× L

T

��

L× L

T

��L× L

[·,·]��L

Figure 4.1: Diagram of interval constructor

L× L

r1×r1

��

r2×r2

��M ×M

T

��

M ×M

T

��M ×M

H��L

Figure 4.2: Diagram of intuitive idea

81

4. EXTENSION METHOD VIA E-OPERATORS

Definition 4.1 Let MEL with respect to (r1, r2, s). A mapping � : M×M −→ L

is called an extension operator on M (e-operator for short) if it is isotonic and

satisfies, for each a, b ∈M and for each x ∈ L, the following conditions:

r1(a� b) = a ∧M b and r2(a� b) = a ∨M b (4.3)

r1(x)� r2(x) = x (4.4)

In other words, if M E L with respect to (r1, r2, s) (by Definition 2.12, there

are two retractions r1, r2 : L −→ M with the same pseudo-inverse s : M −→ L

such that s◦r1 6 idL 6 s◦r2), then the e-operator � describes an isotonic way to

relate retractions r1 and r2 with the meet and join operators of M , respectively,

by (4.3).

Example 4.1 Given a bounded lattice 〈L,∧L,∨L, 0L, 1L〉 and its interval version

〈L,∧,∨, [0, 0], [1, 1]〉, as described at the beginning of this section LEL with respect

to (r1, r2, s). Considering the interval operations

[x, y] ∧ [a, b] = [x ∧L a, y ∧L b] and [x, y] ∨ [a, b] = [x ∨L a, y ∨L b].

by Definition 4.1, the mapping � : L× L −→ L defined by

a� b = [a ∧L b, a ∨L b]

for each a, b ∈ L, is trivially an e-operator on L.

In what follows we present an example of e-operator for finite lattices.

Example 4.2 Let M and L be bounded lattices depicted in Figure 4.3. Consider

the functions s : M −→ L and r1, r2 : L −→M defined by:

• s(1) = a, s(2) = b, s(3) = c, s(4) = d and s(5) = e;

• r1(s(x)) = x for any x ∈ M , r1(x1) = r1(x2) = r1(x3) = r1(x4) = 1,

r1(x5) = r1(x6) = r1(x7) = 2, r1(x8) = 3 and r1(x9) = 4; and

82

• r2(s(x)) = x for any x ∈ M , r2(x1) = 2, r2(x2) = r2(x5) = 3, r2(x3) =

r2(x6) = 4 and r2(x4) = r2(x7) = r2(x8) = r2(x9) = 5.

It is not hard to see that M E L with respect to (r1, r2, s). Now define � :

M ×M −→ L by

• x� x = s(x) and x� y = y � x for each x, y ∈M ; and

• 1 � 2 = x1, 1 � 3 = x2, 1 � 4 = x3, 1 � 5 = x4, 2 � 3 = x5, 2 � 4 = x6,

2� 5 = x7, 3� 4 = x7, 3� 5 = x9 and 4� 5 = x9.

Thus, r1(3�4) = r1(x7) = 2 = 3∧M 4 and r2(3�4) = r2(x7) = 5 = 3∨M 4 and so

Condition (4.3) is satisfied for this pair of values. Analogously, r1(x7)� r2(x7) =

2� 5 = x7 and so Condition (4.4) is satisfied for x7. By similar calculations all

remaining cases can be checked and the isotonicity of � shown. Therefore, it can

be stated that � is an e-operator on M .

M

◦

◦ ◦

◦

◦

5

3 4

2

1

@@@

���

���

@@@

L◦

◦

◦ ◦

◦

◦ ◦

◦◦ ◦

◦ ◦

◦

◦

b

c d

e

x9x8

x7

x5 x6

x4x2 x3

x1

a

@@@

@@@

@@@

���

���

����

BBBB

AAAAAAAAAAA

�����

��

HHHHH

HH

���

���

���

@@@

@@@

���

Figure 4.3: Hasse diagrams of lattices M and L

Remark 4.1 It is worth noting that given two retractions r1, r2 : L −→ M with

the same pseudo-inverse s : M −→ L such that s ◦ r1 6 idL 6 s ◦ r2 then it can

be easily concluded that r1 6 r2.

The following lemma provides us useful properties concerning the mapping �defined above.

83

4. EXTENSION METHOD VIA E-OPERATORS

Lemma 4.1 Consider MEL with respect to (r1, r2, s) and let � be an e-operator

on M . Then, for all a, b ∈M and x, y ∈ L, the following properties hold:

1. a 6M b if and only if r1(a� b) = a and r2(a� b) = b;

2. For every a ∈M we have s(a) = a� a;

3.

r1(x) 6M r1(y) and r2(x) 6M r2(y)⇔ x 6L y; (4.5)

4. r1(x) = r1(y) and r2(x) = r2(y) if and only if x = y;

5. � is commutative.

Proof:

1. Straightforward from (4.3) and the fact that, if a 6M b then a ∧M b = a

and a ∨M b = b.

2. By (4.4), it follows that s(a) = r1(s(a))� r2(s(a)) = a� a.

3. Since r1 and r2 are homomorphisms, we only need to prove the right side.

If r1(x) 6M r1(y) and r2(x) 6M r2(y) then by isotonicity of �, r1(x) �r2(x) 6M r1(y)� r2(y) and hence by Equation (4.4) we have x 6L y.

4. Straightforward from the previous property.

5. By Equation (4.3), r1(a� b) = a∧M b = b∧M a = r1(b� a) and r2(a� b) =

a ∨M b = b ∨M a = r2(b� a). Thus, by the previous item, a� b = b� a.

�

Lemma 4.2 Let M E L with respect to (r1, r2, s) and � : M ×M → L be an

e-operator. Thus, for each a, b ∈M ,

1. If a� b = 0L then a = 0M and b = 0M ;

2. If a� b = 1L then a = 1M and b = 1M ;

Proof:

84

1. Supposing a� b = 0L we have that a ∧M b = r1(a� b) = r1(0L) = 0M and

a ∨M b = r2(a� b) = r1(0L) = 0M , what means that a = 0M and b = 0M .

2. Analogous to item 1.

Proposition 4.1 Consider M E L with respect to (r1, r2, s) and take a mapping

� : M ×M −→ L satisfying Equation (4.3).Then � is an e-operator if and only

if the Equation (4.5) is satisfied.

Proof: (⇒) By Lemma 4.1.

(⇐) Suppose that y 6M z. We shall prove that � is isotonic and satisfies (4.4).

By Equation (4.3), r1(x� y) = x ∧M y 6M x ∧M z = r1(x� z) and r2(x� y) =

x ∨M y 6M x ∨M z = r2(x � z). So, by Equation (4.5) we have x � y 6L x � z.

Analogously it is possible to prove that y � x 6L z � x. So � is isotonic in each

component.

Moreover, by Equation (4.3), r1(r1(x) � r2(x)) = r1(x) ∧M r2(x) = r1(x) and

r2(r1(x) � r2(x)) = r1(x) ∨M r2(x) = r2(x). So, by Equation (4.5) (item 3 of

Lemma 4.1), it follows that r1(x)� r2(x) = x for all x ∈ L.

�

4.2 T-norms

Theorem 4.1 Let M E L with respect to (r1, r2, s) and � be an e-operator on

M . Thus, given a t-norm T on M , the function TE� : L2 −→ L defined by

TE� (x, y) = T (r1(x), r1(y))� T (r2(x), r2(y)) (4.6)

is a t-norm on L.

Proof: Firstly, note that r1 6 r2 so, by monotonicity of T , it follows that

T (r1(x), r1(y)) 6M T (r2(x), r2(y)) (4.7)

Let x, y, z ∈ L. Thus, we have:

85

4. EXTENSION METHOD VIA E-OPERATORS

– (Commutativity)

TE� (x, y) = T (r1(x), r1(y))� T (r2(x), r2(y))

= T (r1(y), r1(x))� T (r2(y), r2(x))

= TE� (y, x)

– (Associativity)

TE� (x, TE� (y, z)) =

= T (r1(x), r1(T (r1(y), r1(z))�T (r2(y), r2(z))))�T (r2(x), r2(T (r1(y), r1(z))�T (r2(y), r2(z)))) by eq. (5.14)

= T (r1(x), T (r1(y), r1(z))∧MT (r2(y), r2(z)))�T (r2(x), T (r1(y), r1(z))∨MT (r2(y), r2(z))) by eq. (4.3)

= T (r1(x), T (r1(y), r1(z)))�T (r2(x), T (r2(y), r2(z))) by eq. (5.16)

= T (T (r1(x), r1(y)), r1(z))�T (T (r2(x), r2(y)), r2(z)) by assoc. of T

= T (r1(T (r1(x), r1(y))�T (r2(x), r2(y))), r1(z))�T (r2(T (r1(x), r1(y))�T (r2(x), r2(y))), r2(z)) by eq. (4.3) and (5.16)

= TE� (TE� (x, y), z) by eq. (5.14)

– (Monotonicity)

Suppose that x 6L y. Thus, r1(x) 6M r1(y) and r2(x) 6M r2(y). So, by

monotonicity of T , T (r1(x), r1(z)) 6M T (r1(y), r1(z))) and T (r2(x), r2(z)) 6MT (r2(y), r2(z))). Thus, by isotonicity of �, T (r1(x), r1(z)) � T (r2(x), r2(z)) 6LT (r1(y), r1(z)))� T (r2(y), r2(z))). Therefore, TE� (x, z) 6L TE� (y, z).

– (Boundary Condition)

By Equations (4.4) and (5.14) we have

TE� (x, 1L) = T (r1(x), r1(1L))� T (r2(x), r2(1L)) = r1(x)� r2(x) = x

�

Example 4.3 Considering the lattices and operators defined in the Example 4.1,

86

if T is a t-norm on L then, according to Theorem 4.1 it follows that

TE� ([x, y], [a, b]) = T (r1([x, y]), r1([a, b]))� T (r2([x, y]), r2([a, b]))

= [T (x, a), T (y, b)]

Thus, by Proposition 4.10 in Bedregal et al. [2006b] the extension of T from

L to L (i.e. TE� ) is a t-norm on L.

Lemma 4.3 Under the same conditions as in Theorem 4.1, for any n ∈ N,

(TE� )n(x) = T n(r1(x))� T n(r2(x)).

Proof: We will prove it by induction on n. Trivially, (TE� )0(x) = 1L =

1M � 1M = T 0(r1(x)) � T 0(r2(x)). Assume as inductive hypothesis (IH) that

(TE� )k(x) = T k(r1(x))� T k(r2(x)). Then,

(TE� )k+1(x) = TE� ((TE� )k(x), x) by eq. (2.3)

= TE� (T k(r1(x))� T k(r2(x)), x) by IH

= T (r1(Tk(r1(x))� T k(r2(x))), r1(x))�

T (r2(Tk(r1(x))� T k(r2(x))), r2(x)) by eq. (5.14)

= T (T k(r1(x)) ∧M T k(r2(x)), r1(x))�T (T k(r1(x)) ∨M T k(r2(x)), r2(x)) by eq. (4.3)

= T (T k(r1(x)), r1(x))� T (T k(r2(x)), r2(x))

= T k+1(r1(x))� T k+1(r2(x)) by eq. (2.3)

�

Proposition 4.2 Under the same conditions as in Theorem 4.1, and considering

pseudo-quasi-metrics dM and dL for M and L respectively, it holds that:

1. If T is (d2M , dM)-continuous, r1 and r2 are (dL, dM)-continuous and � is

(d2M , dL)-continuous then TE� is (d2L, dL)-continuous;

2. If T is Archimedean, r−11 (0M) = r−12 (0M) = {0L} and r−11 (1M) = r−12 (1M)

= {1L} then TE� is Archimedean;

3. If T is dM -nilpotent, r1 and r2 are (dL, dM)-continuous, � is (d2M , dL)-

continuous, r−11 (0M) = r−12 (0M) = {0L} and r−11 (1M) = r−12 (1M) = {1L}then TE� is dL-nilpotent;

87

4. EXTENSION METHOD VIA E-OPERATORS

4. If T is idempotent then TE� is idempotent; and

5. If T is positive then TE� is positive.

Proof:

1. Since TE� = �◦ (T ×T )◦ (r1×r1×r2×r2)◦Id2L2 , T is (d2M , dM)-continuous,

r1 and r2 are (dL, dM)-continuous and � is (d2M , dL)-continuous, then the

result is straightforward from Propositions 2.7, 2.8 and 2.9 and from the

fact that trivially Id2L2 is (d2L, (d2L)2)-continuous, so it can be concluded that

TE� is (d2L, dL)-continuous.

2. Note that for any x ∈M\{0M , 1M}, Tm+1(x) 6M Tm(x). In fact,

Tm+1(x) = T (Tm(x), x) 6M Tm(x)

since T (x, y) 6M x for all x, y ∈M . Therefore, it can be inferred that

Tm(x) 6M T n(x), with n 6 m and n,m ∈ N (4.8)

Suppose that T is Archimedean. Thus, for each x, y ∈ L\{0L, 1L} by hy-

pothesis r1(x), r2(x) ∈M\{0M , 1M}, and therefore there are n,m ∈ N such

that T n(r1(x)) 6M r1(y) and Tm(r2(x)) 6M r2(y). Letting n 6 m we have

(TE� )m(x) = Tm(r1(x))� Tm(r2(x)) by Lemma 4.3

6L T n(r1(x))� T n(r2(x)) by (4.8)

6L r1(y)� r2(y) = y by (4.4)

3. If T is dM -nilpotent, r1 and r2 are (dL, dM)-continuous and � is (d2M , dL)-

continuous, then by the first item TE� is (d2L, dL)-continuous and so it just

rests to prove that each x ∈ L\{0L, 1L} is a nilpotent element of TE� . Let

x ∈ L\{0L, 1L} then by hypothesis r1(x), r2(x) ∈ M\{0L, 1L} and since

T is dM -nilpotent, there exists m,n ∈ N such that Tm(r1(x)) = 0M and

T n(r2(x)) = 0M . So, by Equation (4.8), T k(r1(x)) = 0M and T k(r2(x)) =

0M for k = max(m,n). Therefore, by Lemma 4.3, (TE� )k(x) = T k(r1(x))�T k(r2(x)) = 0M � 0M = 0L.

88

4. Let x ∈ L, then

TE� (x, x) = T (r1(x), r1(x))� T (r2(x), r2(x)) by eq. (5.14)

= r1(x)� r2(x) since T is idempotent

= x by eq. (4.4)

5. If TE� (x, y) = 0L then by Equation (5.14) T (r1(x), r1(y))�T (r2(x), r2(y)) =

0L. Thus,

0M = r2(0L) = r2(T (r1(x), r1(y))� T (r2(x), r2(y)))

= T (r1(x), r1(y)) ∨M T (r2(x), r2(y)) by eq. (4.3)

= T (r2(x), r2(y))

As T is positive, or r2(x) = 0M or r2(y) = 0M . If r2(x) = 0M then since r1 6

r2, r1(x) = 0M and hence r1(x) = r1(0L) and r2(x) = r2(0L). Therefore, by

Lemma 4.1, x = 0L. Analogously, if r2(y) = 0M it is possible to prove that

y = 0L.

�

4.3 T-conorms and Fuzzy Negations

Using the same idea as in the Theorem 4.1 it is also possible to extend t-

conorms and fuzzy negations.

Proposition 4.3 Let M EL with respect to (r1, r2, s) and � be an e-operator on

M . Thus,

1. If S is a t-conorm on M then

SE�(x, y) = S(r1(x), r1(y))� S(r2(x), r2(y)) (4.9)

is a t-conorm on L.

2. If N is a fuzzy negation on M then

NE� (x) = N(r1(x))�N(r2(x)) (4.10)

89

4. EXTENSION METHOD VIA E-OPERATORS

is a fuzzy negation on L. Moreover,

(a) If N is involutive (i.e., it satisfies (N3)) then NE� is also involutive.

(b) If a is an equilibrium point of fuzzy negation N then s(a) is an equi-

librium point of NE� .

Proof:

1. Analogous to the proof of Theorem 4.1.

2. Let be x, y ∈ L such that x 6L y. Thus, by monotonicity of r1 and r2,

we have r1(x) 6M r1(y) and r2(x) 6M r2(y) implying that N(r1(y)) 6MN(r1(x)) and N(r2(y)) 6M N(r2(x)) respectively. Hence, by isotonicity of

�, we have NE� (y) = N(r1(y))�N(r2(y)) 6L N(r1(x))�N(r2(x)) = NE

� (x).

Moreover, it follows that

NE� (0L) = N(r1(0L))�N(r2(0L)) = N(0M)�N(0M) = 1M � 1M = 1L

and

NE� (1L) = N(r1(1L))�N(r2(1L)) = N(1M)�N(1M) = 0M � 0M = 0L

Therefore, it can be concluded that NE� is a fuzzy negation on L.

(a) Now, suppose that N is involutive. Thus, for each x ∈ L, we have

NE� (NE

� (x)) = NE� (N(r1(x))�N(r2(x)))

= N(r1(N(r1(x))�N(r2(x))))

�N(r2(N(r1(x))�N(r2(x))))

= N(N(r2(x)))�N(N(r1(x)))

= r2(x)� r1(x)

= r1(x)� r2(x) = x

90

(b) We shall prove that NE� (s(a)) = s(a). Thus,

NE� (s(a)) = N(r1(s(a)))�N(r2(s(a)))

= N(a)�N(a)

= a� a = s(a)

�

One advantage of using extension method via e-operator for extending fuzzy

negations instead of extension method vie retractions is reveled by item 2(a) of

Proposition 4.3 which shows that strong (strict) negations are preserved by this

method.

4.4 De Morgan Triples

Another advantage of extension method via e-operators is that using it we can

extend De Morgan triples for no trivial cases what does not hold for extension

method via retractions as we have seen in Section 3.4.

Proposition 4.4 Let M EL with respect to (r1, r2, s) and � be an e-operator on

M . If 〈T, S,N〉 is a De Morgan triple on M then 〈TE� , SE� , NE� 〉 is a De Morgan

triple.

Proof: Let us recall that NE� (x) = N(r1(x)) � N(r2(x)) and SE�(x, y) =

S(r1(x), r1(y))� S(r2(x), r2(y)) for all x, y ∈ L. Thus, we have

91

4. EXTENSION METHOD VIA E-OPERATORS

NE� (TE� (x, y)) =

= NE� (T (r1(x), r1(y))� T (r2(x), r2(y)))

= N(r1(T (r1(x), r1(y))� T (r2(x), r2(y))))

�N(r2(T (r1(x), r1(y))� T (r2(x), r2(y))))

= N(T (r1(x), r1(y)))�N(T (r2(x), r2(y))) by Lemma 4.1(1)

= S(N(r1(x)), N(r1(y)))� S(N(r2(x)), N(r2(y))) by item 1 def. 4.1.

= S(N(r2(x)), N(r2(y)))� S(N(r1(x)), N(r1(y))) by commut. of �= S(r1(N(r1(x))�N(r2(x))), r1(N(r1(y))�N(r2(y))))

�S(r2(N(r1(x))�N(r2(x))), r2(N(r1(y))�N(r2(y)))) by Lemma 4.1(1)

= S(r1(NE� (x)), r1(N

E� (y)))� S(r2(N

E� (x)), r2(N

E� (y)))

= SE�(NE� (x), NE

� (y))

Analogously, it can be verified that NE� (SE�(x, y)) = TE� (NE

� (x), NE� (y)).

�

It is clear that it is possible to extend De Morgan T -semitriples (or De Morgan

S-semitriples) and triples 〈F,R,N〉 where F is a t-subnorm, R a t-subconorm and

N a fuzzy negation. So, proof of following corollaries are obvious.

Corollary 4.1 Let M E L with respect to (r1, r2, s) and � be an e-operator on

M . If 〈T, S,N〉 is a De Morgan T -semitriple ( De Morgan S-semitriples) on M

then 〈TE� , SE� , NE� 〉 is a De Morgan T -semitriple (or De Morgan S-semitriples)

on L.

Corollary 4.2 Let M E L with respect to (r1, r2, s) and � be an e-operator on

M . If 〈F,R,N〉 is a De Morgan triple on M then 〈FE� , R

E�, N

E� 〉 is a De Morgan

triple on L.

4.5 Extension and Automorphisms

In Section 3.5 we have discussed about relations between extension (using the

method via retractions) of fuzzy connectives and its conjugates. In fact, being

M a (r, s)-sublattice of L, ρ an automorphism on M and T a t-norm (or a t-

conorm S) on M , the results have shown that we have only (T ρ)E 6 (TE)ψ

92

((Sρ)E > (SE)ψ) where ψ is a suitable automorphism on L. This fact happens due

to the extension ρE (via retractions) of automorphism ρ is not an automorphism

on L in general.

Now we investigate this issue again but using extension method via e-operators,

i.e. given M E L with respect to (r1, r2, s), a t-norm T on M and ρ an automor-

phism on M , is the t-norm (T ρ)E generated from the extension of the conjugate

of T equal to the conjugate of the extension of T? In other words, is there an

automorphism ψ on L defined from ρ such that identity (T ρ)E = (TE� )ψ holds?

The main advantage of extension method via e-operators is that now extension

of an automorphism ρ is also an automorphism as one can see in Proposition 4.5.

Definition 4.2 Let MEL with respect to (r1, r2, s) and � be an e-operator on M .

If ρ is an automorphism on M , its extension is given by ρE(x) = ρ◦r1(x)�ρ◦r2(x),

for all x ∈ L.

The following proposition establishes conditions under which ρE is an auto-

morphism on L.

Proposition 4.5 Let M EL with respect to (r1, r2, s) and � be an e-operator on

M and ρ : M −→M an automorphism. Thus,

1. ρE is an automorphism on L;

2. The inverse of ρE is given by

(ρE)−1(x) = ρ−1 ◦ r1(x)� ρ−1 ◦ r2(x)

for all x ∈ L, i.e., (ρE)−1 = (ρ−1)E.

Proof:

1. By Proposition 2.2, we shall prove that ρE is bijective and x 6L y if and only

if ρE(x) 6L ρE(y). Note that ρE is naturally surjective since ρ, r1, r2 and

� are surjective. It remains to prove that ρE is injective. If ρE(x) = ρE(y)

then (ρ ◦ r1(x)) � (ρ ◦ r2(x)) = (ρ ◦ r1(y)) � (ρ ◦ r2(y)). Thus, r1(ρ ◦r1(x))� (ρ◦r2(x)) = r1(ρ◦r1(y))� (ρ◦r2(y)) and hence, by Equation (4.3),

93

4. EXTENSION METHOD VIA E-OPERATORS

ρ ◦ r1(x) = ρ ◦ r1(y). So, r1(x) = ρ−1(ρ(r1(x))) = ρ−1(ρ(r1(y))) = r1(y).

Analogously, it is possible to prove that r2(x) = r2(y). Therefore, by item

4 of Lemma 4.1 we have that x = y, that is, ρE is bijective.

Now, if x 6L y then r1(x) 6M r1(y) and r2(x) 6M r2(y) what allows us

to conclude that ρ(r1(x)) 6M ρ(r1(y)) and ρ(r2(x)) 6M ρ(r2(y)). Since �is isotonic then (ρ ◦ r1(x)) � (ρ ◦ r2(x)) 6L (ρ ◦ r1(y)) � (ρ ◦ r2(y)) and so

ρE(x) 6L ρE(y).

On the other hand, if ρE(x) 6L ρE(y) then (ρ ◦ r1(x)) � (ρ ◦ r2(x)) 6L(ρ ◦ r1(y))� (ρ ◦ r2(y)). Thus, r1(ρ ◦ r1(x))� (ρ ◦ r2(x)) 6L r1(ρ ◦ r1(y))�(ρ ◦ r2(y)) and therefore, by Equation (4.3), ρ ◦ r1(x)) 6M ρ ◦ r1(y)). So,

r1(x) = ρ−1(ρ(r1(x))) 6M ρ−1(ρ(r1(y))) = r1(y). Analogously, it is possible

to prove that r2(x) 6M r2(y). Therefore, by item 3 of Lemma 4.1 it follows

that x 6L y.

2. Now, we will prove that (ρE)−1 : L −→ L given by (ρE)−1(x) = (ρ−1 ◦r1(x))� (ρ−1 ◦ r2(x)) for all x ∈ L, is the inverse of ρE. We shall prove that

ρE ◦ (ρE)−1 = idL and (ρE)−1 ◦ ρE = idL. But

ρE ◦ (ρE)−1(x) = (ρ ◦ r1((ρE)−1(x)))� (ρ ◦ r2((ρE)−1(x)))

= (ρ ◦ r1((ρ−1 ◦ r1(x))� (ρ−1 ◦ r2(x))))�(ρ ◦ r2((ρ−1 ◦ r1(x))� (ρ−1 ◦ r2(x))))

= ρ(ρ−1 ◦ r1(x))� ρ(ρ−1 ◦ r2(x)) by Lemma 4.1(1)

= r1(x)� r2(x)

= x by eq. (4.4)

Analogously, it can be proved that (ρE)−1 ◦ ρE = idL.

�

Lemma 4.4 Let M E L with respect to (r1, r2, s) and � be an e-operator on M .

Thus, given an automorphism ρ : M −→M , ρE satisfies the following properties:

1. For all x ∈M we have that ρE(x� x) = ρ(x)� ρ(x);

2. ρ ◦ r1 = r1 ◦ ρE and ρ−1 ◦ r1 = r1 ◦ (ρE)−1;

94

3. ρ ◦ r2 = r2 ◦ ρE and ρ−1 ◦ r2 = r2 ◦ (ρE)−1.

Proof:

1. By item 2. of Lemma 4.1 we know that x� x = s(x) for all x ∈M . Thus

ρE(x� x) = ρ(r1(x� x))� ρ(r2(x� x))

= ρ(r1(s(x)))� ρ(r2(s(x)))

= ρ(x)� ρ(x)

2. Since r1 6 r2 and ρ is an automorphism, then ρ(r1(x)) 6M ρ(r2(x)) for all

x ∈ L and hence r1(ρ(r1(x)) � ρ(r2(x))) = ρ(r1(x)) by item 1. of Lemma

4.1. Thus, r1 ◦ ρE(x) = r1(ρ(r1(x))� ρ(r2(x))) = ρ(r1(x)) for all x ∈ L.

Analogously, one can prove that ρ−1 ◦ r1 = r1 ◦ (ρE)−1.

3. Similar to item 2.

�

The following theorem gives an answer to the problem presented at the be-

ginning of this section.

Theorem 4.2 Let M E L with respect to (r1, r2, s) and � be an e-operator on

M . Then, given an automorphism ρ and a t-norm T both defined on M , it is

possible to define an automorphism ψ on L such that (T ρ)E = (TE� )ψ.

Proof: Take as ψ the function ρE as given in Definition 4.2. Note that

r1(ψ(x)) 6M r2(ψ(x)) for all x ∈ L and hence

T (r1(ψ(x)), r1(ψ(y))) 6M T (r2(ψ(x)), r2(ψ(y)))

Thus, by item 1. of Lemma 4.1 we have that

r1(T (r1(ψ(x)),r1(ψ(y)))�T (r2(ψ(x)),r2(ψ(y))))=T (r1(ψ(x)),r1(ψ(y))) (4.11)

95

4. EXTENSION METHOD VIA E-OPERATORS

and

r2(T (r1(ψ(x)),r1(ψ(y)))�T (r2(ψ(x)),r2(ψ(y))))=T (r2(ψ(x)),r2(ψ(y))) (4.12)

Therefore, for all x, y ∈ L it follows that

(TE� )ψ(x, y) =

= ψ−1(TE� (ψ(x), ψ(y)))

= ψ−1(T (r1(ψ(x)), r1(ψ(y)))� T (r2(ψ(x)), r2(ψ(y))))

= ρ−1(r1(T (r1(ψ(x)), r1(ψ(y)))� T (r2(ψ(x)), r2(ψ(y)))))�ρ−1(r2(T (r1(ψ(x)), r1(ψ(y)))� T (r2(ψ(x)), r2(ψ(y))))

= ρ−1(T (r1(ψ(x)), r1(ψ(y))))� ρ−1(T (r2(ψ(x)), r2(ψ(y)))) by (4.11) and (4.12)

= ρ−1(T (ρ(r1(x)), ρ(r1(y))))� ρ−1(T (ρ(r2(x)), ρ(r2(y)))) by Lemma 4.4

= T ρ(r1(x), r1(y))� T ρ(r2(x), r2(y))

= (T ρ)E(x, y)

�

Results similar to those proved above can be demonstrated to t-conorms and

fuzzy negations, i.e., taking a t-conorm S and a fuzzy negation N both defined

on M , under the same conditions as in Theorem 4.2 we have that (Sρ)E = (SE�)ψ

and (Nρ)E = (NE� )ψ.

Proposition 4.6 Let M and L be two bounded lattices such that M E L with

respect to (r1, r2, s) and � be an e-operator on M . If 〈T, S,N〉 is a De Morgan

triple on M and ρ is an automorphism on M , then 〈(TE� )ρE, (SE�)ρ

E, (NE

� )ρE〉 is

a De Morgan triple on L.

Proof: Analogous to the proof of Theorem 7 in da Costa et al. [2011].

�

96

4.6 On Extension of n-dimensional T-norms

The n-dimensional fuzzy sets theory has been studied as a way to generalize

the fuzzy sets theory valued to the simplex Ln([0, 1]) = {x = (x1, x2, . . . , xn) ∈[0, 1]n | x1 6 x2 6 · · · 6 xn} for a fixed n ∈ N−{0} (see Shang et al. [2010]). Thus,

it is natural to think about the fuzzy operators (t-norms, t-conorms and fuzzy

negations) on Ln([0, 1]). For n = 2, a good formalization about interval-valued

fuzzy logic is given by Deschrijver and partners in Deschrijver [2006, 2008, 2011].

Recent studies for arbitrary n has been done by Bedregal et al. [2006a] where it is

done a formalization of n-dimensional aggregation functions, particulary t-norms,

fuzzy negations and automorphisms on Ln([0, 1]).

Based on this framework, an interesting issue that arises is that generalizing

lattice-valued aggregation functions to higher dimension using a bounded lattice

L instead of [0, 1] on the definition of Ln([0, 1]).

One can naturally define a lattice version of the set Ln([0, 1]), namely

Ln(L) = {x = (x1, x2, . . . , xn) ∈ Ln | x1 6L x2 6L · · · 6L xn} (4.13)

where L is a bounded lattice.

For each x,y ∈ Ln(L) we define by

x ∧ y = (x1 ∧L y1, x2 ∧L y2, . . . , xn ∧L yn)

and

x ∨ y = (x1 ∨L y1, x2 ∨L y2, . . . , xn ∨L yn)

the meet and join operations on Ln(L), respectively.

Denote /x/ = (x, x, . . . , x) for each x ∈ L. Thus, /0L/ and /1L/ are a

bottom and a top element of Ln(L). As an easy exercise one can prove that

〈Ln(L),∧,∨, /0L/, /1L/〉 is a bounded lattice.

A partial order on Ln(L) is given by

x 6 y ⇔ xi 6L yi for each i = 1, 2, . . . , n

The remainder of this section is devoted to define t-norms on Ln(L) (called

97

4. EXTENSION METHOD VIA E-OPERATORS

n-dimensional t-norms on L) constructed from t-norms on L, and to provide

a generalization of the extension proposed in Theorem 4.1 for n-dimensional t-

norms.

Proposition 4.7 Bedregal et al. [2006a] Let T1, T2, . . . , Tn : [0, 1]× [0, 1]→ [0, 1]

be t-norms such that T1 6 T2 6 · · · 6 Tn. Then

˜T1 · · ·Tn(x,y) = (T1(x1, y1), . . . , Tn(xn, yn)) (4.14)

is an n-dimensional t-norm. In case that T1 = T2 = · · · = Tn we denote ˜T1 · · ·Tnby TT .

A similar result can be easily shown considering a bounded lattice L instead

of [0, 1] in Proposition 4.7.

We decided to work in this section just with TT n-dimensional t-norm, but

every result presented here remains valid for ˜T1 · · ·Tn.

Corollary 4.3 Let M E L with respect to (r1, r2, s), � be an e-operator on M

and T be a t-norm on M . Then TTE� : Ln(L)2 −→ Ln(L) given by

TTE� (x,y) = (TE� (x1, y1), TE� (x2, y2), . . . , T

E� (xn, yn))

is a n-dimensional t-norm on Ln(L).

Proof: Straightforward from Theorem 4.1 and Proposition 4.7.

�

Proposition 4.8 Let M be a (r, s)-sublattice of L. Then

1. Ln(M) is a (r, s)-sublattice of Ln(L);

2. If M is a lower (upper) (r1, s)-sublattice of L then Ln(M) is a lower (upper)

(r1, s)-sublattice of Ln(L);

3. If M E L with respect to (r1, r2, s) then Ln(M) E Ln(L) with respect to

(r1, r2, s) where r1, r2 and s are suitable homomorphisms defined from r1, r2

and s respectively.

98

Proof:

1. Since M is a (r, s)-sublattice of L, by Definition 2.10 there is a retraction

r : L −→M with a pseudo-inverse s : M −→ L such that r◦s = idM . Define

a n-dimensional function r : Ln(L) −→ Ln(M) given for each x ∈ Ln(L) by

r(x) = (r(x1), r(x2), . . . , r(xn)) (4.15)

We claim that r is a n-dimensional retraction such that its pseudo-inverse

is an n-dimensional function s : Ln(M) −→ Ln(L) given by

s(x) = (s(x1), s(x2), . . . , s(xn))

for each x ∈ Ln(M).

Indeed, it is clear that r and s are n-dimensional homomorphisms since r

and s are. Moreover, for each x ∈ Ln(M), we have

r ◦ s(x) = r(s(x1), s(x2), . . . , s(xn))

= (r(s(x1)), r(s(x2)), . . . , r(s(xn)))

= (x1, x2, . . . , xn) = x

Thus r ◦ s = idLn(M) and hence Ln(M) is a (r, s)-sublattice of Ln(L) by

Definition 2.10.

2. If M is a lower (r1, s)-sublattice of L then s ◦ r1 6 idL. We shall prove that

s ◦ r1 6 idLn(L). Thus, for each x ∈ Ln(L) it follows that

s ◦ r1(x) = s(r1(x1), r1(x2), . . . , r1(xn))

= (s(r1(x1)), s(r1(x2)), . . . , s(r1(xn)))

6 (x1, x2, . . . , xn)

Analogously, one can prove that Ln(M) is an upper (r2, s)-sublattice of

Ln(L) assuming that M is an upper (r2, s)-sublattice of L.

3. Suppose thatMEL. Thus, there are a lower and an upper retractions r1 and

r2 from L onto M with the same pseudo-inverse s : M −→ L. Therefore, by

99

4. EXTENSION METHOD VIA E-OPERATORS

items 1. and 2. it is easy to check that r1(x)=(r1(x1), r1(x2), . . . , r1(xn)) and

r2(x) = (r2(x1), r2(x2), . . . , r2(xn)) for each x ∈ Ln(L) are n-dimensional

lower and upper retractions with the same n-dimensional pseudo-inverse

s(x) = (s(x1), s(x2), . . . , s(xn)) for each x ∈ Ln(M) according to which it

can be inferred that Ln(M)E Ln(L) with respect to (r1, r2, s).

�

Proposition 4.9 Let M EL with respect to (r1, r2, s) and let � be an e-operator

on M . The function �n : Ln(M)× Ln(M) −→ Ln(L) given by

a�n b = (a1 � b1, a2 � b2, . . . , an � bn) (4.16)

for all a,b ∈ Ln(M) is an e-operator on Ln(M).

Proof: Straightforward from Proposition 4.8 and from the fact that � is an

e-operator on M .

�

Corollary 4.4 Let M E L with respect to (r1, r2, s) and � be an e-operator on

M . If T is a t-norm on Ln(M) then the function TE� : Ln(L)× Ln(L) −→ Ln(L)

given by

TE�(x,y) = T(r1(x), r1(y))�n T(r2(x), r2(y))

for all x,y ∈ Ln(L), is a t-norm on Ln(L).

Proof: Straightforward from Theorem 4.1 and Propositions 4.8 and 4.9.

�

Theorem 4.3 Let M E L with respect to (r1, r2, s), � be an e-operator on M

and T be a t-norm on M . Then (TT )E� = TTE� .

100

Proof: Take x,y ∈ Ln(L). Then,

(TT )E�(x,y) = TT (r1(x), r1(y))�n T(r2(x), r2(y)) by (5.14)

= TT ((r1(x1), . . ., r1(xn)),(r1(y1), . . . ,r1(yn))�n

TT ((r2(x1), . . . , r2(xn)), (r2(y1), . . . , r2(yn)) by eq.(4.15)

= (T (r1(x1), r1(y1)), . . . , T (r1(xn), r1(yn)))�n

(T (r2(x1), r2(y1)), . . . , T (r2(xn), r2(yn))) by eq. (4.14)

= (T (r1(x1), r1(y1))� T (r2(x1), r2(y1)), . . . ,

T (r1(xn), r1(yn))� T (r2(xn), r2(yn)) by eq. (4.16)

= (TE� (x1, y1), . . . , TE� (xn, yn)) by eq. (5.14)

= TTE� (x,y) by eq. (4.14)

�

Let CM be the set of all t-norms T on M (similar to CL, CLn(M) and CLn(L)).

The theorem above shows that the following diagram is commutative:

CMTE� //

TT

��

CL

TTE�

��CLn(M)

(TT )E� // CLn(L)

A generalization to higher dimension of the concepts of t-conorm and fuzzy

negation may be done as we did here for t-norms. Results similar to those in

Corollary 4.4 and Theorem 4.3 can be proved for n-dimensional t-conorms and

n-dimensional fuzzy negations as done in Section 4.3.

Proposition 4.10 Let L be a bounded lattice. For all n ∈ N\{0} we have

Lm(L) E Ln(L) with respect to some (r1, r2, s) when n = 2m. Moreover, the

mapping � : Lm(L)× Lm(L) −→ Ln(L) defined by

(x1, . . . , xm)� (y1, . . . , ym) = (x1 ∧L y1, x1 ∨L y1, . . . , xm ∧L ym, xm ∨L ym)

is an e-operator.

101

4. EXTENSION METHOD VIA E-OPERATORS

Proof: We shall present a n-dimensional lower retraction r1 : Ln(L) −→ Lm(L),

a n-dimensional upper retraction r2 : Ln(L) −→ Lm(L) and a n-dimensional

pseudo-inverse s : Lm(L) −→ Ln(L) such that s ◦ r1 6 idLn(L) 6 s ◦ r2. Let

r1(x1, . . . , xn) = (x1, x3, x5, x7, . . . , xn−1)

and

r2(x1, . . . , xn) = (x2, x4, x6, x8 . . . , xn)

for all (x1, . . . , xn) ∈ Ln(L). It is clear that r1 and r2 are homomorphisms.

Moreover, the function given by s(x1, x2, . . . , xm) = (x1, x1, x2, x2, . . . , xm, xm) is

a homomorphism such that

s ◦ r1(x1, . . . , xn) = s(x1, x3, . . . , xn−1)

= (x1, x1, x3, x3, . . . , xn−1, xn−1)

6 (x1, x2, x3, x4, . . . , xn)

andr1 ◦ s(x1, x2, . . . , xm) = r1(x1, x1, x2, x2, . . . , xm, xm)

= (x1, x2, . . . , xm)

= idLm(L)(x1, x2, . . . , xm)

Therefore, r1 is a n-dimensional lower retraction which pseudo-inverse is s.

Analogously, it can be proved that r2 is a n-dimensional upper retraction with

pseudo-inverse s.

On the other hand,

1. clearly � is isotonic;

2. r1((x1, . . . , xm)� (y1, . . . , ym)) = r1(x1 ∧L y1, x1 ∨L y1, . . . , xm ∧L ym, xm ∨Lym) = (x1 ∧L y1, . . . , xm ∧L ym) = (x1, . . . , xm) ∧Lm(L) (y1, . . . , ym);

3. r2((x1, . . . , xm)� (y1, . . . , ym)) = (x1, . . . , xm) ∨Lm(L) (y1, . . . , ym); and

4. r1(x1, . . . , xn)�r2(x1, . . . , xn) = (x1, x3, . . . , xn−1)�(x2, x4, . . . , xn) = (x1∧Lx2, x1 ∨L x2, . . . , xn−1 ∧L xn, xn−1 ∨L xn) = (x1, x2, . . . , xn).

Therefore, � is an e-operator.

102

�

Notice that there are other possibilities for m and n such that Lm(L)ELn(L).

For example, m = 5 and n = 8. In this case r1(x1, . . . , x8) = (x1, x3, x5, x7, x8),

r2(x1, . . . , x8) = (x1, x2, x4, x6, x8) and s(x1, . . . , x5) = (x1, x2, x2, x3, x3, x4, x4,

x5).

Corollary 4.5 Let T be a t-norm on Lm(L) and n = 2m. The function TE� :

Ln(L)× Ln(L) −→ Ln(L) given by

TE�(x,y) = T(r1(x), r1(y))� T(r2(x), r2(y))

for all x,y ∈ Ln(L), is a t-norm on Ln(L).

Proof: Straightforward from Theorem 4.1 and Proposition 4.10.

�

4.7 Final Remarks

Now, considering the same symbols as in Section 3.7 and the following ab-

breviations shown in Table 4.1 we present here the main properties preserved by

extension method via e-operators.

Notice that we have not studied the extension of implications and negations

obtained from fuzzy connectives using method via e-operators what we pretend

to do as soon as possible.

103

4. EXTENSION METHOD VIA E-OPERATORS

Abbreviation Property

AC Archimedean

NI Nilpotent

IDN Idempotent

ZD Zero divisor

CL Cancellation law

C Continuity

P Positive

EC Relation between extension and conjugate

RE Relation between NKE�

and (NK)E� with K a fuzzy connective

Table 4.1: Table of Abbreviations

t-norm and t-conorm AC NI IDN ZD CL C P EC

TE�√ √ √ √ √ √ √

(T ρ)E� = (TE� )ψ

SE�√ √ √ √ √ √ √

(Sρ)E� = (SE�)ψ

Table 4.2: Properties preserved by TE� and SE�

fuzzy negation Strong Strict Eq. P. EC RE

NE�

√ √ √(Nρ)E� = (NE

� )ψ –

(NT )E� ? ? ? ? ?

(NS)E� ? ? ? ? ?

(NI)E� ? ? ? ? ?

Table 4.3: Properties preserved by NE�

104

Chapter 5

On Restricted Equivalence

Functions

It is well known that fuzzy sets theory provides an interesting framework for

several areas of knowledge. Particularly, regarding to the treatment of images

(pattern recognition, image segmentation, etc.) there is a huge amount of papers

which present important results and methods considering fuzzy sets valued on

[0, 1] (see for example Bustince et al. [2007]; Chaira and Ray [2003, 2005]; der

Weken et al. [2004]; Lopez-Molina et al. [2011]).

On image processing, a problem is to provide a good measure tool for making a

global comparison of images. In this scope, Bustince et al. introduced in Bustince

et al. [2006] the notion of restricted equivalence functions on [0, 1] (for short REF)

as a particular case of equivalence functions defined in Fodor and Roubens [1994].

Also, in Bustince et al. [2006] it is presented a theorem that characterizes REF

via implications, i.e. a way to construct these kind of functions using fuzzy

implications.

It is natural to generalize concepts related to image processing for lattices

in order to obtain a much more general framework than [0, 1]. In particular

bounded lattices are of interest since intensities in images can be considered as

taking values in such lattices.

Notice that, in particular, restricted equivalence functions are fuzzy operators

so, we can apply our two methods to extend them and see which one works better.

105

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

We start discussing about how to define REF on bounded lattices in order to

provide a construction method of these functions via implications (in Section 5.5,

we study similar issues for REF on L([0, 1])). We also work on defining restricted

dissimilarity functions and normal Ee,N -functions on bounded lattice. Finally, we

extend these functions via retractions and e-operators, in Sections 5.6 and 5.7.

5.1 Restricted Equivalence Functions on L

The problem of global comparison of two images has been studied by several

researchers in image processing (see Baddeley [1992]; Bustince et al. [2006, 2007];

der Weken et al. [2004]). One of the most used tools for this global comparison

is the equivalence functions introduced in Fodor and Roubens [1994].

Definition 5.1 A function EF : [0, 1]2 → [0, 1] is called equivalence function if

the following properties hold:

1. EF (x, y) = EF (y, x) for all x, y ∈ [0, 1];

2. EF (0, 1) = EF (1, 0) = 0;

3. EF (x, x) = 1 for all x ∈ [0, 1];

4. If x 6 x′ 6 y′ 6 y then EF (x, y) 6 EF (x′, y′).

As a particular class of these kind of functions, in Bustince et al. [2006] is

defined the notion of restricted equivalence functions.

Definition 5.2 A function REF : [0, 1]2 → [0, 1] is called a restricted equivalence

function if it satisfies the following conditions:

(1) REF (x, y) = REF (y, x) for all x, y ∈ [0, 1];

(2) REF (x, y) = 1 if and only if x = y;

(3) REF (x, y) = 0 if and only if x = 1 and y = 0, or x = 0 and y = 1;

(4) REF (x, y) = REF (N(x), N(y)) for all x, y ∈ [0, 1], N being a strong negation

on [0, 1];

(5) For all x, y, z ∈ [0, 1] such that x 6 y 6 z then REF (x, z) 6 REF (x, y) and

REF (x, z) 6 REF (y, z).

106

It is clear that every restricted equivalence function is an equivalence function

in the sense of Definition 5.1. But the reciprocal of this affirmation is not true.

For instance, the function EF : [0, 1]2 → [0, 1] given by

EF (x, y) =

{0, if x = 0 and y = 1 or x = 1 and y = 0;

1, otherwise.

for all x, y ∈ [0, 1], is an equivalence function but it is not a restricted equivalence

function (see Example 2 in Bustince et al. [2006]).

Naturally, concept of equivalence functions can be generalized for bounded

lattices as follows.

Definition 5.3 Let L be a bounded lattice. A function EF : L2 → L is called an

equivalence if it satisfies the following conditions:

(F1) EF (x, y) = EF (y, x) for all x, y ∈ L;

(F2) EF (0L, 1L) = EF (1L, 0L) = 0L;

(F3) EF (x, x) = 1L for all x ∈ L;

(F4) If x 6L y 6L z then EF (x, y) 6L EF (x, z).

The main goal of this section is presenting a formalization of the concept of

restricted equivalence functions for a bounded lattice L. A first problem arises

from the fact that, in general, it does not need to exist a strong negation for a given

lattice L. On the other hand, for many applications, specially in image processing,

it is crucial that an analog of (4) in Definition 5.2 holds, since it ensures the fact

that a given property is preserved when the negative of an image instead of the

image itself is considered. For this reason, we introduce the following definition.

Definition 5.4 Let N be a frontier negation on L. A function REF : L2 → L is

called a restricted equivalence function on L with respect to N , or just an L-REF

with respect to N , if it satisfies, for all x, y, z ∈ L, the following conditions:

(L1) REF (x, y) = REF (y, x);

107

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

(L2) REF (x, y) = 1L if and only if x = y;

(L3) REF (x, y) = 0L if and only if x = 1L and y = 0L, or x = 0L and y = 1L;

(L4) REF (x, y) = REF (N(x), N(y));

(L5) if x 6L y 6L z then REF (x, z) 6L REF (x, y).

Notice for a given lattice L a frontier negation always exists, as Example 2.9

shows. On the other hand, the requirement of N being a frontier negation can

no be weakened since otherwise a contradiction between (L3) and (L4) arises.

In any cases, and with an eye on possible applications, we will mostly deal with

REF’s defined with respect to a strong negation.

Example 5.1 Let L be a lattice with at least three elements, and take x0 ∈L\{0L, 1L}. Then we can define

REF (x, y) =

0L if x = y;

1L if {x, y} = {0L, 1L};x0 otherwise.

which is a restricted equivalence function with respect to any frontier negation N .

Notice that from (L4), (L5) and (L1) it is also possible to conclude that

REF (x, z) 6L REF (y, z) whenever x 6L y 6L z.

Example 5.2 Let M be a bounded lattice (see Figure 2.5) and N1 be a strong

M-negation as in Example 2.8. Thus, a function REF : M2 → M as defined in

the Table 5.1 is a L-REF with respect to N1 in the sense of Definition 5.4.

REF 0L x y 1L

0L 1L x y 0Lx x 1L x yy y x 1L x1L 0L y x 1L

Table 5.1: Restricted equivalence function on lattice M

Notice that in this case, the mapping N(x) = REF (0L, x) defines a strong

negation on the lattice M .

108

Proposition 5.1 Let REF be an L-REF with respect to N and ψ be an auto-

morphism on L. Then ψ ◦REF also is a restricted equivalence function also with

respect to N .

Proof: Analogous to [Bustince et al., 2006, Prop. 3].

�

Proposition 5.2 Let f : L2 → L be a commutative function and REF be an

L-REF with respect to a strong L-negation N . Then the function REFf : L2 → L

defined by

REFf (x, y) =

{REF (x, y) if x ¨ y

f(x, y) otherwise(5.1)

is an L-REF with respect to N .

Proof: Straightforward.

�

5.1.1 Restricted Equivalence Functions and Negations

The remark in Example 5.2 is just a particular case of the following general

result.

Proposition 5.3 Let REF be a restricted equivalence function on the lattice L

with respect to some frontier negation N . Then, the mapping

N0(x) = REF (0L, x)

is also a frontier negation on L.

Proof: That N0 satisfies (N1) is trivial. Let’s prove that (N2) holds for it.

Indeed, for each x, y ∈ L such that x 6L y since 0L 6L x 6L y by (L5) it follows

that N0(y) = REF (0L, y) 6L REF (0L, x) = N0(x).

�

109

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

Example 5.3 Notice that sometimes a given function can be a L-REF with re-

spect to some negation but not with respect to another one. For instance, consider

the Example 5.2. In this case, following the notation of Proposition 5.3, we have

that N0(0L) = 1L; N0(x) = x, N0(y) = y and N0(1L) = 0L, which is different

from the negation N1 considered for the definition of the REF. But for this N0

we see that REF (0L, x) = x 6= y = REF (1L, x), so property (L4) does not hold.

Corollary 5.1 Let REF be a restricted equivalence function on L with respect

to some frontier negation N and let N0 be defined as in Proposition 5.3. Then,

N0 = g ◦N , with g : L→ L defined as g(x) = REF (x, 1L).

Proof: From (L4) in the definition of restricted equivalence functions, we see

that N0(x) = REF (0L, x) = REF (1L, N(x)). From the symmmetry of REF the

first part of the statement follows.

�

Corollary 5.2 Let L be a lattice and REF be a restricted equivalence function on

L with respect to a frontier negation N such that g(x) = REF (x, 1L) is injective.

Then N is unique in the sense that if, for every x, y ∈ L, the identity REF (x, y) =

REF (N ′(x), N ′(y)) holds for some other frontier negation N ′, it follows that

N = N ′.

Proof: LetN ′ be a frontier negation such thatREF (x, y) = REF (N ′(x), N ′(y))

holds for every x, y ∈ L. Then, we have that REF (0L, x) = g(N ′(x)), so, as g is

injective, N ′(x) = g−1(REF (0L, x)) for every x ∈ L. In the same way we have

also that N(x) = g−1(REF (0L, x)). Thus, the result follows.

�

Example 5.4 Let’s consider again the lattice L of Example 5.2. Take as REF

the function defined in Example 5.1 with x0 = y. This is a restricted equiva-

lence function with respect to any strong negation defined on L, so in partic-

ular, for N = N1. But, with the notation of Proposition 5.3, we have that

N0(x) = REF (0L, x) = y = REF (0L, y) = N0(y), so N0 is not injective and

hence it is not strong.

110

Theorem 5.1 Let L be a lattice and N a strong L-negation. Then the mapping

REF (x, y) =

{1 if x = y ,

N((x ∨L y) ∧L (N(x) ∨L N(y))) otherwise.

is a restricted equivalence function with respect to N .

Proof: (L1) and (L2) are obvious.

(L3) REF (0L, 1L) = N((0L∨1L)∧ (N(0L)∨N(1L)))= N(1L∧1L) = N(1L) = 0L

and the same is valid for REF (1L, 0L). On the other hand, if REF (x, y) = 0L,

as N is a frontier negation, this means that (x ∨ y) ∧ (N(x) ∨ N(y)) = 1L. So

x ∨ y = 1L and N(x) ∨ N(y) = 1L. From the first identity it follows that either

x or y or both are equal to 1L whereas from the second identity either x = 0L or

y = 0L or both are equal to 0L. So the only possibility is {x, y} = {0, 1}.(L4) is straightforward.

(L5) Take x, y, z ∈ L such that x ≤L y ≤L z. Then REF (x, y) = N(y ∧ N(x))

and REF (x, z) = N(z ∧ N(x)). Since y ∧ N(x) ≤L z ∧ N(x), it follows that

REF (x, z) ≤L REF (x, y), as we wanted to see.

�

5.2 Characterization Theorem for L-REF

We present in this section a generalized method of constructing L-REF’s based

on fuzzy implications (see Theorem 7 of Bustince et al. [2008]). We start doing a

axiomatization of the concept of implications on bounded lattices.

The following theorem is a weak version of [Bustince et al., 2006, Theorem 7]

for the framework of bounded lattices.

Theorem 5.2 Let N : L2 → L be a strong L-negation and M : L2 → L be a

function such that, for all x, y ∈ L, it holds:

(M1) M(x, y) = M(y, x);

(M2) M(x, 1L) = x;

(M3) M(x, y) = 1L if and only if x = y = 1L;

(M4) M(x, y) = 0L if and only if x = 0L or y = 0L.

111

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

If there exist a function I : L2 → L satisfying (FPA), (OP), (CP) for N and (P)

then the function REF : L2 → L defined by

REF (x, y) = M(I(x, y), I(y, x)) (5.2)

is a L-REF with respect to N .

Proof: We shall prove that the conditions (L1) − (L5) hold. It is clear that

(M1) implies (L1). The proof for the others properties is given as follow:

(L2)

Suppose that REF (x, y) = 1L. Thus, by (5.2) we have that M(I(x, y), I(y, x)) =

1L and hence I(x, y) = 1L and I(y, x) = 1L by (M3). Therefore, by (OP ) it

follows that x 6L y and y 6L x, i.e. x = y.

Reciprocally, if x = y, that is x 6L y and y 6L x then I(x, y) = I(y, x) = 1L by

(OP ). Thus, it is easy to see that REF (x, y) = 1L.

(L3)

If REF (x, y) = 0L then M(I(x, y), I(y, x)) = 0L which allow us to conclude that

either I(x, y) = 0L or I(y, x) = 0L since M satisfies (M3). Thus, by (P ) we have

that either x = 1L and y = 0L or x = 1L and y = 0L.

On the other hand, if x = 1L and y = 0L then I(x, y) = 0L by (P ). Therefore,

REF (x, y) = M(I(x, y), I(y, x)) = M(0L, I(y, x)) = 0L by (M3). Analogously it

can be proved that REF (x, y) = 0L whenever x = 0L and y = 1L.

(L4)

For all x, y ∈ L we have that

REF (N(x), N(y)) = M(I(N(x), N(y)), I(N(y), N(x))) by (5.2)

= M(I(y, x), I(x, y)) by (CP )

= M(I(x, y), I(y, x)) by (M1)

= REF (x, y)

(L5)

Given x, y, z ∈ L such that x 6L y 6L z we have that I(x, y) = 1L and I(x, z) =

1L. Thus

REF (x, y) = M(I(x, y), I(y, x)) = M(1L, I(y, x)) = I(y, x) (5.3)

112

and

REF (x, z) = M(I(x, z), I(z, x)) = M(1L, I(y, x)) = I(z, x) (5.4)

Since y 6L z then by (FPA) it follows that I(z, x) 6L I(y, x) for all x ∈ L.

Therefore, considering this fact, by Equations (5.3) and (5.4) it can be concluded

that REF (x, z) 6L REF (x, y).

�

Example 5.5 Let L be the bounded lattice shown in Figure 2.5. If I : L2 → L

is the function defined in the Example 2.12 and M : L2 → L is a function given

by M(x, y) = min{x, y} for all x, y ∈ L then, by Theorem 5.2, REF (x, y) =

M(I(x, y), I(x, y)) is a restricted equivalence function on L (see Table 5.2).

REF 0L a b c d e 1L

0L 1L e b c d a 0a e 1L e e e e ab b e 1L e e e bc c e e 1L e e cd d e e e 1L e de a e e e e 1L e

1L 0L a b c d e 1L

Table 5.2: Restricted equivalence function on lattice L

It is worth noting that the reciprocal of Theorem 5.2 does not hold, in gen-

eral. In other words, given an L-REF with respect to a strong L-negation N

and a function M : L2 → L satisfying (M1), (M2) and (M3) it is not always

possible to define a function I : L2 → L which satisfy (FPA), (OP ), (CP ), (P )

and REF (x, y) = M(I(x, y), I(y, x)). This is due to there is not a specific way

to define restricted equivalence function for pairs (x, y) ∈ L2 such that x ‖ y(incomparable elements of L). According to properties (L2) and (L3) we can just

infer that 0L <L REF (x, y) <L 1L if x ‖ y.

It means that determinate how to define a general condition for L-REF on

incomparable elements of L2 in order to make the reciprocal of the Theorem 5.2

holds constitute a very interesting open problem to be studied.

113

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

Proposition 5.4 Let REF be an L-REF related to a strong L-negation N . Then

the function IREF : L2 → L defined by

IREF (x, y) =

{1L if x 6L y

REF (x, y) otherwise(5.5)

satisfies the properties (OP), (CP) and (P).

Proof: .

(OP )

Let x, y ∈ L. If x 6L y then, by definition, it is trivial that IREF (x, y) = 1L.

Reciprocally, if IREF (x, y) = 1L then either x 6L y or REF (x, y) = 1L and

y <L x or x ‖ y. But, note that if REF (x, y) = 1L then x = y by (L2) which is

a contradiction with both y <L x and x ‖ y. Therefore, it can be concluded that

x 6L y.

(CP )

Note that if x 6L y we have N(y) 6L N(x) and hence IREF (x, y) = 1L =

IREF (N(y), N(x)) by definition of IREF . Beyond that we can have y <L x or x ‖ ywhich implies that N(x) <L N(y) and N(x) ‖ N(y) respectively by Proposition

2.12. Thus in both cases it follows that IREF (N(y), N(x)) = REF (N(y), N(x)) =

REF (N(x), N(y)) and IREF (x, y) = REF (x, y) what allow us to conclude that

IREF (N(y), N(x)) = IREF (x, y) since REF (N(x), N(y)) = REF (x, y) by (L4).

(P )

If IREF (x, y) = 0L then either REF (x, y) = 0L and y <L x or REF (x, y) = 0L

and x ‖ y. But, the second case is a contradiction by (L3). Hence we must have

REF (x, y) = 0L and y <L x and again by (L3) it follows that x = 1L and y = 0L.

Reciprocally, if x = 1L and y = 0L then IREF (x, y) = REF (x, y) = REF (1L, 0L) =

0L.

�

Remark 5.1 Notice that if REF and IREF are functions as in Proposition 5.4

and M : L2 → L is a function as in Theorem 5.2 (satisfying also M(x, x) = x

for all x ∈ L ) then the Identity (5.2) holds.

114

Indeed, let x, y ∈ L and suppose that they are comparable (i.e. x ¨ y). In this

case, if x 6L y then IREF (x, y) = 1L and IREF (y, x) = REF (y, x) = REF (x, y).

Thus

REF (x, y) = IREF (y, x) = M(1L, IREF (y, x)) = M(IREF (x, y), IREF (y, x))

Analogously, one can prove that (5.2) holds if y 6L x.

On the other hand, if x ‖ y then IREF (x, y) = REF (x, y) and IREF (y, x) =

REF (y, x) = REF (x, y). Therefore

REF (x, y) = M(REF (x, y), REF (y, x)) = M(IREF (x, y), IREF (y, x))

Example 5.6 Here we show that the reciprocal of Theorem 5.2 does not hold

in general. Actually, we would like to highlight the fact that the property (FPA)

fails for the function IREF defined in (5.5) (by Theorem 7 in Bustince et al.

[2006] it known that IREF should satisfy (FPA), (OP), (CP) and (P)). To do

so, consider the bounded lattice M (see Figure 1) and the restricted equivalence

function defined in Example 5.2. In this case, the function IREF in (5.5) is defined

as in Table 5.3. Note that y < 1M but IREF (1M , x) = y and IREF (y, x) = x which

means that IREF (1M , x) ‖ IREF (y, x) since x ‖ y.

IREF 0L x y 1L

0L 1L 1L 1L 1Lx x 1L x 1Ly y x 1L 1L1L 0L y x 1L

Table 5.3: The function IREF on lattice M

Taking into account the important fact highlighted above, we can see that the

problem of proving the reciprocal of Theorem 5.2 is reduced to seek conditions

under which the function IREF as in Proposition 5.4 satisfies (FPA). Clearly

if all the elements of the bounded lattice L are comparable then the problem is

sold, it means:

115

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

Theorem 5.3 Let L be a total bounded lattice and suppose that M : L2 → L is

a function satisfying (M1), (M2), (M3) and M(x, x) = x for all x ∈ L. Thus a

function REF : L2 → L is an L-REF for a strong L-negation N if and only if

there exist a function IREF : L2 → L satisfying (FPA), (OP), (CP) for N and

(P) in such a way that the Equation (5.2) holds.

Proof: Let REF be a L-REF with respect to a strong negation N . Define:

IREF (x, y) =

{1L if x 6L y ;

REF (x, y) otherwise.

From Proposition 5.4 and Remark 5.1 we can conclude that IREF satisfies (OP ),

(CP ), (P ) and REF (x, y) = M(IREF (x, y), IREF (y, x)) for all x, y ∈ L. Thus, it

remains to prove that property (FPA) holds.

Take y ∈ L and x, z ∈ L such that x ≤L z. We want to see that IREF (z, y) ≤LIREF (x, y). There are three possibilities.

(i) If y ≤L x ≤L z we have that IREF (z, y) = REF (z, y) and IREF (x, y) =

REF (x, y). But by (L5), REF (z, y) ≤L REF (x, y) and hence IREF (z, y) �IREF (x, y).

(ii) When x ≤L y ≤L z it follows that IREF (z, y) = REF (z, y) ≤L 1L =

IREF (x, y).

(iii) If x ≤L z ≤L y then IREF (z, y) = 1L = IREF (x, y).

The reciprocal is straightforward from Theorem 5.2.

�

5.3 Restricted Dissimilarity Functions

The task of providing suitable ways to define metric functions to measure the

similarity or dissimilarity between two images is a very studied problem. In this

sense, Bustince et al. [2008] introduce the concept of restricted dissimilarity func-

tions valued on [0, 1] based on the concept of dissimilarity proposed by Baddeley

[1992]. Moreover, in Bustince et al. [2008] is proposed a method to construct

these kind of functions from restricted equivalence functions.

116

In this section we generalize the definition of restricted dissimilarity functions

given in Bustince et al. [2008] for bounded lattices and prove some related results.

Definition 5.5 Let L be a bounded lattice. A function dL : L2 → L is called

a restricted dissimilarity function on L (for short L-RDF) if it satisfies, for all

x, y, z ∈ L, the following conditions:

(D1) dL(x, y) = dL(y, x);

(D2) dL(x, y) = 1L if and only if either x = 1L and y = 0L or x = 0L and y = 1L;

(D3) dL(x, y) = 0L if and only if x = y;

(D4) if x 6L y 6L z then dL(x, y) 6L dL(x, z) and dL(y, z) 6L dL(x, z).

Theorem 5.4 Let REF : L2 → L be an L-REF with respect to N . If N ′ a

strong L-negation (not necessarily equal to N) then, the function defined by

dL(x, y) = N(REF (x, y)) for all x, y ∈ L (5.6)

is a restricted dissimilarity function.

Proof: Analogously to the proof of Theorem 5 in Bustince et al. [2008].

�

Corollary 5.3 Under the same conditions of Theorem 5.11, it holds that

dL(x, y) = dL(N(x), N(y)) for all x, y ∈ L (5.7)

Proof: Straightforward from Theorem 5.11 and (L4).

�

Lemma 5.1 Let G : L2 → L be a function satisfying for all x, y ∈ L(G1) G(x, y) = G(y, x);

(G2) G(x, 0L) = x;

(G3) G(x, 1L) = 1L.

Thus, if N is a strong L-negation on L then the function M : L2 → L given

117

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

by M(x, y) = N(G(N(x), N(y))) satisfies (M1), (M2) and (M3). Moreover, the

following equation holds:

N(M(x, y)) = M(N(x), N(y)) (5.8)

Proof: It is easy to see that property (M1) and Equation (5.8) hold (since

N is involutive). It remains to prove (M2) and (M3). In other words, we shall

prove that 1L is a neutral element and 0L is a annihilator of M . Thus, for all

x ∈ L, we have

M(x, 1L) = N(G(N(x), N(1L))) = N(G(N(x), 0L)) = N(0L) = 1L

and

M(x, 0L) = N(G(N(x), N(0L))) = N(G(N(x), 1L)) = N(N(x)) = x

�

The following theorem is a version for bounded lattices of Theorem 6 in

Bustince et al. [2008] (only the sufficiency part).

Theorem 5.5 Let G : L2 → L be a function satisfying (G1), (G2) and (G3).

Given a function dL : L2 → L, if there exists a function I : L2 → L for

which the properties (FPA), (OP ), (CP ) and (P ) hold and such that dL(x, y) =

G(N(I(x, y)), N(I(y, x))) for all x, y ∈ L, then dL is a restricted dissimilarity

function satisfying Equation (5.7).

Proof: Given G satisfying properties (G1), (G2) and (G3) by Lemma 5.1 the

function M : L2 → L defined by M(x, y) = N(G(N(x), N(y))) for all x, y ∈ Lis such that (M1), (M2) and (M3) hold. Thus, since I is a function satisfying

(FPA), (OP ), (CP ) and (P ) then by Theorem 5.2 a function REF : L2 → L

given by REF (x, y) = M(I(x, y), I(y, x)) for all x, y ∈ L is a restricted equiva-

lence function.

118

Moreover,

REF (x, y) = M(I(x, y), I(y, x))

= N(G(N(I(x, y)), N(I(y, x))))

= N(dL(x, y))

Since N is involutive then dL(x, y) = N(REF (x, y)) and hence by Theorem 5.11

we can affirm that dL is a restricted dissimilarity function. Also, it is clear that

dL satisfies Equation (5.7).

�

5.4 Normal Ee,N-functions on L

Another concept that can naturally be generalized for bounded lattices are

normal EN -functions (see Bustince et al. [2008]) where N is a strong fuzzy nega-

tion. It is important to say that in this section we are considering only strong

negations N which have at least one equilibrium point (as it can be seen in Section

2.2.2 there are strong negations that do not have any equilibrium point).

Definition 5.6 Let e be an equilibrium point of a strong L-negation N . A func-

tion Ee,N : L→ L is called a normal Ee,N -function associated to N (L-NEF, for

short) if it satisfies the following conditions:

1. Ee,N(x) = 1L if and only if x = e;

2. Ee,N(x) = 0L if and only if x = 0L or x = 1L;

3. For all x, y ∈ L such that either e 6L x 6L y or y 6L x 6L e it follows

Ee,N(y) 6L Ee,N(x);

4. Ee,N(x) = Ee,N(N(x)) for all x ∈ L.

As done for restricted dissimilarity function we present now a way to construct

normal Ee,N -function on bounded lattices from restricted equivalence functions.

119

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

Theorem 5.6 Let N be a strong L-negation and e be an equilibrium point of

N . If REF : L2 → L is an L-REF then the function given by Ee,N(x) =

REF (x,N(x)) for all x ∈ L is a Ee,N -function.

Proof:

1. Suppose that Ee,N(x) = 1L. Thus REF (x,N(x)) = 1L and by (L2) we

have x = N(x). Then x = e. Reciprocally, if x = e then Ee,N(e) =

REF (e,N(e)) = REF (e, e) = 1L.

2. If Ee,N(x) = 0L then REF (x,N(x)) = 0L. Hence either x = 1L and

N(x) = 0L or x = 0L and N(x) = 1L, i.e. x = 1L or x = 0L. On the other

hand, if x = 0L then Ee,N(x) = REF (0L, N(0L)) = 0L and if x = 1L then

Ee,N(x) = REF (1L, N(1L)) = 0L.

3. Note that if e 6L x 6L y then N(y) 6L N(x) 6L N(e). Since N(e) = e

we have that N(y) 6L N(x) 6L x 6L y and hence REF (y,N(y)) 6LREF (x,N(x)) by (L5). Thus Ee,N(y) 6L Ee,N(x).

Suppose now that y 6L x 6L e. In this case, we have that x 6L y 6LN(x) 6L N(y) and again by (L5) it can concluded that REF (y,N(y)) 6LREF (x,N(x)), i.e. Ee,N(y) 6L Ee,N(x).

4. For all x ∈ L we have

Ee,N(N(x)) = REF (N(x), N(N(x)))

= REF (N(x), x)

= REF (x,N(x))

= Ee,N(x)

�

Corollary 5.4 If dL is a restricted dissimilarity function on L then the function

given by Ee,N(x) = N(dL(x,N(x))), for all x, y ∈ L, is a normal Ee,N -function.

Proof: Straightforward from Theorem 5.11 and 5.12.

�

120

Theorem 5.7 Let M : L2 → L a function satisfying (M1), (M2) and (M3). If

I : L2 → L satisfies (FPA), (OP ), (CP ) and (P ) then

Ee,N(x) = M(I(x,N(x)), I(N(x), x)) (5.9)

for all x ∈ L is a normal Ee,N -function.

Proof: By Theorem 5.2 we know that REF (x, y) = M(I(x, y), I(y, x)) for all

x, y ∈ L is a L-REF. Thus Ee,N(x) = REF (x,N(x)) = M(I(x,N(x)), I(N(x), x))

is a normal Ee,N -function by Theorem 5.12.

�

Corollary 5.5 Let e be an equilibrium point of the strong L-negation N . Under

the same conditions of Theorem 5.7, we have Ee,N(x) = I(x,N(x)) for all x ∈ Lsuch that e 6L x.

Proof: If e 6L x then N(x) 6L N(e) and hence N(x) 6L e 6L x since

N(e) = e. Thus by (OP ) we have that I(N(x), x) = 1L. Therefore

Ee,N(x) = M(I(x,N(x)), I(N(x), x)) = M(I(x,N(x)), 1L) = I(x,N(x))

by Theorem 5.7 and (M2).

�

5.5 REF on L([0, 1]): Definition and characteri-

zation

One of main tasks related to restricted equivalence functions is providing

a characterization of them, i.e. propose a suitable method to construct these

functions. In this sense, it is presented in Bustince et al. [2006] (see Theorem 7)

a method based on implications (for REFs on [0, 1]). In this section we introduce

a generalization of this method for restricted equivalence functions on L([0, 1]).

121

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

As discussed in Example 2.2 from [0, 1] it is possible to define a set of intervals

L([0, 1]) = {[x, y] | 0 6 x 6 y 6 1} which is endowed with the partial order

[a, b] 62 [c, d]⇔ a 6 c and b 6 d (5.10)

for all a, b, c, d ∈ [0, 1], is a bounded lattice with bottom and top elements being

[0, 0] and [1, 1] respectively. There are several works in the literature discussing

about fuzzy operators on L([0, 1]) and its generalizations (see Bedregal et al.

[2012b]; Bilgic and Turksen [1994]; Dimuro et al. [2011]; M et al. [1996]).

Though the relation 62 generates just a partial order on L([0, 1]) it is possible

to extend this partial order to a linear order (total order) . Bustince et al. in

Bustince et al. [2012] introduced some ways to make this extension.

Definition 5.7 Bustince et al. [2012] Let (L([0, 1]),�) be a poset1. The order

� is called an admissible order if

1. � is a linear order on L([0, 1]);

2. for all [a, b], [c, d] ∈ L([0, 1]), [a, b] � [c, d] whenever [a, b] 62 [c, d].

Example 5.7 It is possible to prove that the order defined by [a, b] �Lex1 [c, d] if

and only if either a < c or a = c and b 6 d is an admissible order in L([0, 1])

motivated by the lexicographical order in R2.

Definition 5.8 Jurio et al. [2009] A function REFIV : L([0, 1])2 → L([0, 1])

is called a interval valued restricted equivalence function if it satisfies for all

X, Y, Z ∈ L([0, 1]):

1. REFIV (X, Y ) = REFIV (Y,X);

2. REFIV (X, Y ) = [1, 1] if and only if X = Y ;

3. REFIV (X, Y ) = [0, 0] if and only if either X = [1, 1] and Y = [0, 0] or

X = [0, 0] and Y = [1, 1];

1A non-empty set P endowed with a partial order 6P is called a partial order set or forshort a poset.

122

4. REFIV (X, Y ) = REFIV (N(X), N(Y )) with N a frontier negation;

5. if X 6 Y 6 Z then REF (X,Z) 6 REF (X, Y ).

Definition 5.9 Baczynski and Jayaram [2008] A function IIV : L([0, 1])2 →L([0, 1]) is called a interval fuzzy implication if it satisfies for all X, Y, Z,K ∈L([0, 1]) the following conditions:

(FPA) X 6 Z implies IIV (X, Y ) ≥ IIV (Z, Y ) (first place antitonicity);

(SPI) Y 6 K implies IIV (X, Y ) 6 IIV (X,K) (second place isotonicity);

(RB) IIV (X, [1, 1]) = [1, 1] (right corner condition);

(LB) IIV ([0, 0], Y ) = [1, 1] (left corner condition);

(CC3) IIV ([1, 1], [0, 0]) = [0, 0].

Extra properties for a given fuzzy implication IIV on L([0, 1]):

(OP) IIV (X, Y ) = [1, 1] if and only if X 6 Y (the ordering property);

(CP) IIV (X, Y ) = IIV (N(Y ), N(X)) with a strong negation N (contraposition);

(P) IIV (X, Y ) = [0, 0] if and only if X = [1, 1] and Y = [0, 0] (positive).

Theorem 5.8 Let � be an admissible order in L([0, 1]) and MIV : L([0, 1])2 →L([0, 1]) be a function satisfying (M1), (M2), (M3), (M4) and MIV (X,X) = X

for each X ∈ L([0, 1]). Thus REFIV : L([0, 1])2 → L([0, 1]) is an L-REF if and

only if there exists a function IIV : L([0, 1])2 → L([0, 1]) satisfying (FPA), (OP ),

(CP ), (P ) and such that REFIV (X, Y ) = MIV (IIV (X, Y ), IIV (Y,X))

Remark 5.2 It is worth noting that considering an admissible order (linear or-

der) on L([0, 1]) is an essential hypothesis for the Theorem 5.8 holds. For in-

stance, assuming the partial order 62 on L([0, 1]) and let REF : [0, 1]→ [0, 1] be

given by REF (x, y) = 1− |x− y|, which is a restricted equivalence function with

respect to the strong negation n(x) = 1 − x for all x ∈ [0, 1] (see Bustince et al.

[2006], example 1). Then

REFIV ([a, b], [c, d]) = [min{REF (a, c), REF (b, d)},max{REF (a, c), REF (b, d)}](5.11)

123

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

is a restricted equivalence function on (L([0, 1]),62). Nevertheless, if IIV is the

interval version of IREF defined in (5.5) then IIV does not satisfy property (FPA).

Indeed, taking X = [0.4, 0.7], Y = [0.5, 0.8] and Z = [0.5, 0.6] we have that

X 62 Y but REFIV (X,Z) = [0.9, 0.9] and REFIV (Y, Z) = [0.8, 1] which are not

comparable with respect to 62.

5.6 Extension of REF via Retractions

In this section we are interested in applying the method via retractions for

extending equivalence functions motivated by the fact that images can be repre-

sented by lattices (see Palmeira et al. [submitted 2013a]). For instance, imagine

you have an equivalence function E that makes a good comparison between im-

ages A and B. If you extend both images and want to make a new comparison

you should have a function E ′ to make it. One possible solution is extending E

in such way to preserve its properties. In this sense, our extension method can

be a suitable alternative.

Theorem 5.9 Let M be a (r, s)-sublattice of L and EF : M2 → M an equiva-

lence function. Then the function

EFE(x, y) = s(EF (r(x), r(y))) (5.12)

for each pair (x, y) ∈ L2 is an equivalence function that extends EF from M to

L.

Proof: It is clear that EFE satisfies (F1) since EF is an equivalence function.

Moreover, for all x ∈ L we have

EFE(x, x) = s(EF (r(x), r(x))) = s(1M) = 1L

and

EFE(0L, 1L) = s(EF (r(0L), r(1L))) = s(EF (0M , 1L)) = s(0M) = 0L

Analogously, one can prove that EFE(1L, 0L) = 0L. Thus, (F2) and (F3) hold.

124

It remains to prove (F4). To do so, note that for all x, y, z ∈ L such that x 6Ly 6L z then r(x) 6M r(y) 6M r(z) and hence EF (r(x), r(y)) 6M EF (r(x), r(z)).

Thus

EFE(x, y) = s(EF (r(x), r(y))) 6L s(EF (r(x), r(z))) = EFE(x, z).

�

Notice that Definition 5.4 refines Definition 5.3 since it imposes some new

constraints. Beyond property (L4), Definition 5.4 requires also that the unique

elements assigned to 0L by REF are (0L, 1L) and (1L, 0L) and that REF is

evaluated as 1L only for pairs with the same value in both coordinates. Thus,

to show that we can extend restricted equivalence functions using the method

provided by Propositions 3.1 and 3.3.

If M is a (r, s)-sublattice of L , N is a fuzzy negation on M and REF :

M2 → M a restricted equivalence function with respect to N then the function

REFE : L2 → L given by REFE(x, y) = s(REF (r(x), r(y))) satisfies property

(L4)

REFE(NE(x), NE(y)) = s(REF (r(NE(x)), r(NE(y))))

= s(REF (r(s(N(r(x)))), r(s(N(r(y))))))

= s(REF (N(r(x)), N(r(y))))

= s(REF (r(x), r(y)))

= REFE(x, y)

Moreover, if r is such that r(x) = 0L if and only if x = 0L and r(x) = 1L if and only

if x = 1L then supposing REFE(x, y) = 0L we have that s(REF (r(x), r(y))) =

0L. Since s is an injective function it follows that REF (r(x), r(y)) = 0M and

hence r(x) = 1M and r(y) = 0M or r(x) = 0M and r(y) = 1M by (L3), what allow

us to conclude that x = 1L and y = 0L or x = 0L and y = 1L, i.e. REFE satisfies

property (L3) (the necessity side of (L3) is shown in Theorem 5.9). Nevertheless

it does not satisfies (L2) in general as we can see in the following example.

Example 5.8 Let M and L be the bounded lattices pictured in Figure 2.5. Func-

tion s : M → L defined by s(0M) = 0L, s(x) = b, s(y) = d and s(1M) = 1L is a

125

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

pseudo-inverse of retraction

r(t) =

0M , t = 0L;

1M , t = 1L;

y, t = d;

x, otherwise.

and hence M is a (r, s)-sublattice of L. Then, considering the REF as in Ex-

ample 5.2, its extension to L is such that REFE(e, c) = s(REF (r(e), r(c))) =

s(REF (x, x)) = s(1M) = 1L showing that REFE does not satisfies (L2).

The above discussion allows us to enunciate the following proposition (exten-

sion of a weak version of restricted equivalence functions).

Proposition 5.5 Let M be a (r, s)-sublattice of L and REF : M2 → M a re-

stricted equivalence function. Then, function REFE : L2 → L given by REFE(x, y) =

s(REF (r(x), r(y))) for all x, y ∈ L satisfies

(L1) REFE(x, y) = REFE(y, x);

(WL2) REFE(x, x) = 1L;

(L4) REFE(x, y) = REFE(NE(x), NE(y)), being N a fuzzy negation on M ;

(L5) if x 6L y 6L z then REF (x, z) 6L REF (x, y).

Moreover, if r(x) = 0M iff x = 0L and r(x) = 1M iff x = 1L then

(L3) REFE(x, y) = 0L if and only if x = 1L and y = 0L or x = 0L and y = 1L.

Definition 5.10 Let L be an arbitrary bounded lattice. A function REF : L2 →L satisfying properties (L1), (WL2), (L3), (L4) and (L5) is called a weak re-

stricted equivalence function.

126

5.7 Extension of REF via e-operators

As we have seen in previous section the extension method proposed in Palmeira

and Bedregal [2012] is not efficient enough to extend restricted equivalence func-

tions. So, let’s apply the extension method via e-operators to verify if we can use

it to extend restricted equivalence functions.

Theorem 5.10 Let M E L with respect to (r1, r2, s), � : M ×M → L be an

e-operator and REF : M2 → M a restricted equivalence function. Then the

function REFE� : L2 → L given by

REFE� (x, y) = REF (r1(x), r1(y))�REF (r2(x), r2(y)) (5.13)

for all x, y ∈ L, is a restricted equivalence function on L.

Proof: It easy to see that REFE� is commutative since � is a commutative

operator and hence (L1) holds.

(L2)

Note that by Lemma 4.1 (item 2) and (L2) we have thatREFE� (x, x) = REF (r1(x), r1(x))�

REF (r2(x), r2(x)) = 1M �1M = s(1M) = 1L. On the other hand, supposing that

REFE� (x, y) = 1L it follows that REF (r1(x), r1(y))�REF (r2(x), r2(y)) = 1L and

then by Lemma 4.2 we have thatREF (r1(x), r1(y)) = 1M andREF (r2(x), r2(y)) =

1M . Since REF is a restricted equivalence function then r1(x) = r1(y) and

r2(x) = r2(y). Therefore, by Lemma 4.1 we can conclude that x = y.

(L3)

It is clear that

REFE� (1L, 0L) = REF (r1(1L), r1(0L))�REF (r2(1L), r2(0L))

= REF (1M , 0M)�REF (1M , 0L) = 0M � 0M

= s(0M) = 0L

Similarly REFE� (0L, 1L) = 0L is proved.

Conversely supposing REFE� (x, y) = 0L we have that REF (r1(x), r1(y)) �

REF (r2(x), r2(y)) = 0L and thenREF (r1(x), r1(y)) = 0M andREF (r2(x), r2(y)) =

0M by Lemma 4.2. Hence by Lemma 4.1 we have

127

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

(i) r1(x) = 1M and r1(y) = 0M ; or

(ii) r1(x) = 0M and r1(y) = 1M ; and

(iii) r2(x) = 1M and r2(y) = 0M ; or

(iv) r2(x) = 0M and r2(y) = 1M .

Note that (i) and (iv) can not happen simultaneously since otherwise we should

have 0M = r2(x) < r1(x) = 1M what is a contradiction with the fact that

r1(x) 6M r2(x) for every x ∈ L. A similar argument works to show that (ii) and

(iii) can not happen simultaneously. It remains to analyze two other possibilities:

(i) and (iii)

In this case, by Lemma 4.1, item 4, we have that{r1(x) = 1M = r1(1L)

r2(x) = 1M = r2(1L)⇒ x = 1L

and {r1(y) = 0M = r1(0L)

r2(y) = 0M = r2(0L)⇒ y = 0L

Analogously it can be proved that (ii) and (iv) imply x = 0L and y = 1L.

(L4)

Let N be a frontier negation on M . We shall prove that identity REFE� (x, y) =

REFE� (NE

� (x), NE� (y)) holds for each x, y ∈ L. Indeed, we have that

REFE� (NE

� (x), NE� (y)) =

= REF (r1(NE(x)), r1(N

E(y)))�REF (r2(NE(x)), r2(N

E(y))) (5.14)

Since by Lemma 4.1 item 1

REF (r1(NE(x)), r1(N

E(y))) =

= REF (r1(N(r1(x))�N(r2(x))), r1(N(r1(y))�N(r2(y))))

= REF (N(r2(x)), N(r2(y)))(5.15)

128

and

REF (r2(NE(x)), r2(N

E(y))) =

= REF (r2(N(r1(x))�N(r2(x))), r2(N(r1(y))�N(r2(y))))

= REF (N(r1(x)), N(r1(y)))(5.16)

by commutativity of � and Identity (5.14) it follows that

REFE� (NE

� (x), NE� (y)) = REF (N(r1(x)), N(r1(y)))�REF (N(r2(x)), N(r2(y)))

= REF (r1(x), r1(y))�REF (r2(x), r2(y))

= REFE� (x, y)

(L5)

Let x, y, z ∈ L such that x 6L y 6L z. In this case we have r1(x) 6Mr1(y) 6M r1(z) and r2(x) 6M r2(y) 6M r2(z) and hence REF (r1(x), r1(z)) 6MREF (r1(x), r1(y)) and REF (r2(x), r2(z)) 6M REF (r2(x), r2(y)). Then, by iso-

tonicity of � we have that

REFE� (x, z) = REF (r1(x), r1(z))�REF (r2(x), r2(z))

6L REF (r1(x), r1(y))�REF (r2(x), r2(y))

= REFE� (x, y)

�

5.7.1 Extension of Natural Negation of REF

Now we turn our attention for extending a special class of fuzzy negations

constructed from restricted equivalence functions as defined in Section 5.1.1.

Notice that if M is a (r1, r2, s)-sublattice of L, given a restricted equivalence

function REF on M its extension using method via e-operators is naturally

an extension for the negation obtained from this REF , i.e. an extension of

NREF (x) = REF (0M , x) which is given by

(NREF )E�(y) = NREF (r1(y))�NREF (r2(y))

= REF (0M , r1(y))�REF (0M , r2(y))

129

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

for all y ∈ L. This is obviously a fuzzy negation on L (see Section 4).

On the other hand, we have that

NREFE�(y) = REFE

� (0L, y)

= REF (r1(0L), r1(y))�REF (r2(0L), r2(y))

= REF (0M , r1(y))�REF (0M , r2(y))

So, it follows that (NREF )E� = NREFE�.

In a similar way we can also extend the class of fuzzy negations obtained from

weak restricted equivalence functions using extension method via retractions.

Proposition 5.6 Let M be a (r, s)-sublattice of L. If REF is a restricted equiv-

alence function on M then NREFE(x) = s(REF (0L, r(x))) for each x ∈ L is a

fuzzy negation which satisfies (NREF )E = NREFE .

Proof: Straght.

�

The idea behind Proposition 5.6 is that since it is not possible to extend REF’s

using extension method via retractions, negations generated from its extension

can be extended.

5.8 Extension of Restricted Dissimilarity Func-

tion

As we have done for restricted equivalence functions, here we apply both exten-

sion methods via retractions and via e-operators to extend restricted dissimilarity

functions.

Definition 5.11 Let L be a bounded lattice. A function dL : L2 → L is called

a restricted dissimilarity function on L (for short L-RDF) if it satisfies, for all

x, y, z ∈ L, the following conditions:

(D1) dL(x, y) = dL(y, x);

(D2) dL(x, y) = 1L if and only if either x = 1L and y = 0L or x = 0L and y = 1L;

130

(D3) dL(x, y) = 0L if and only if x = y;

(D4) if x 6L y 6L z then dL(x, y) 6L dL(x, z) and dL(y, z) 6L dL(x, z).

As we have done for restricted equivalence functions, here we apply both

extension methods via retractions and via e-operators to extend restricted dis-

similarity functions.

Proposition 5.7 Let M be a (r, s)-sublattice of L and suppose r is such that

r(x) = 0M if and only if x = 0L and r(x) = 1M if and only if x = 1L. If dM :

M2 → M a restricted dissimilarity function then, dEM(x, y) = s(dM(r(x), r(y)))

for each x, y ∈ L satisfies (D1), (D2) and (D4). Moreover, dL(x, x) = 0L for

every x ∈ L.

Proof: By Proposition 5.5 it is clear that (D1) holds. Moreover, for every

x ∈ L it follows that dEM(x, x) = s(dM(r(x), r(x))) = s(0M) = 0L. So, it remains

to prove (D2) and (D4).

Supposing dEM(x, y) = 1L then we have that s(dM(r(x), r(y))) = 1L implying

dM(r(x), r(y)) = r(1L) = 1M and hence either r(x) = 1M and r(y) = 0M or

r(x) = 0M and r(y) = 1M . Since r(x) = 0M if and only if x = 0L and r(x) = 1M

if and only if x = 1L then either x = 1L and y = 0L or x = 0L and y = 1L. On the

other hand, dEM(1L, 0L) = s(dM(r(1L), r(0M))) = s(dM(1M , 0M)) = s(1M) = 1L.

By item 1 it holds that dEM(0L, 1L) = 0L. Therefore, dEM satisfies (D2).

Now, take x, y, z ∈ L such that x 6L y 6L z. In this case, we have that

r(x) 6M r(y) 6M r(z) and hence dM(x, y) 6L dM(x, z). Then

dEM(x, y) = s(dM(r(x), r(y))) 6L s(dM(r(x), r(z))) = dEM(x, z)

Analogously, one can prove that dEM(y, z) 6L dEM(x, z) what allows us to say that

(D4) holds.

�

In other words, Proposition 5.7 says that extension method via retractions is

not efficient to extend restricted dissimilarity functions, just a weak version of this

kind of function what also happened for extending restricted equivalence functions

131

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

using same method as we have seen in Section 5.6. But, using extension method

via e-operators it is possible extend restricted dissimilarity functions successfully.

Proposition 5.8 Let M E L with respect to (r1, r2, s), � : M × M → L be

an e-operator and dM : M2 → M a restricted dissimilarity function. Then the

function (dM)E� : L2 → L given by

(dM)E�(x, y) = dM(r1(x), r1(y))� dM(r2(x), r2(y)) (5.17)

for all x, y ∈ L, is a restricted dissimilarity function on L.

Proof: Similar to proof of Theorem 4.14.

�

The following theorem proposes a way to construct restricted dissimilarity

functions from restricted equivalence functions.

Theorem 5.11 Let REF : L2 → L be an L-REF with respect to N . If N ′ a

frontier L-negation (not necessarily equal to N) then, the function defined by

dL(x, y) = N(REF (x, y)) for all x, y ∈ L (5.18)

is a restricted dissimilarity function.

Corollary 5.6 Let M E L with respect to (r1, r2, s), � : M × M → L be an

e-operator and REF : M2 →M a restricted equivalence function. If N is a fuzzy

negation on M then dL : L2 → L given by dL(x, y) = NE� (REFE

�(x, y)) for all

x, y ∈ L, is restricted dissimilarity function.

Proof: Straightforward from Proposition 4.3, Theorem 5.10 and Theorem 5.11.

�

132

5.9 Extension of Normal Ee,N-functions

Proposition 5.9 Let M be a (r, s)-sublattice of L and e be an equilibrium point

of a strong negation N : M →M . If Ee,N : M →M is a a normal Ee,N -function

associated to N then the function given by (Ee,N)E(x) = s(Ee,N(r(x))) for each

x ∈ L satisfies

1. (Ee,N)E(s(e)) = 1L;

2. if r is a lower retraction and (Ee,N)E(x) = 1L then s(e) 6L x;

3. if r is an upper retraction and (Ee,N)E(x) = 1L then x 6L s(e);

4. if r(x) = 0M if and only if x = 0L and r(x) = 1M if and only if x = 1L then

(Ee,N)E(x) = 0L if and only if x = 0L or x = 1L;

5. For all x, y ∈ L such that either s(e) 6L x 6L y or y 6L x 6L s(e) it

follows (Ee,N)E(y) 6L (Ee,N)E(x);

6. (Ee,N)E(x) = (Ee,N)E(NE(x)) for all x ∈ L;

Proof: First of all, notice that if e is an equilibrium point of N then NE(s(e)) =

s(N(r(s(e)))) = s(N(e)) = s(e) what means that s(e) is an equilibrium point of

NE. Taking this into account it follows that:

1. (Ee,N)E(s(e)) = s(Ee,N(r(s(e)))) = s(Ee,N(e)) = s(1M) = 1L;

2. If (Ee,N)E(x) = 1L for a x ∈ L then s(Ee,N(r(x))) = 1L what implies

Ee,N(r(x)) = 1M and hence by Definition 5.6 we have r(x) = e. Since r is

a lower retraction s ◦ r 6 idL then s(e) = s(r(x)) 6L x.

3. Analogous to previous item, considering that idL 6 s ◦ r if r is an upper

retraction.

4. Suppose (Ee,N)E(x) = 0L for a x ∈ L. Thus, = s(Ee,N(r(x))) = 0L which

means that Ee,N(r(x)) = 0M implying r(x) = 0M or r(x) = 1M . Hence

x = 0L or x = 1L. On the other hand, it is clear that (Ee,N)E(0L) =

(Ee,N)E(1L) = 0L.

133

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

5. If s(e) 6L x 6L y then e 6M r(x) 6M r(y). In this case, we have

that (Ee,N)E(y) = s(Ee,N(r(y))) 6L s(Ee,N(r(x))) = (Ee,N)E(x) since

Ee,N(y) 6L Ee,N(x). Analogously one can prove the same thesis supposing

y 6L x 6L s(e).

6. For each x ∈ L it follows

(Ee,N)E(NE(x)) = s(Ee,N(r(NE(x))))

= s(Ee,N(r(s(N(r(x))))))

= s(Ee,N(N(r(x))))

= s(Ee,N(r(x)))

= (Ee,N)E(x)

�

Remark 5.3 Proposition 5.9 shows that again extension method via retraction

is not efficient for extending normal Ee,N -functions since property 1 of Defini-

tion 5.6 is not satisfied by (Ee,N)E, just a weakened version of it (items 2 and

3 of Proposition 5.9) as also happened for restricted equivalence functions and

restricted dissimilarity functions.

Proposition 5.10 Let M E L with respect to (r1, r2, s), � : M × M → L be

an e-operator and e be an equilibrium point of strong negation N : M → M . If

Ee,N : M → M is a normal Ee,N -function then its extension (Ee,N)E� : L → L

given by

(Ee,N)E�(x) = Ee,N(r1(x))� Ee,N(r2(x))

for each x ∈ L, is a normal Es(e),NE�

-function.

Proof: It is clear that s(e) is an equilibrium point of NE� since e is an equilib-

rium point of N . Moreover,

1. if (Ee,N)E�(x) = 1L then Ee,N(r1(x))�Ee,N(r2(x)) = 1L and by Lemma 4.2

we have Ee,N(r1(x)) = 1M and Ee,N(r2(x)) = 1M which means that r1(x) =

e and r2(x) = e. Hence s(e) = s(r1(x)) 6L x and x 6L s(r2(x)) = s(e)

134

allowing us to conclude that x = s(e). On the other hand,

(Ee,N)E�(s(e)) = Ee,N(r1(s(e)))� Ee,N(r2(s(e)))

= Ee,N(e)� Ee,N(e)

= 1M � 1M = s(1M) = 1L

Therefore, (Ee,N)E�(x) = 1L if and only if x = s(e).

2. Note that

(Ee,N)E�(0L) = Ee,N(r1(0L))� Ee,N(r2(0L))

= Ee,N(0M)� Ee,N(0M)

= 0M � 0M = s(0M) = 0L

Similarly, one can verify that (Ee,N)E�(1L) = 0L.

Now, suppose (Ee,N)E�(x) = 0L, i.e. Ee,N(r1(x))� Ee,N(r2(x)) = 0L. Thus,

Ee,N(r1(x)) = 0M and Ee,N(r2(x)) = 0M by Lemma 4.2 and hence either

r1(x) = 0M or r1(x) = 1M or r2(x) = 0M or r2(x) = 1M which implies that

x = 0L or x = 1L.

3. Supposing s(e) 6L x 6L y it is easy to see that e 6M r1(x) 6M r1(y) and

e 6M r2(x) 6M r2(y), then Ee,N(r1(y)) 6M Ee,N(r1(x)) andEe,N(r2(y)) 6MEe,N(r2(x)) and hence Ee,N(r1(y))�Ee,N(r2(y)) 6L Ee,N(r1(x))�Ee,N(r2(x))

since � is isotonic. Therefore, (Ee,N)E�(y) = Ee,N(r1(y)) � Ee,N(r2(y)) 6LEe,N(r1(x))� Ee,N(r2(x)) = (Ee,N)E�(x).

4. Finally, take x ∈ L. We know that r1(x) 6M r2(x) and hence N(r2(x)) 6MN(r1(x)) and by Lemma 4.1 we have that r1(N(r1(x)) � N(r2(x))) =

N(r2(x)) and r2(N(r1(x))�N(r2(x))) = N(r1(x)). Therefore,

(Ee,N)E�(NE� (x)) =

135

5. ON RESTRICTED EQUIVALENCE FUNCTIONS

= Ee,N(r1(NE� (x)))� Ee,N(r2(N

E� (x))

= (Ee,N(r1(N(r1(x))�N(r2(x)))))� (Ee,N(r2(N(r1(x))�N(r2(x)))))

= Ee,N(N(r2(x)))� Ee,N(N(r1(x)))

= Ee,N(N(r1(x)))� Ee,N(N(r2(x)))

= (Ee,N)E�(x)

�

Theorem 5.12 Let N be a strong L-negation and e be an equilibrium point

of N . If REF : L2 → L is an L-REF then the function given by Ee,N(x) =

REF (x,N(x)) for all x ∈ L is a Ee,N -function.

Corollary 5.7 Let M E L with respect to (r1, r2, s), � : M × M → L be an

e-operator and REF : M2 → M a restricted equivalence function. If e is an

equilibrium point of strong negation N on M then function Es(e),NE�

: L2 → L

given by Es(e),NE�

(x) = REFE�(x,NE

� (x)) for all x ∈ L, is a normal Es(e),NE�

-

function.

Proof: Straightforward from Proposition 4.3 and Theorem 5.12.

�

136

Chapter 6

Remarks and Further Works

In this chapter we make a brief discussion of the results achieved in this thesis

and the prospects for future work. In addition, we summarize the major published

papers.

We have presented here two different methods to extend fuzzy operators. As

we have seen, the first method, named extension method via retractions, make

a generalization of the method proposed by Saminger-Platz et al. [2008] consid-

ering a more general environment provided by the notion (r, s)-sublattice, i.e.

considering that we should look to lattice M as a sublattice of another lattice L

through an algebraic immersion using retractions. This objective was successfully

achieved for t-norms, t-conorms, fuzzy negations and fuzzy implications even if,

in some cases, it has been necessary to add some particular constraints (eg, for

t-norms is necessary to require to r be an upper retraction).

However, even with these limitations, the extension method via retractions

presents some advantages with relation to the method proposed by Saminger-

Platz:

• Extension method provided by Saminger-Platz is a particular case of our

method via retraction (see Remark 3.1.1);

• Since we can identify lattice M as a (r, s)-sublattice of L in more than one

different way (as we know, a pseudo-inverse s can not be unique) then it is

possible to choose the most appropriate way to make the extension of the

operator, depending on the need presented;

137

6. REMARKS AND FURTHER WORKS

• M may not be a complete sublattice of L as required by Saminger-Platz

et al. [2008];

• Allows a greater variation of possibilities of extension of operators, since in

(3.1) it is considered only a specific retraction r(x) = sup{y ∈M | y 6L x}.

Furthermore, as is expected, we would like also this extension method could

be able to preserve the main properties that the extended operator has. However,

some results shown that not all properties of fuzzy operators are preserved by this

extension method. As we can see in Example 3.2, for no trivial cases the cancel-

lation law for triangular norms (and of course for t-conorms also) is not preserved

by extension method via retractions. This happens due to retraction r can not

be injective, since if it was, then this retraction is obviously an isomorphism (i.e.

a trivial case). Moreover, we also have seen that strong (strict) negations are not

preserved by extension method via retraction as well. For a given implications I,

property (CP) fails for its extension IE, for instance.

Aware of limitations presented by extension method via retraction, we started

investigating another way to make extension of fuzzy operators which could be

more efficient in preserving properties still considering the framework provided

by the notion of (r, s)-sublattice even if the new method had no more relation

to the method proposed by Saminger-Platz et al. [2008]. So, with inspiration

stemming from the interval operator – a kind of application very efficient in

carrying on algebraic informations from a lattice L to its interval representant L– we have proposed extension method via e-operators. The results presented by

this method were qualitatively good in terms of the preservation of properties of

extended fuzzy operators, overcoming all the limitations presented by the other

method (considering the questions studied till now) to t-norms, t-conorms and

negations. Unfortunately, due to lack of time, it was not possible to present here a

study of the application of the extension method via e-operators for implications,

but we strongly believed that the good results obtained for the other operators

studied here are also repeated for implications.

The following tables make a global comparison between results obtained for

each fuzzy connective using both methods considering the tables presented at the

end of Chapthers 3 and 4.

138

Abbreviation Property

AC Archimedean

NI Nilpotent

IDN Idempotent

ZD Zero divisor

CL Cancellation law

C Continuity

P Positive

EC Relation between extension and conjugate

RE Relation between NKE and (NK)E with K a fuzzy connective√Property preserved by this extension method√

Property not preserved by this extension method

? We have no results about this property

Table 6.1: Table of Abbreviations

t-norm AC NI IDN ZD CL C P EC

TE√ √ √ √

× ×√

(T ρ)E 6 (TE)ψ

TE�√ √ √ √ √ √ √

(T ρ)E� = (TE� )ψ

Table 6.2: Comparing results of TE and TE�

t-conorm AC NI IDN ZD CL C P EC

SE√ √ √ √

× ×√

(Sρ)E > (SE)ψ

SE�√ √ √ √ √ √ √

(Sρ)E� = (SE�)ψ

Table 6.3: Comparing results of SE and SE�

fuzzy implication (RB) (LB) (CC4) (NP) (EP) (IP) (OP) (P) (IBL) (CP)

IE√ √ √

×√ √

× ×√ √

IE� ? ? ? ? ? ? ? ? ? ?

Table 6.4: Comparing results of IE and IE�

fuzzy negation Strong Strict Eq. P. EC

NE × ×√

(Nρ)E = (NE)ψ

NE�

√ √ √(Nρ)E� = (NE

� )ψ

Table 6.5: Comparing results of NE and NE�

In order to investigate the behavior of extension methods proposed in this

thesis to other fuzzy operators outside the scope of fuzzy logic, we started apply-

139

6. REMARKS AND FURTHER WORKS

ing these methods for extending functions used to compare images, as we have

seen in Chapter 5. We did an interesting study on restricted equivalence func-

tions (REF), restricted d issimilarity functions (RDF) and normal Ee,N -functions

(NEF) valued on bounded lattices. Then, we applied our extension methods for

extension these functions and results were fateful. Again, extension method via

retractions presented limitations for extending REF, RDF and NEF while the

method via e-operators confirmed its effectiveness.

It is noteworthy that even extension method via e-operators has been achieved

better and more consistent results then extension method via retractions, this

method has an value due to it robustness. As observed by Saminger-Platz et al.

[2008] about their extension method, our method proposed in (3.3) seek to present

the most drastic way to extend fuzzy operators by means this method takes the

smallest possible extension.

Beside our published or submitted paper for main important journals concern-

ing to fuzzy sets Palmeira et al. [2012b],Palmeira and Bedregal [2012],Palmeira

et al. [submitted 2013a],Palmeira et al. [submitted 2013b],Palmeira et al. [sub-

mitted 2013c] we also published some papers in national and international con-

ferences.

It is also important to highlight that at the beginning of our doctoral studies

we have also worked on fuzzy topology and fuzzy homotopy and we had the

interest of considering extension of fuzzy homotopies between fuzzy connectives,

but due to the lack of time to make this investigation we have put this researches

in stand by. However, we would like to present here our main publications on

this for being a guide for further works.

1. E.S. Palmeira, B. Bedregal e R.H.N Santiago: Homotopia Intervalar. In:

TEMA - Tendencias em Matematica Aplicada e Computacional, v 12, p.

145–156, 2011.

2. E.S. Palmeira and B. Bedregal: On F-homotopy and F-fundamental Group.

In: 30th Annual Conference of the North American Fuzzy Information Pro-

cessing Society, El Paso, TX. Annals of the NAFIPS, 2011;

Undoubtedly, due to lack of studies on extension of fuzzy operators in the

literature and considering the originality of our extension methods, we believe

140

possess on our hands a tool with great potential for producing new scientific

articles in this area of knowledge. Thus, as a first step in the direction of the

production of future works, we are interested in applying the extension method via

e-operators to other fuzzy operators we have been extend using extension method

via retractions (implications, subclasses of negations and t-subnorms, see Akella

[2007]; Durante and Sempi [2005]; Ouyang et al. [2008]). Furthermore, we would

like to investigate the behavior of both methods for extending other operators of

our interests such as n-norms, nullnorms, copulas, additive generators of t-norms,

among others. It is also worth mentioning that we are carrying out research in

partnership with professor Radko Mesiar about the extension of n-dimensional

fuzzy implications.

141

6. REMARKS AND FURTHER WORKS

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150

Index

REFf , 109

(S,N)-implication, 33

(dX , dY )-continuous, 22

(r, s)-sublattice, 15

lower, 19

upper, 19

(r1, r2, s)-sublattice, 19

/x/, 97

Aut(L), 14

IT , 34

IREF , 114

L-NEF, 119

L-RDF, 117

L([0, 1]), 11

Ln(L), 97

Ln([0, 1]), 97

M < L, 19

M > L, 19

M m L, 19

M l L, 19

M E L, 19

N0, 109

NI , 34

NS, 30

NT , 28

N⊥, 25

N>, 26

REF , 107

REFIV , 123

¨, 11

62, 122

x, 97

TTE� , 98

TT , 98

r, 99

s, 99

�, 82

�n, 100

‖, 11

�, 122

�Lex1, 122

˜T1 · · ·Tn, 98

dL, 117

dL-nilpotent, 24

Alg-homomorphism, 12

Action of an automorphism, 14

Admissible order, 122

Alg-lattice, 10

Archimedean, 23

Automorphism, 12

151

INDEX

Bottom element, 10

Cancellation law, 46

Characterization Theorem of REF, 111

Complete lattice, 11

Conjugate, 14

Contrapositivity property, 32

De Morgan S-semitriple, 40

De Morgan T -semitriple, 40

De Morgan triple, 38

e-operator, 82

n-dimensional, 100

Epimorphism, 12

Equilibrium point, 26

Equivalence function

on [0, 1], 106

Exchange principle, 32

Extension method via e-operator

for De Morgan triple, 91

for negation, 89

for t-conorm, 89

for t-norm, 85

Extension method via retraction

for De Morgan T -semitriple, 68

for implication, 55

for negation, 50

for t-conorm, 46

for t-norm, 43

Extension operator, 82

n-dimensional, 100

Fuzzy implication, 31

Fuzzy Negation, 25

Homomorphism, 12

Idempotent, 24

Identity principle, 32

Implication, 31

positive, 32

Interval constructor, 80

interval fuzzy implication, 123

Interval order, 122

Involution property, 25

Isomorphism, 12

Iterative Boolean law, 32

L-REF, 107

Lattice, 10

bounded, 10

Left boundary condition, 32

Left contraposition law, 32

Left neutrality principle, 32

Left ordering property, 32

Lexicographical order, 122

Linear order, 122

Lower (r, s)-sublattice, 19

Monomorphism, 12

n-dimensional t-norm, 98

Natural negation

of I, 34

of S, 30

of T , 28

of REF, 109

Negation, 25

frontier, 25

strict, 25

152

INDEX

strong, 25

Normal Ee,N -function, 119

Ord-homomorphism, 12

Ord-lattice, 10

Ordering property, 32

Ordinary sublattice, 15

Positive, 24

Pre-metrics, 21

Pseudo-inverse, 15

n-dimensional, 99

Pseudo-quasi-metrics, 21

R-implication, 34

Restricted dissimilarity function, 117

Restricted equivalence function

interval-valued, 122

lattice-valued, 107

on [0, 1], 106

Retract, 15

lower, 17

upper, 17

Retraction, 15

lower, 17

n-dimensional, 99

upper, 17

Right boundary condition, 31

Right contraposition law, 32

S-implication, 33

t-conorm, 24

positive, 24

t-norm, 23

positive, 25

t-subconorm, 37

t-subnorm, 36

Top element, 10

Total order, 122

Triangular conorm, 24

Triangular norm, 23

Upper (r, s)-sublattice, 19

Weak first place antitonicity, 31

Weak left ordering property, 32

Weak second place isotonicity, 31

153