fuzzy logical algebras and their...

The Scientific World Journal

Fuzzy Logical Algebras and Their Applications

Guest Editors: Jianming Zhan, Bijan Davvaz, Wieslaw A. Dudek, Young Bae Jun, and Hee Sik Kim

EditorialFuzzy Logical Algebras and Their Applications

Jianming Zhan,1 Bijan Davvaz,2 Wieslaw A. Dudek,3 Young Bae Jun,4 and Hee Sik Kim5

1Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China2Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran3Institute of Mathematics and Computer Science, Wroclaw University of Technology, 50-370 Wroclaw, Poland4Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea5Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea

Correspondence should be addressed to Jianming Zhan; [email protected]

Received 28 January 2015; Accepted 28 January 2015

Copyright © 2015 Jianming Zhan et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is well known that an important task of the artificial intel-ligence is to make a computer simulate a human being indealing with certainty and uncertainty in information. Logicgives a technique for laying the foundations of this task.Information processing dealing with certain information isbased on the classical logic. Nonclassical logic includingmany-valued logic and fuzzy logic takes the advantage of theclassical logic to handle information with various facets ofuncertainty, such as fuzziness and randomness. Therefore,nonclassical logic has become a formal and useful tool forcomputer science to deal with fuzzy information and uncer-tain information.

For investigation of several properties, such logics havebeen represented as algebras, that is, sets with one, two,or more algebraic operations satisfying some conditionsinspired by these logics. This inspiration is illustrated by thesimilarities between the names. We have BCK-algebras anda BCK positive logic, BCI-algebras and a BCI positive logic,and BL-algebras and a basic logic.

In many cases, the connection between such algebras andtheir corresponding logics is much stronger. In this case, onecan give a translation procedure which translates all well-formed formulas and all theorems of a given logic L intoterms and theorems of the corresponding algebras. In somecases, one can give also an inverse translation. In this case,we say that the given logic and class of such algebras areisomorphic. Nevertheless, the study of algebras motivated bylogics is interesting and very useful also in the casewhen thesestructures are not isomorphic.

An important class of algebras inspired by logic is BL-algebras introduced by Hajek in order to provide an algebraicproof of the completeness theorem of Basic Logic. A classicalexample of a BL-algebra is the interval [0, 1] endowed withthe structure induced by a continuous t-norm. Other impor-tant examples of BL-algebras are MV-algebras introducedearlier as amodel of some infinitely valued Łukasiewicz logic.All these algebras are also strongly connected with residuatedlattices. Hence, the filter theory plays an important role inthe study of these algebras. From logic point of view, variousfilters correspond to various sets of provable formulae in therespective logic.

Below we present several papers on filters in various alge-bras. In one of these papers, it is proved that there is a one-to-one correspondence between the set of all vague filters and allvague congruence of a lattice implication algebra. In another,the relations among fuzzy t-filters on residuated lattices aredescribed. Another paper is devoted to the filter topology onlattice implication algebras. In further articles is characterizedthe relationship between themain types of fuzzy filters in BE-algebras and EQ-algebras. The role of ideals and soft sets inBL-algebras is described in another two papers.

In papers on pseudo-weak-R0 algebras, the most sim-plified axioms system is presented and it is proved thatpseudo-weak-R0 algebras are categorically isomorphic topseudo-IMTL algebras. Pseudo-R0 algebras are categoricallyisomorphic to pseudo-NM algebras.

Several presented papers are devoted to the possibleapplications of algebras inspired by logic and fuzzy sets.

Hindawi Publishing Corporatione Scientific World JournalVolume 2015, Article ID 682648, 2 pageshttp://dx.doi.org/10.1155/2015/682648

http://dx.doi.org/10.1155/2015/682648

2 The Scientific World Journal

For example, a robust fuzzy logic controller is proposed forstabilization and disturbance rejection in nonlinear controlsystems of a particular type. The dynamic feedback con-troller is designed as a combination of a control law thatcompensates for nonlinear terms in a control system anda dynamic fuzzy logic controller that addresses unknownmodel uncertainties and an unmeasured disturbance. Amathematical derivation is used to prove that the controlleris able to achieve asymptotic stability by processing statemeasurements. Properties of some hesitant triangular fuzzyaggregation operators based on Bonferroni means also arediscussed. In one paper, the method of a construction of aranking of the physical matches is proposed. This method isbased on the fuzzy clustering analysis.

Decisions are often made by many decision expertshaving preference of risks. In view of the grey situationgroup decision-making problems, a grey situation groupdecision-making method on the basis of prospect theory isshown. The method takes the positive and negative idealsituation distance as reference points, defines positive andnegative prospect value function, and introduces decisionexperts’ risk preference into grey situation decision-makingto make the final decision more in line with decision experts’psychological behavior. One possibility of determination ofthe weight of each decision expert sets up comprehensiveprospect value matrix for decision experts’ evaluation, andfinally determination of the optimal situation is presented.

Jianming ZhanBijan Davvaz

Wieslaw A. DudekYoung Bae JunHee Sik Kim

Research ArticleA New Fuzzy System Based on Rectangular Pyramid

Mingzuo Jiang,1 Xuehai Yuan,1,2 Hongxing Li,1 and Jiaxia Wang3

1Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China2Foundation Department, Dalian University of Technology, Panjin 124221, China3Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Xuehai Yuan; [email protected]

Received 10 July 2014; Accepted 7 September 2014

Academic Editor: Jianming Zhan

Copyright © 2015 Mingzuo Jiang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new fuzzy system is proposed in this paper. The novelty of the proposed system is mainly in the compound of the antecedents,which is based on the proposed rectangular pyramidmembership function instead of t-norm. It is proved that the system is capableof approximating any continuous function of two variables to arbitrary degree on a compact domain.Moreover, this paper providesone sufficient condition of approximating function so that the new fuzzy system can approximate any continuous function of twovariables with bounded partial derivatives. Finally, simulation examples are given to show how the proposed fuzzy system can beeffectively used for function approximation.

1. Introduction

Fuzzy system has been the subject of numerous researchesin more than three decades. It has been successfully appliedto a great variety of different processes such as control engi-neering, signal processing, information processing, machineintelligence, decision making, management, finance, medi-cine, and robotics [1–9].

Motivated by successful applications of fuzzy system,there have been a number of works aiming at improvingthe structure and performance of fuzzy system. Yuan et al.[10] put forward a parameter singleton fuzzifier method.Celikyilmaz and Turksen [11] presented a new fuzzy systemmodeling approach based on improved fuzzy functions tomodel systems with continuous output variable. Adaptivefuzzy systems are also developed well. In [12], a dynamicrule base which allows the fuzzy sets to dynamically changeor move with the inputs was used for the construction offuzzy system. Márquez et al. [13] proposed adaptive t-normsfor the antecedents connection and adaptive defuzzificationmethods. Moreover, fuzzy system can be integrated withsome other techniques, for example, neurocontrol [14–17],genetic algorithms and particle swarm optimization [18–23],and fuzzy sliding-mode control [24–26]. This paper focuses

on the structure design of conventional dual-input single-output Mamdani fuzzy system to improve its approximationproperty and simplify its structure. A new kind of fuzzy sys-tem based on the proposed rectangular pyramidmembershipfunction is established.Themodel of the rectangular pyramidfuzzy system (RPFS) is introduced mainly by replacing thecompound of the two rule antecedents using t-norm withthe rectangular pyramid membership function of the inputvector. With the help of rectangular pyramid membershipfunction, the fuzzy system structure becomes simple andeasily realized.

In most applications of fuzzy systems, the main designobjective can be considered as problems of functions approx-imation. So the study on approximation theory of fuzzy sys-tems is very important and necessary. Wang [27] used Stone-Weierstrass Theorem to prove the approximation capabilityof a common kind of fuzzy systems. Based on the aboveresearch, Wang and Mendel [28] proposed fuzzy basicfunctions to explain the approximation property of fuzzysystems. In [29], the approximation properties of MIMOfuzzy systems are discussed based on its fuzzy basic functions.Castro [30] proved the approximation properties of the fuzzysystems with a wide class of fuzzy logics and membershipfunctions. Mao et al. [31] addressed whether a fuzzy system


http://dx.doi.org/10.1155/2015/682989


with weaker constraints to its membership functions can be auniversal approximator. Li [32] found out that the commonlyused fuzzy system algorithms can be regarded as someinterpolation functions. Generally speaking, fuzzy systemscan approximate any continuous function on any compactdomain, which explains the ability of fuzzy controller inachieving satisfactory performance in applications. So beforethe application of one kind of fuzzy systems, it is helpfulto know clearly whether they are universal approximators.Note that the features of RPFS are mainly determined by theproperties of its fuzzy basic functions. We will first give ananalysis of the properties of fuzzy basic functions of RPFS andthen discuss the approximation property of RPFS.

Sufficient conditions of fuzzy systems lead to the fol-lowing practical result: the derived formulas can calculatethe numbers of input fuzzy sets and fuzzy rules needed tosatisfy any given approximation accuracy. Ying [33] gavesufficient conditions for general fuzzy system. Chen [34]established sufficient conditions for two classes of fuzzylogic controllers in [33]. In [35], the sufficient conditions forBoolean fuzzy systems were proposed by using WeierstrassTheorem. Zeng et al. [36] and Liu et al. [37]made a systematicand comparative study on sufficient conditions for differentfuzzy systems. In a constructive way, we have found one suffi-cient condition on the premise that RPFS can uniformlyapproximate any real continuous function on a compactdomain to any degree of accuracy.

The structure of this paper is as follows. After theintroduction, themodel of the conventional fuzzy system andits approximation theory are given in Section 2.The construc-tion of RPFS is introduced in Section 3. Some definitionsand properties of RPFS are given in Section 4 in order todescribe the approximation capability of RPFS. One sufficientcondition of RPFS is given in Section 5. After the above dis-cussion,we provide some simulation results of approximatingfunctions to evaluate the approximation performance ofRPFS in Section 6. Conclusions are made in the last section.

2. Preliminaries

In this section, we will introduce the model of conventionaldual-input single-output fuzzy system. Furthermore, we willdescribe the ability of the system in approximating any con-tinuous function on an arbitrary compact domain.

Conventional fuzzy system consists of four principalcomponents: fuzzifier, fuzzy rule base, fuzzy inference engine,and defuzzifier. The fuzzy rule base contains informationof how to infer new control actions. The fuzzy inferenceengine is a reasoning mechanism which performs inferenceprocedure on the fuzzy rules and derives reasonable controlactions. It is the central part of a fuzzy system. The fuzzifi-cation interface (or fuzzifier) defines a mapping from a real-valued space to a fuzzy space, and the defuzzification inter-face (or defuzzifier) defines a mapping from a fuzzy spaceto a real-valued space. The fuzzifier converts a crisp value toa fuzzy number while the defuzzifier converts the inferredfuzzy conclusion to a crisp value.

Consider a dual-input single-output fuzzy system: 𝑈 ×𝑉 → 𝑊, where 𝑈 × 𝑉 ⊂ 𝑅2 is the input space and𝑊 ⊂ 𝑅 isthe output space, respectively. The fuzzy rule base consists of𝑀𝑁 rules in the following form:

𝑅𝑖𝑗: IF 𝑥 is 𝐴1

𝑖, 𝑦 is 𝐴2

𝑗, THEN 𝑧 is 𝐵

𝑖𝑗, (1)

where 𝑖 = 1, 2, . . . ,𝑀; 𝑗 = 1, 2, . . . , 𝑁; 𝑥 and 𝑦 are the inputvariables of the fuzzy system; 𝑧 is the output variable of thefuzzy system. 𝐴1

𝑖⊂ 𝑈, 𝐴2

𝑗⊂ 𝑉, and 𝐵

𝑖𝑗⊂ 𝑊 are linguistic

terms characterized by fuzzy membership functions 𝐴1𝑖(𝑥),

𝐴2

𝑗(𝑦), and 𝐵

𝑖𝑗(𝑧), respectively. 𝑧

𝑖𝑗is the point in𝑊 at which

𝐵𝑖𝑗(𝑧) achieves its maximum value. Assume that the fuzzifier

is the singleton fuzzifier method. Under the commonly usedfuzzy engine and defuzzifier method, the final output ofthe conventional dual-input single-output fuzzy system isderived as follows:

𝑧 =

∑𝑀

𝑖=1∑𝑁

𝑗=1𝐴1

𝑖(𝑥) 𝐴2

𝑗(𝑦) 𝑧𝑖𝑗

∑𝑀

𝑖=1∑𝑁

𝑗=1𝐴1

𝑖(𝑥) 𝐴2

𝑗(𝑦)

. (2)

The below theorem [27] gives the basic approximationproperty of conventional dual-input single-output fuzzy sys-tem.

Theorem 1. For any given real continuous function 𝑔 (𝑥, 𝑦) onthe compact set 𝑈 × 𝑉 ⊂ 𝑅2 and arbitrary 𝜀 > 0, there exists afuzzy system 𝑓 (𝑥, 𝑦) such that

sup(𝑥,𝑦)∈𝑈×𝑉

𝑔 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) < 𝜀. (3)

3. Construction of Rectangular PyramidFuzzy System

In this section, we applied the definition of rectangularpyramid membership function instead of t-norm to theconstruction of fuzzy system to achieve high accuracy ofthe output of given data. Moreover, the structure of fuzzysystem is simplified. A detailed description of RPFS is given.The discussion is limited to dual-input single-output systems.RPFS consists of four principle parts which are similar tothose of conventional fuzzy systems: fuzzifier, fuzzy rule base,fuzzy inference engine, and defuzzifier. For this class of fuzzysystems, they can be constructed by Sections 3.1–3.3. Thestructure of RPFS is presented in Figure 1 and the sketchof rectangular pyramid membership function is shown inFigure 2.

3.1. Fuzzifier. The fuzzification interface can translate inputvalues into linguistic terms which are characterized by therectangular pyramidmembership functions.Denote by (𝑥, 𝑦)the input vector and by 𝑧 the output variable. Let [𝑎, 𝑏]×[𝑐, 𝑑]be the universe of the input vector and [𝑚, 𝑛] the universe ofthe output variable.The universe [𝑎, 𝑏]×[𝑐, 𝑑] is equidistantlydivided by peak points (𝑥

𝑖, 𝑦𝑗) (𝑖 = 1, 2, . . . ,𝑀; 𝑗 =

1, 2, . . . , 𝑁) of the rectangular pyramid membershipfunctions. Consider one subset [𝑥

𝑖, 𝑥𝑖+1] × [𝑦

𝑗, 𝑦𝑗+1

] in

The Scientific World Journal 3

Knowledgebase

Inferenceengine

Fuzzificationinterface

Controllerinputs

Fuzzyinputs

Fuzzyoutputs

Controlleroutputs

Defuzzificationinterface

Figure 1: The structure of RPFS.

0

0.2

0.4

0.6

0.8

1

10.5

0

−0.5

−1

10.5

0−0.5−1

Figure 2: Rectangular pyramid membership function.

[𝑎, 𝑏] × [𝑐, 𝑑]; for simplicity of discussion, we construct,respectively, the rectangular pyramid membership functions𝑃𝑖𝑗(𝑥, 𝑦), 𝑃

𝑖.𝑗+1(𝑥, 𝑦), 𝑃

𝑖+1,𝑗+1(𝑥, 𝑦), and 𝑃

𝑖+1,𝑗(𝑥, 𝑦) (on the

small area [𝑥𝑖, 𝑥𝑖+1] × [𝑦

𝑗, 𝑦𝑗+1

]) of the peak points (𝑥𝑖, 𝑦𝑗),

(𝑥𝑖, 𝑦𝑗+1

), (𝑥𝑖+1, 𝑦𝑗+1

), and (𝑥𝑖+1, 𝑦𝑗) as follows.

At point (𝑥𝑖, 𝑦𝑗), when 𝑥

𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗≤ 𝑦 ≤ 𝑦

𝑗+

((𝑦𝑗+1

− 𝑦𝑗)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖), we have

𝑃𝑖𝑗(𝑥, 𝑦) =

𝑥𝑖+1

− 𝑥

𝑥𝑖+1

− 𝑥𝑖

; (4)

when 𝑥𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗+ ((𝑦𝑗+1

− 𝑦𝑗)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖) ≤

𝑦 ≤ 𝑦𝑗+1

, we have

𝑃𝑖𝑗(𝑥, 𝑦) =

𝑦𝑗+1

− 𝑦

𝑦𝑗+1

− 𝑦𝑗

. (5)

At point (𝑥𝑖, 𝑦𝑗+1

), when 𝑥𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗≤ 𝑦 ≤ 𝑦

𝑗+1+ ((𝑦𝑗−

𝑦𝑗+1

)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖), we have

𝑃𝑖,𝑗+1

(𝑥, 𝑦) =

𝑦 − 𝑦𝑗

𝑦𝑗+1

− 𝑦𝑗

; (6)


𝑖+1, 𝑦𝑗+1

+ ((𝑦𝑗−𝑦𝑗+1

)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖) ≤

𝑦 ≤ 𝑦𝑗+1

, we have

𝑃𝑖,𝑗+1

(𝑥, 𝑦) =𝑥𝑖+1

− 𝑥

𝑥𝑖+1

− 𝑥𝑖

. (7)

At point (𝑥𝑖+1, 𝑦𝑗+1

), when 𝑥𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗≤ 𝑦 ≤ 𝑦

𝑗+

((𝑦𝑗+1

− 𝑦𝑗)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖), we have

𝑃𝑖+1,𝑗+1

(𝑥, 𝑦) =

𝑦 − 𝑦𝑗

𝑦𝑗+1

− 𝑦𝑗

; (8)


𝑖+1, 𝑦𝑗+ ((𝑦𝑗+1

− 𝑦𝑗)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖) ≤

𝑦 ≤ 𝑦𝑗+1

, we have

𝑃𝑖+1,𝑗+1

(𝑥, 𝑦) =𝑥 − 𝑥𝑖

𝑥𝑖+1

− 𝑥𝑖

. (9)

At point (𝑥𝑖+1, 𝑦𝑗), when 𝑥

𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗≤ 𝑦 ≤ 𝑦

𝑗+1+ ((𝑦𝑗−

𝑦𝑗+1

)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖), we have

𝑃𝑖+1,𝑗

(𝑥, 𝑦) =𝑥 − 𝑥𝑖

𝑥𝑖+1

− 𝑥𝑖

; (10)


𝑖+1, 𝑦𝑗+1

+ ((𝑦𝑗−𝑦𝑗+1

)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖) ≤

𝑦 ≤ 𝑦𝑗+1

, we have

𝑃𝑖+1,𝑗

(𝑥, 𝑦) =

𝑦𝑗+1

− 𝑦

𝑦𝑗+1

− 𝑦𝑗

. (11)

The singleton fuzzifier method is adopted in this step. Itmaps a real-valued point (𝑥, 𝑦) ∈ [𝑎, 𝑏] × [𝑐, 𝑑] into a fuzzysingleton set 𝑃∗ which has membership value 1 at (𝑥, 𝑦) and0 at the other points in [𝑎, 𝑏] × [𝑐, 𝑑]; that is,

𝑃∗(𝑥, 𝑦) = {

1, if (𝑥, 𝑦) = (𝑥, 𝑦)0, otherwise.

(12)

3.2. Fuzzy Inference. The fuzzy inference engine is a decision-making mechanism that employs fuzzy rules from the fuzzyrule base to determine a mapping from the fuzzy sets in theinput space to the fuzzy sets in the output space.

The conventional fuzzy rule base consists of a set of lin-guistic rules in the form of (1). In this paper, we consider thefuzzy rules which are different from those of the conventionalfuzzy system in the following form:

𝑅𝑖𝑗: IF (𝑥, 𝑦) is 𝑃

𝑖𝑗,

THEN 𝑧 is 𝐶𝑖𝑗(𝑖 = 1, 2, . . . ,𝑀; 𝑗 = 1, 2, . . . , 𝑁) ,

(13)

where 𝑃𝑖𝑗are the fuzzy set characterized by rectangular pyra-

mid membership function 𝑃𝑖𝑗(𝑥, 𝑦). The difference between

RPFS and the conventional fuzzy system mainly lies in the


compound of the antecedents. In the conventional fuzzysystem, fuzzy intersections (t-norms) for connective “and”are used. Then, the compound fuzzy proposition

IF 𝑥 is 𝐴1𝑖, 𝑦 is 𝐴2

𝑗(14)

is interpreted as a fuzzy set 𝐴1𝑖∩ 𝐴2

𝑗in 𝑈 × 𝑉 with a mem-

bership function

𝜇𝐴1

𝑖∩𝐴2

𝑗

(𝑥, 𝑦) = 𝑡 [𝜇𝐴1

𝑖

(𝑥) , 𝜇𝐴2

𝑗

(𝑦)] , (15)

where 𝑡 : [0, 1] × [0, 1] → [0, 1] is any t-norm.As the fuzzy rule base consists of a set of rules, the

relationship among these rules is an interesting question.Important properties of a set of conventional rules arecompleteness, consistency, and continuity. Note that the formof fuzzy rules in this paper has been changed; it is necessaryto know whether the changed rules have the similar goodcharacter as the conventional rules.

Definition 2. A set of fuzzy IF-THEN rules is complete if, forany (𝑥, 𝑦) ∈ 𝑈 × 𝑉, there exists at least one rule in the fuzzyrule base, say rule 𝑅st (in the form of (13)), such that

𝑃st (𝑥, 𝑦) ̸= 0 ∀𝑖 = 1, 2, . . . ,𝑀; 𝑗 = 1, 2, . . . , 𝑁. (16)

Intuitively, the completeness of a set of rulesmeans that at anypoint in the input space there exists at least one rule makingthe membership value of the IF part of the rule at this pointnonzero.

Definition 3. A set of fuzzy IF-THEN rules is consistent ifthere are no rules with the same IF parts but different THENparts.

Definition 4. A set of fuzzy IF-THEN rules is continuous ifthere do not exist such neighboring rules whose THEN partfuzzy sets have empty intersection.

It is obvious that the rule base of RPFS has properties ofcompleteness, consistency, and continuity.

For fuzzy rule base containing more than one rule, thekey question is how to infer with a set of rules. There aretwo ways: composition-based inference and individual-rulebased inference. If all the𝑀𝑁 fuzzy rules are firstly composedinto a new fuzzy rule which is then used to generate a fuzzyconsequence in accordance with the given fuzzy antecedents,this is the so-called composition-based inference. Alterna-tively, each of the 𝑀𝑁 fuzzy rules is individually used togenerate a fuzzy consequence in accordance with the givenfuzzy antecedent. The resulting 𝑀𝑁 fuzzy consequences arethen composed into a new fuzzy consequence. This is theso-called individual-rule based inference. In this paper, wefollow the widely used composition-based inference. So eachfuzzy rule can be written as

𝑅𝑖𝑗(𝑥, 𝑦, 𝑧) = 𝜃 (𝑃

𝑖𝑗(𝑥, 𝑦) , 𝐶

𝑖𝑗(𝑧)) = 𝑃

𝑖𝑗(𝑥, 𝑦) 𝐶

𝑖𝑗(𝑧) , (17)

where 𝜃 is the Mamdani product implication operator fromthe fuzzy antecedent 𝑃

𝑖𝑗to the fuzzy consequence 𝐶

𝑖𝑗. Based

on composition-based inference, we have

𝑅 =

𝑀

⋃

𝑖=1

𝑁

⋃

𝑗=1

𝑅𝑖𝑗. (18)

Therefore, the membership function of 𝑅 can be expressed as

𝑅 (𝑥, 𝑦, 𝑧) =

𝑀

⋁

𝑖=1

𝑁

⋁

𝑗=1

[𝑃𝑖𝑗(𝑥, 𝑦) 𝐶

𝑖𝑗(𝑧)] . (19)

𝑅 composes the 𝑀𝑁 fuzzy rules into a single fuzzy rule viathe union operator. To derive the expression of the output,we need to construct a set transformation 𝑇 : 𝑃∗ → 𝐶∗,𝐶∗= [(𝑃∗× 𝑍) ∩ 𝑅]

𝑧,

𝐶∗(𝑧) = ⋁

(𝑥 ,𝑦)∈𝑋×𝑌

[𝑃∗(𝑥, 𝑦) ∧ 𝑅 (𝑥

, 𝑦, 𝑧)]

= 𝑅 (𝑥, 𝑦, 𝑧) .

(20)

3.3. Defuzzifier. In this step,we need a defuzzification processto get a crisp decision. Among the commonly used defuzzi-fication strategies, fuzzy centroid-defuzzification methodyields superior results [38]. Through this method, we get anequation of output variable

𝑧 =

∫𝑛

𝑚𝑧𝐶∗(𝑧) 𝑑𝑧

∫𝑛

𝑚𝐶∗ (𝑧) 𝑑𝑧

=

∫𝑛

𝑚𝑧 [⋁𝑀

𝑖=1⋁𝑁

𝑗=1(𝑃𝑖𝑗(𝑥, 𝑦) 𝐶

𝑖𝑗(𝑧))] 𝑑𝑧

∫𝑛

𝑚[⋁𝑀

𝑖=1⋁𝑁


𝑖𝑗(𝑧))] 𝑑𝑧

≈

∑𝑀𝑁

𝑡=1𝑧𝑡[⋁𝑀

𝑖=1⋁𝑁


𝑖𝑗(𝑧))] Δ𝑧

𝑡

∑𝑀𝑁

𝑡=1[⋁𝑀

𝑖=1⋁𝑁


𝑖𝑗(𝑧))] Δ𝑧

𝑡

= (𝑃𝑖,𝑗(𝑥, 𝑦) 𝑧

𝑖,𝑗+ 𝑃𝑖+1,𝑗

(𝑥, 𝑦) 𝑧𝑖+1,𝑗

+𝑃𝑖+1,𝑗+1

(𝑥, 𝑦) 𝑧𝑖+1,𝑗+1

+ 𝑃𝑖,𝑗+1

(𝑥, 𝑦) 𝑧𝑖,𝑗+1

)

× (𝑃𝑖,𝑗(𝑥, 𝑦) + 𝑃

𝑖+1,𝑗(𝑥, 𝑦)

+𝑃𝑖+1,𝑗+1

(𝑥, 𝑦) + 𝑃𝑖,𝑗+1

(𝑥, 𝑦))−1

,

(21)

where 𝑧𝑖1,𝑗1

= 𝑓(𝑥𝑖1

, 𝑦𝑗1

) (𝑖1= 𝑖, 𝑖 + 1; 𝑗

1= 𝑗, 𝑗 + 1) and

𝑓 (𝑥, 𝑦) is a real-valued function defined on a domain𝐷⊂ 𝑅2. From the expressions of the rectangular pyramid

membership functions 𝑃𝑖𝑗(𝑥, 𝑦), 𝑃

𝑖,𝑗+1(𝑥, 𝑦), 𝑃

𝑖+1,𝑗+1(𝑥, 𝑦),


and 𝑃𝑖+1,𝑗

(𝑥, 𝑦) and the above formula, the general structureof RPFS can be represented as follows:

(i) when 𝑥𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗≤ 𝑦 ≤ 𝑦

𝑗+ ((𝑦𝑗+1

−𝑦𝑗)/(𝑥𝑖+1

−

𝑥𝑖)) (𝑥 − 𝑥

𝑖), 𝑦𝑗≤ 𝑦 ≤ 𝑦

𝑗+1+ ((𝑦𝑗− 𝑦𝑗+1

)/(𝑥𝑖+1

−

𝑥𝑖)) (𝑥 − 𝑥

𝑖),

𝑆 (𝑥, 𝑦) =

(𝑥𝑖+1

− 𝑥) (𝑦𝑗+1

− 𝑦𝑗)

(2𝑦 + 𝑦𝑗+1

− 3𝑦𝑗) (𝑥𝑖+1

− 𝑥𝑖)

𝑧𝑖𝑗

+

𝑦 − 𝑦𝑗

2𝑦 + 𝑦𝑗+1

− 3𝑦𝑗

𝑧𝑖,𝑗+1

+

𝑦 − 𝑦𝑗

2𝑦 + 𝑦𝑗+1

− 3𝑦𝑗

𝑧𝑖+1,𝑗+1

+

(𝑥 − 𝑥𝑖) (𝑦𝑗+1

− 𝑦𝑗)

(2𝑦 + 𝑦𝑗+1

− 3𝑦𝑗) (𝑥𝑖+1

− 𝑥𝑖)

𝑧𝑖+1,𝑗

;

(22)

(ii) when 𝑥𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗+ ((𝑦𝑗+1

− 𝑦𝑗)/(𝑥𝑖+1

−

𝑥𝑖)) (𝑥 − 𝑥

𝑖) ≤ 𝑦 ≤ 𝑦

𝑗+1, 𝑦𝑗≤ 𝑦 ≤ 𝑦

𝑗+1+ ((𝑦𝑗−

𝑦𝑗+1

)/(𝑥𝑖+1

− 𝑥𝑖)) (𝑥 − 𝑥

𝑖),

𝑆 (𝑥, 𝑦) =

(𝑥𝑖+1

− 𝑥𝑖) (𝑦𝑗+1

− 𝑦)

(2𝑥 + 𝑥𝑖+1

− 3𝑥𝑖) (𝑦𝑗+1

− 𝑦𝑗)

𝑧𝑖𝑗

+

(𝑥𝑖+1

− 𝑥𝑖) (𝑦 − 𝑦

𝑗)

(2𝑥 + 𝑥𝑖+1

− 3𝑥𝑖) (𝑦𝑗+1

− 𝑦𝑗)

𝑧𝑖,𝑗+1

+𝑥 − 𝑥𝑖

2𝑥 + 𝑥𝑖+1

− 3𝑥𝑖

𝑧𝑖+1,𝑗+1

+𝑥 − 𝑥𝑖

2𝑥 + 𝑥𝑖+1

− 3𝑥𝑖

𝑧𝑖+1,𝑗

;

(23)

(iii) when 𝑥𝑖≤ 𝑥 ≤ 𝑥

𝑖+1, 𝑦𝑗+ ((𝑦𝑗+1

− 𝑦𝑗)/(𝑥𝑖+1

−

𝑥𝑖)) (𝑥 − 𝑥

𝑖) ≤ 𝑦 ≤ 𝑦

𝑗+1, 𝑦𝑗+1

+ ((𝑦𝑗− 𝑦𝑗+1

)/(𝑥𝑖+1

−

𝑥𝑖)) (𝑥 − 𝑥

𝑖) ≤ 𝑦 ≤ 𝑦

𝑗+1,

𝑆 (𝑥, 𝑦) =

𝑦 − 𝑦𝑗+1

−2𝑦 + 3𝑦𝑗+1

− 𝑦𝑗

𝑧𝑖𝑗

+

(𝑦𝑗+1

− 𝑦𝑗) (𝑥𝑖+1

− 𝑥)

(−2𝑦 + 3𝑦𝑗+1

− 𝑦𝑗) (𝑥𝑖+1

− 𝑥𝑖)

𝑧𝑖,𝑗+1

+

(𝑦𝑗+1

− 𝑦𝑗) (𝑥 − 𝑥

𝑖)

(−2𝑦 + 3𝑦𝑗+1

− 𝑦𝑗) (𝑥𝑖+1

− 𝑥𝑖)

𝑧𝑖+1,𝑗+1

+

𝑦𝑗+1

− 𝑦

−2𝑦 + 3𝑦𝑗+1

− 𝑦𝑗

𝑧𝑖+1,𝑗

;

(24)

(xi+1,

(xi+1, yj)(xi, yj)

(xi, yj+1) yj+1)

I

II

III

IV

Figure 3:The division of small rectangular area [𝑥𝑖, 𝑥𝑖+1]×[𝑦𝑗, 𝑦𝑗+1

]

of 𝑆 (𝑥, 𝑦).

otherwise

𝑆 (𝑥, 𝑦) =𝑥𝑖+1

− 𝑥

−2𝑥 + 3𝑥𝑖+1

− 𝑥𝑖

𝑧𝑖𝑗

+𝑥𝑖+1

− 𝑥

−2𝑥 + 3𝑥𝑖+1

− 𝑥𝑖

𝑧𝑖,𝑗+1

+

(𝑥𝑖+1

− 𝑥𝑖) (𝑦 − 𝑦

𝑗)

(−2𝑥 + 𝑥𝑖+1

− 𝑥𝑖) (𝑦𝑗+1

− 𝑦𝑗)

𝑧𝑖+1,𝑗+1

+

(𝑥𝑖+1

− 𝑥𝑖) (𝑦𝑗+1

− 𝑦)

(−2𝑥 + 3𝑥𝑖+1

− 𝑥𝑖) (𝑦𝑗+1

− 𝑦𝑗)

𝑧𝑖+1,𝑗

.

(25)

The four conditions correspond to the division domains I,II, III, and IV in Figure 3, respectively. In the rest of thispaper, 𝑆 (𝑥, 𝑦) will be used to represent the output of RPFSand 𝑃

𝑖𝑗(𝑥, 𝑦) the rectangular pyramid membership function

at peak point (𝑥𝑖, 𝑦𝑗).

4. Approximation Property of RectangularPyramid Fuzzy System

In this section, in order to introduce the basic approximationproperty of RPFS, the definition and properties of fuzzy basicfunctions are firstly given. From the expressions of the outputof RPFS, it is easy to find that 𝑆 (𝑥, 𝑦) can be represented by alinear combination of one kind of functions; for example,

𝜑𝑖,𝑗(𝑥, 𝑦)

=

𝑃𝑖,𝑗(𝑥, 𝑦)

𝑃𝑖,𝑗(𝑥, 𝑦) + 𝑃

𝑖+1,𝑗(𝑥, 𝑦) + 𝑃

𝑖+1,𝑗+1(𝑥, 𝑦) + 𝑃

𝑖,𝑗+1(𝑥, 𝑦)

.

(26)

Thus, this kind of functions can be defined as fuzzy basicfunctions of RPFS. The exact definition is as follows.


Definition 5. Define fuzzy basic functions of RPFS as

𝜑𝑖2,𝑗2

(𝑥, 𝑦)

=

𝑃𝑖2,𝑗2

(𝑥, 𝑦)

𝑃𝑖,𝑗(𝑥, 𝑦) + 𝑃

𝑖+1,𝑗(𝑥, 𝑦) + 𝑃

𝑖+1,𝑗+1(𝑥, 𝑦) + 𝑃

𝑖,𝑗+1(𝑥, 𝑦)

,

(27)

where 𝑖2= 𝑖, 𝑖 + 1; 𝑗

2= 𝑗, 𝑗 + 1. According to Definition 5, it

is obvious that

𝜑𝑖,𝑗(𝑥, 𝑦) + 𝜑

𝑖+1,𝑗(𝑥, 𝑦) + 𝜑

𝑖+1,𝑗+1(𝑥, 𝑦) + 𝜑

𝑖,𝑗+1(𝑥, 𝑦) = 1.

(28)

Now the output of RPFS can be represented as

𝑆 (𝑥, 𝑦) = 𝜑𝑖,𝑗(𝑥, 𝑦) 𝑧

𝑖,𝑗+ 𝜑𝑖+1,𝑗

(𝑥, 𝑦) 𝑧𝑖+1,𝑗

+ 𝜑𝑖+1,𝑗+1

(𝑥, 𝑦) 𝑧𝑖+1,𝑗+1

+ 𝜑𝑖,𝑗+1

(𝑥, 𝑦) 𝑧𝑖,𝑗+1

,

(29)

where 𝜑𝑖2,𝑗2

(𝑥𝑖3

, 𝑦𝑗3

) = 1 (𝑖3= 𝑖, 𝑖 + 1; 𝑗

3= 𝑗, 𝑗 + 1), when

𝑖2= 𝑖3and 𝑗2= 𝑗3, and, inversely, 𝜑

𝑖2,𝑗2

(𝑥𝑖3

, 𝑦𝑗3

) = 0, when𝑖2

̸= 𝑖3or 𝑗2

̸= 𝑗3.

From the above analysis, we can conclude as follows.The basic idea of RPFS is a kind of piecewise interpolationfunction with the conditions

𝑆 (𝑥𝑖, 𝑦𝑗) = 𝑓 (𝑥

𝑖, 𝑦𝑗) (𝑖 = 1, 2, . . . ,𝑀; 𝑗 = 1, 2, . . . , 𝑁) .

(30)

Fuzzy basic functions play the same role in RPFS as the inter-polation basis functions do in computational mathematics.The output of RPFS is a weighted sumwith the correspondingfuzzy basic function values as the weights on [𝑥

𝑖, 𝑥𝑖+1] ×

[𝑦𝑗, 𝑦𝑗+1

].Denote by 𝐶(𝐷) the collection of all the continuous

functions mapping 𝐷 into the real numbers. The distancebetween 𝑓 and 𝑔 in 𝐶(𝐷) can be measured as 𝑓 − 𝑔

=

sup(𝑥,𝑦)∈𝐷

𝑓 (𝑥, 𝑦) − 𝑔 (𝑥, 𝑦). The problem of approxima-

tion can be described as follows: given 𝑓 (𝑥, 𝑦) ∈ 𝐶(𝐷) andany 𝜀 > 0, is it possible for RPFS to approximate the function𝑓 (𝑥, 𝑦) on an arbitrary compact domain 𝐷 to 𝜀 level? Thefollowing theorem addresses the above posed problem.

Theorem 6. For any given real continuous function 𝑓 (𝑥, 𝑦)on the compact set 𝐷 ⊂ 𝑅2 and arbitrary 𝜀 > 0, there exists aRPFS 𝑆 (𝑥, 𝑦) such that

𝑆 − 𝑓 = sup

(𝑥,𝑦)∈𝐷

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) < 𝜀. (31)

Proof. Without losing generality, the proof is discussed onone partition [𝑥

𝑖, 𝑥𝑖+1] × [𝑦

𝑗, 𝑦𝑗+1

] of the domain𝐷. We can

now prove the theorem with the help of the interpolationproperty of RPFS. From (30), we have

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)

=𝑆 (𝑥, 𝑦) − 𝑆 (𝑥

𝑖, 𝑦𝑗) + 𝑓 (𝑥

𝑖, 𝑦𝑗) − 𝑓 (𝑥, 𝑦)

≤𝑆 (𝑥, 𝑦) − 𝑆 (𝑥

𝑖, 𝑦𝑗)+𝑓 (𝑥𝑖, 𝑦𝑗) − 𝑓 (𝑥, 𝑦)

.

(32)

Using LagrangeMean ValueTheorem, there exist (𝜉1, 𝜂1) and

(𝜉2, 𝜂2) which both belong to (𝑥

𝑖, 𝑥𝑖+1) × (𝑦

𝑗, 𝑦𝑗+1

) such that

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)

=

𝜕𝑆

𝜕𝑥

(𝜉1,𝜂1)

(𝑥 − 𝑥𝑖) +

𝜕𝑆

𝜕𝑦

(𝜉1,𝜂1)

(𝑦 − 𝑦𝑗)

+

𝜕𝑓

𝜕𝑥

(𝜉2,𝜂2)

(𝑥 − 𝑥𝑖) +

𝜕𝑓

𝜕𝑦

(𝜉2,𝜂2)

(𝑦 − 𝑦𝑗)

≤ (

𝜕𝑆

𝜕𝑥

(𝜉1,𝜂1)

+

𝜕𝑆

𝜕𝑦

(𝜉1,𝜂1)

+

𝜕𝑓

𝜕𝑥

(𝜉2,𝜂2)

+

𝜕𝑓

𝜕𝑦

(𝜉2,𝜂2)

) ℎ,

(33)

where ℎ = max{|𝑥 − 𝑥𝑖|, |𝑦 − 𝑦

𝑗|}. As (| (𝜕𝑆/𝜕𝑥)|

(𝜉1,𝜂1)| +

| (𝜕𝑆/𝜕𝑦)(𝜉1,𝜂1)| + | (𝜕𝑓/𝜕𝑥)

(𝜉2,𝜂2)| + | (𝜕𝑓/𝜕𝑦)(𝜉

2,𝜂2)|) is a

constant, when ℎ is sufficiently small, it is evident that for any𝜀 > 0 the following inequality can be obtained:

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) < 𝜀 ∀ (𝑥, 𝑦) ∈ 𝐷

. (34)

Then𝑆 − 𝑓

= sup(𝑥,𝑦)∈𝐷

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) < 𝜀. (35)

The proof is completed.

In conclusion, Theorem 6 shows that RPFS is capable ofapproximating any real continuous function of two variableson a compact set to arbitrary accuracy.

5. Sufficient Condition of Approximation ofRectangular Pyramid Fuzzy System

In this section, we will establish one sufficient conditionon the premise that RPFS can approximate any continu-ous functions of two variables on a compact domain (inSection 4). It is impossible to give a formula of the neededrule number of RPFS to satisfy the required approximationaccuracy for all continuous functions. However, for a specialclass of continuous functions, this is possible. In the belowtheorem, continuous functions with bounded partial deriva-tives are approximated by RPFS. Now, we present the mainresult of this section.

Theorem 7. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → [𝑚, 𝑛] be a continuousfunction which satisfies 𝑓

𝑥(𝑥, 𝑦) ≤ 𝑀

1and 𝑓

𝑦(𝑥, 𝑦) ≤ 𝑀

2,

where𝑀1and𝑀

2are both constants. For any approximation


error bound 𝜀 > 0, there exists a RPFS 𝑆 (𝑥, 𝑦) that satisfies𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)

< 𝜀 (∀ (𝑥, 𝑦) ∈ [𝑎, 𝑏] × [𝑐, 𝑑]) when

𝑁1= max (𝑁,𝑀) >

4𝑀1(𝑏 − 𝑎) + 4𝑀

2(𝑑 − 𝑐)

𝜀. (36)

Proof. Note that, from (28), we have

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)

=𝜑𝑖𝑗(𝑥, 𝑦) 𝑓

𝑖𝑗+ 𝜑𝑖+1,𝑗

(𝑥, 𝑦) 𝑓𝑖+1,𝑗

+ 𝜑𝑖+1,𝑗+1

(𝑥, 𝑦) 𝑓𝑖+1,𝑗+1

+𝜑𝑖,𝑗+1

(𝑥, 𝑦) 𝑓𝑖,𝑗+1

− 𝑓 (𝑥, 𝑦)

=𝜑𝑖𝑗(𝑥, 𝑦) (𝑓

𝑖𝑗− 𝑓 (𝑥, 𝑦))

+ 𝜑𝑖+1,𝑗

(𝑥, 𝑦) (𝑓𝑖+1,𝑗

− 𝑓 (𝑥, 𝑦))

+ 𝜑𝑖+1,𝑗+1

(𝑥, 𝑦) (𝑓𝑖+1,𝑗+1

− 𝑓 (𝑥, 𝑦))

+ 𝜑𝑖,𝑗+1

(𝑥, 𝑦) (𝑓𝑖,𝑗+1

− 𝑓 (𝑥, 𝑦)).

(37)

Applying the triangle inequality and Lagrange MeanValue Theorem, we get, in light of 𝜑

𝑖1,𝑗1

(𝑥, 𝑦) ≤ 1

(𝑖1= 𝑖, 𝑖 + 1; 𝑗

1= 𝑗, 𝑗 + 1),

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)

≤𝑓𝑖𝑗− 𝑓 (𝑥, 𝑦)

+𝑓𝑖+1,𝑗

− 𝑓 (𝑥, 𝑦)

+𝑓𝑖+1,𝑗+1

− 𝑓 (𝑥, 𝑦)+𝑓𝑖,𝑗+1

− 𝑓 (𝑥, 𝑦)

=𝑓𝑥(𝜉

1, 𝜂

1) (𝑥 − 𝑥

𝑖) + 𝑓𝑦(𝜉

1, 𝜂

1) (𝑦 − 𝑦

𝑗)

+𝑓𝑥(𝜉

2, 𝜂

2) (𝑥 − 𝑥

𝑖+1) + 𝑓𝑦(𝜉

2, 𝜂

2) (𝑦 − 𝑦

𝑗)

+𝑓𝑥(𝜉

3, 𝜂

3) (𝑥 − 𝑥

𝑖+1) + 𝑓𝑦(𝜉

3, 𝜂

3) (𝑦 − 𝑦

𝑗+1)

+𝑓𝑥(𝜉

4, 𝜂

4) (𝑥 − 𝑥

𝑖) + 𝑓𝑦(𝜉

4, 𝜂

4) (𝑦 − 𝑦

𝑗+1)

≤𝑀1(𝑥 − 𝑥

𝑖) + 𝑀

2(𝑦 − 𝑦

𝑗)

+𝑀1(𝑥 − 𝑥

𝑖+1) + 𝑀

2(𝑦 − 𝑦

𝑗)

+𝑀1(𝑥 − 𝑥

𝑖+1) + 𝑀

2(𝑦 − 𝑦

𝑗+1)

+𝑀1(𝑥 − 𝑥

𝑖) + 𝑀

2(𝑦 − 𝑦

𝑗+1).

(38)

It is simple to show that

𝑥 − 𝑥𝑖 ≤

𝑏 − 𝑎

𝑀,

𝑥 − 𝑥𝑖+1 ≤

𝑏 − 𝑎

𝑀,

𝑦 − 𝑦𝑗

≤𝑑 − 𝑐

𝑁,

𝑦 − 𝑦𝑗+1

≤𝑑 − 𝑐

𝑁.

(39)

1.5

1

0.5

01

0.5

0

−0.5

−1 −1−0.5

0

0.5

1

Figure 4: Membership functions of rule antecedents of RPFS in thesimulation.

So let

𝑆 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) ≤ 4𝑀1

𝑏 − 𝑎

𝑀+ 4𝑀2

𝑑 − 𝑐

𝑁< 𝜀, (40)

where (𝜉1, 𝜂

1), (𝜉2, 𝜂

2), (𝜉3, 𝜂

3), and (𝜉

4, 𝜂

4) ∈ (𝑥

𝑖, 𝑦𝑗) ×

(𝑥𝑖+1, 𝑦𝑗+1

). Simplifying the above formula, we obtain

𝑁1= max (𝑁,𝑀) >

4𝑀1(𝑏 − 𝑎) + 4𝑀

2(𝑑 − 𝑐)

𝜀. (41)

The proof is completed.

6. Simulation

In this section, we compare the approximation performanceof RPFS with the conventional fuzzy system (using Gaussianmembership functions) mentioned in Section 2 by approxi-mating four typical functions as follows:

𝑓1(𝑥, 𝑦) = 0.52 + 0.1𝑥

3+ 0.28𝑦

3− 0.06𝑥𝑦,

(𝑥, 𝑦) ∈ [−1, 1] × [−1, 1] ;

𝑓2(𝑥, 𝑦) = 𝑥

3+ 𝑦3, (𝑥, 𝑦) ∈ [−1, 1] × [−1, 1] ;

𝑓3(𝑥, 𝑦) = sin (𝑥 + 𝑦) , (𝑥, 𝑦) ∈ [−1, 1] × [−1, 1] ;

𝑓4(𝑥, 𝑦) = 0.5 sin (𝑥𝑦) , (𝑥, 𝑦) ∈ [−1, 1] × [−1, 1] .

(42)

For convenience, some notations are stated as follows:System I represents rectangular pyramid fuzzy system andSystem II represents the conventional fuzzy system (men-tioned in Section 2) using Gaussian membership functions.Let the distances between the peak points of the two fuzzy sys-tems be 0.2, and the distances between the sample points arechosen as 0.01.Themembership functions of rule antecedentsof RPFS are given in Figure 4. The original and simulationsurfaces and the approximation error surfaces are shown inFigure 5. The max approximation errors and the standard


1

0.5

01

0

−1 −1 −0.50 0.5

1

x

−1 −0.50 0.5

1

x

y

10

−1y

z

1

0.5

0

z

Original curve

Simulation curve

(a)

0.04

0.02

0

−0.02

−0.04

z

−1−1−0.5

00 0.5

1

1

x

y

(b)

1

0

−11

0−1 −1 −0.5 0

0.5 1

x

−1 −0.5 00.5 1

x

y

10

−1y

z

1

0

−1

z

Simulation curve

Original curve

(c)

0.04

0.02

0

−0.02

−0.041

0

−1 −1−0.5

00.5

1

x

y

z

(d)

xy

z

0.5

0

−0.51

0−1 −1 −0.5

0 0.51

xy

z

0.5

0

−0.51

0−1 −1 −0.5

0 0.51

Simulation curve

Original curve

(e)

0.02

0.01

0

−0.01

−0.021

0

−1−1

−0.50

0.51

x

y

z

(f)

Figure 5: Continued.


2

0

10

−1 −1 −0.50 0.5

1

xy

z

Simulation curve

−2

2

0

1

0−1 −1 −0.5

0 0.51

xy

z

−2

Original curve

(g)

0.2

0.1

0

−0.1

−0.21

0

−1−1

−0.50

0.51

xy

z

(h)

Figure 5: (a) The original and simulation surfaces of 𝑓1. (b) The approximation error surface of 𝑓

1. (c) The original and simulation surfaces

of 𝑓2. (d)The approximation error surface of 𝑓

2. (e) The original and simulation surfaces of 𝑓

3. (f) The approximation error surface of 𝑓

3. (g)

The original and simulation surfaces of 𝑓4. (h) The approximation error surface of 𝑓

4.

Table 1: The maximum approximation errors of System I andSystem II.

Function ErrorSystem I System II

𝑓1

0.0212 0.0344𝑓2

0.1211 0.2004𝑓3

0.0353 0.0638𝑓4

0.0217 0.0214

Table 2: The standard deviations of System I and System II.

Function Standard deviationSystem I System II

𝑓1

0.0056 0.0097𝑓2

0.0265 0.0457𝑓3

0.0126 0.0247𝑓4

0.0040 0.0083

deviations of System I and System II are presented in Tables 1and 2, respectively.

It can be seen from Figure 5 that the simulation surfacesof RPFS almost coincided with the original surfaces. RPFSachieves a better approximation performance comparingwith the conventional fuzzy system as shown inTables 1 and 2.We can conclude that the proposed fuzzy system improves theapproximation capability of the conventional fuzzy system tosome extent.

7. Conclusions

Anew type of fuzzy system based on the rectangular pyramidmembership function is proposed in this paper. The modelof the system has been introduced mainly by replacing thecompound of the two rule antecedents using t-normwith therectangular pyramid membership function, and the concretederivation process is given. As the application problem offuzzy system is essentially a function approximation problem,it is necessary to know about the approximation capability ofRPFS.We proved that RPFS can approximate any continuousfunction of two variables to arbitrary degree of accuracy onany compact domain. Furthermore, the sufficient conditionwhich reflects the relationship between the rule numberand the approximation accuracy is also given combing theapproximation property of RPFS. Finally, the simulationresults demonstrate that the approximation performance ofRPFS is found to be better than the conventional fuzzy systemin the maximum approximation errors and the standarddeviations. It means that RPFS has successfully improved theperformance of function approximation of the conventionalfuzzy system to some degree.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work is supported by the National Science Foundationof China under Grant no. 90818025 and no. 61074044.


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Research ArticleThe Relations among Fuzzy 𝑡-Filters on Residuated Lattices

Huarong Zhang1,2 and Qingguo Li1

1 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China2Department of Mathematics, China Jiliang University, Hangzhou, Zhejiang 310018, China

Correspondence should be addressed to Qingguo Li; [email protected]

Received 8 July 2014; Accepted 21 September 2014; Published 20 October 2014


Copyright © 2014 H. Zhang and Q. Li. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give the simple general principle of studying the relations among fuzzy t-filters on residuated lattices. Using the general principle,we can easily determine the relations among fuzzy t-filters on different logical algebras.

1. Introduction

Residuated lattices, invented by Ward and Dilworth [1],constitute the semantics of Höhle’s Monoidal Logic [2].Residuated lattices are very useful and are basic algebraicstructures. Many logical algebras, such as Boolean algebras,MV-algebras, BL-algebras, MTL-algebras, Gödel algebras,NM algebras, and R0-algebras, are particular residuatedlattices. Besides their logical interest, residuated lattices havelots of interesting properties. In [3], Idziak proved that thevarieties of residuated lattices are equational.

Filters play a vital role in investigating logical algebras andthe completeness of the corresponding nonclassical logics.From logical points of view, filters correspond to sets ofprovable formulae. At present, the filter theory on differentlogical algebras has been widely studied. Only on residuatedlattices, such literatures are as follows: [4–11]. In [8, 9], Ma etal. found the common features of filters on residuated lattices.They, respectively, proposed the notion of 𝜏-filters and 𝑡-filters on residuated lattices. In [9], Vı́ta studied some basicproperties of 𝑡-filters and gave the simple general frameworkof special types of filters.

After Zadeh [12] proposed the theory of fuzzy sets, it hasbeen applied to many branches in mathematics. The fuzzifi-cation of the filters was originated in 1995 [13]. Subsequently,a large amount of papers about special types of fuzzy filterswas published in many journals on different logical algebras[10, 11, 14–24]. In [23], Vı́ta found the common features offuzzy filters on residuated lattices. He proposed the notionof fuzzy 𝑡-filters and proved its basic properties. However,

the relations among fuzzy 𝑡-filters were not discussed. Usu-ally, when studying the relations among special types of fuzzyfilters, the equivalent characterizations of special types offuzzy filters were firstly discussed. Then, resorting to theproperties of the logical algebras, the relations among specialtypes of fuzzy filters were given. The proofs were tediousin many literatures. The motivation of this paper is to givethe simple general principle of studying the relations amongfuzzy 𝑡-filters on residuated lattices. In contrast to proofs ofparticular results for concrete special types of fuzzy filters,proofs of those general theorems in this paper are simple. Andthe general principle can be applied to all the subvarieties ofresiduated lattices.

2. Preliminary

Definition 1 (see [1, 25]). A residuated lattice is an algebra 𝐿 =(𝐿, ∧, ∨, ⊗, → , 0, 1) such that for all 𝑥, 𝑦, 𝑧 ∈ 𝐿,

(1) (𝐿, ∧, ∨, 0, 1) is a bounded lattice;(2) (𝐿, ⊗, 1) is a commutative monoid;(3) (⊗, → ) forms an adjoint pair; that is, 𝑥 ⊗ 𝑦 ≤ 𝑧 if and

only if 𝑥 ≤ 𝑦 → 𝑧.We denote 𝑥 → 0 = 𝑥∗.

Definition 2 (see [11, 25–30]). Let 𝐿 be a residuated lattice.Then 𝐿 is called

(i) an MTL-algebra if (𝑥 → 𝑦) ∨ (𝑦 → 𝑥) = 1 for all 𝑥,𝑦 ∈ 𝐿 (prelinear axiom);


http://dx.doi.org/10.1155/2014/894346


(ii) an Rl-monoid if 𝑥 ∧ 𝑦 = 𝑥 ⊗ (𝑥 → 𝑦) for all 𝑥, 𝑦 ∈ 𝐿(divisible axiom);

(iii) aHeyting algebra if𝑥⊗𝑦 = 𝑥∧𝑦 for all𝑥,𝑦 ∈ 𝐿, whichis equivalent to an idempotent residuated lattice; thatis, 𝑥 = 𝑥 ⊗ 𝑥 = 𝑥2 for 𝑥 ∈ 𝐿;

(iv) a regular residuated lattice if it satisfies double nega-tion; that is, 𝑥∗∗ = 𝑥 for 𝑥 ∈ 𝐿;

(v) a BL-algebra if it satisfies both prelinear and divisibleaxioms;

(vi) an MV-algebra if it is a regular Rl-monoid;(vii) a Gödel algebra if it is an idempotent BL-algebra;(viii) a Boolean algebra if it is an idempotent MV-algebra;(ix) a R0-algebra (NM algebra) if it satisfies prelinear

axiom, double negation, and (𝑥⊗𝑦 → 0)∨ (𝑥∧𝑦 →𝑥 ⊗ 𝑦) = 1.

Definition 3 (see [25, 31, 32]). Let 𝐿 be a residuated lattice.Then, a nonempty subset 𝐹 of 𝐿 is called a filter if

(1) for all 𝑥 ∈ 𝐹 and 𝑦 ∈ 𝐿, 𝑥 ≤ 𝑦 implies 𝑦 ∈ 𝐹,(2) for all 𝑥, 𝑦 ∈ 𝐹, 𝑥 ⊗ 𝑦 ∈ 𝐹.

Definition 4 (see [5–11]). Let𝐹 be a filter of𝐿.Then,𝐹 is called

(i) an implicative filter if 𝑥 → 𝑥2 ∈ 𝐹 for all 𝑥 ∈ 𝐿,(ii) a regular filter if 𝑥∗∗ → 𝑥 ∈ 𝐹 for all 𝑥 ∈ 𝐿,(iii) a divisible filter if (𝑥 ∧ 𝑦) → (𝑥 ⊗ (𝑥 → 𝑦)) ∈ 𝐹 for

all 𝑥, 𝑦 ∈ 𝐿,(iv) a prelinear filter if (𝑥 → 𝑦) ∨ (𝑦 → 𝑥) ∈ 𝐹 for all 𝑥,𝑦 ∈ 𝐿,

(v) a Boolean filter if 𝑥 ∨ 𝑥∗ ∈ 𝐹 for all 𝑥 ∈ 𝐿,(vi) a fantastic filter if (𝑦 → 𝑥) → (((𝑥 → 𝑦) → 𝑦) →𝑥) ∈ 𝐹 for all 𝑥, 𝑦 ∈ 𝐿,

(vii) an 𝑛-contractive filter if 𝑥𝑛 → 𝑥𝑛+1 ∈ 𝐹 for all 𝑥 ∈ 𝐿,where 𝑥𝑛+1 = 𝑥𝑛 ⊗ 𝑥, 𝑛 ≥ 1.

Remark 5. On residuated lattices, 𝑥 → (𝑦 → 𝑧) = 𝑦 →(𝑥 → 𝑧) holds (see [31]). Using these properties, we havethat 𝐹 is a fantastic filter if ((𝑥 → 𝑦) → 𝑦) → ((𝑦 →𝑥) → 𝑥) ∈ 𝐹.

We now review some fuzzy concepts. A fuzzy set onresiduated lattice is a function 𝜇 : 𝐿 → [0, 1]. For any𝛼 ∈ [0, 1] and an arbitrary fuzzy set 𝜇, we denote the set{𝑥 ∈ 𝐿 | 𝜇(𝑥) ≥ 𝛼} (i.e., the 𝛼-cut) by the symbol 𝜇

𝛼.

Definition 6 (see [10, 11]). A fuzzy set 𝜇 is a fuzzy filter on 𝐿if and only if it satisfies the following two conditions for all𝑥, 𝑦 ∈ 𝐿:

(1) 𝜇(𝑥 ⊗ 𝑦) ≥ min{𝜇(𝑥), 𝜇(𝑦)},(2) if 𝑥 ≤ 𝑦, then 𝜇(𝑥) ≤ 𝜇(𝑦).

In the following, by the symbol 𝑥 we denote the abbrevi-ation of 𝑥, 𝑦, . . .; that is, 𝑥 is a formal listing of variables usedin a given content. By the term 𝑡, it is always meant as a termin the language of residuated lattices.

Definition 7 (see [9]). Let 𝑡 be an arbitrary term on thelanguage of residuated lattices. A filter 𝐹 on 𝐿 is a 𝑡-filter if𝑡(𝑥) ∈ 𝐹 for all 𝑥 ∈ 𝐿.

Definition 8 (see [23]). A fuzzy filter 𝜇 on 𝐿 is called a fuzzy𝑡-filter on 𝐿, if for all 𝑥 ∈ 𝐿 it satisfies 𝜇(𝑡(𝑥)) = 𝜇(1).

Example 9 (see [11]). Fuzzy Boolean filters are fuzzy 𝑡-filtersfor 𝑡 equal to 𝑥 ∨ 𝑥∗.

Example 10 (see [11]). Fuzzy regular filters are fuzzy 𝑡-filtersfor 𝑡 equal to 𝑥∗∗ → 𝑥.

Example 11 (see [11]). Fuzzy fantastic filters are fuzzy 𝑡-filtersfor 𝑡 equal to ((𝑥 → 𝑦) → 𝑦) → ((𝑦 → 𝑥) → 𝑥).

Remark 12. Using the notion of fuzzy 𝑡-filter, fuzzy implica-tive filters are fuzzy 𝑡-filters for 𝑡 equal to 𝑥 → 𝑥2. Fuzzydivisible filters are fuzzy 𝑡-filters for 𝑡 equal to (𝑥 ∧ 𝑦) →(𝑥 ⊗ (𝑥 → 𝑦)) and so forth.

Let us assume that since now, 𝑡 is an arbitrary term inthe language of residuated lattices.We also use another usefulconvention: given a varietyB of residuated lattices, we denoteits subvariety given by the equation 𝑡 = 1 by the symbol B[𝑡];we call this algebra 𝑡-algebra.

The next part of this paper concerns fuzzy quotient con-structions. We recall some known results and constructionsconcerning fuzzy quotients residuated lattices.

Theorem 13 (see [11]). Let 𝜇 be a fuzzy filter on 𝐿 and 𝑥, 𝑦 ∈ 𝐿.For any 𝑧 ∈ 𝐿, we define𝜇𝑥 : 𝐿 → [0, 1],𝜇𝑥(𝑧) = min{𝜇(𝑥 →𝑧), 𝜇(𝑧 → 𝑥)}. Then, 𝜇𝑥 = 𝜇𝑦 if and only if 𝜇(𝑥 → 𝑦) =𝜇(𝑦 → 𝑥) = 𝜇(1).

Theorem 14 (see [11]). Let 𝜇 be a fuzzy filter on 𝐿 and 𝐿/𝜇 :={𝜇𝑥

| 𝑥 ∈ 𝐿}. For any 𝜇𝑥, 𝜇𝑦 ∈ 𝐿/𝜇, we define𝜇𝑥

∧ 𝜇𝑦

= 𝜇𝑥∧𝑦,

𝜇𝑥

∨ 𝜇𝑦

= 𝜇𝑥∨𝑦,

𝜇𝑥

⊗ 𝜇𝑦

= 𝜇𝑥⊗𝑦,

𝜇𝑥

→ 𝜇𝑦

= 𝜇𝑥→𝑦.

Then, 𝐿/𝜇 = (𝐿/𝜇, ∧, ∨, ⊗, → , 𝜇0, 𝜇1) is a residuated latticecalled the fuzzy quotient residuated lattice.

Theorem 15 (quotient characteristics [23]). LetB be a varietyof residuated lattices and 𝐿 ∈ B. Let 𝜇 be a fuzzy filter on 𝐿.Then, the fuzzy quotient 𝐿/𝜇 belongs to B[𝑡] if and only if 𝜇 isa fuzzy 𝑡-filter on 𝐿.

3. The General Principle of the Relationamong Fuzzy 𝑡-Filters and Its Application

In the following, letB be a variety of residuated lattices. 𝐿 ∈ Band 𝜇 is a fuzzy filter on 𝐿.

Theorem 16. Suppose that there are fuzzy 𝑡1-filter and fuzzy

𝑡2-filter on 𝐿 and B[𝑡

1] ⊆ B[𝑡

2]. If 𝜇 is a fuzzy 𝑡

1-filter, then 𝜇

is a fuzzy 𝑡2-filter.


Proof. 𝜇 is a fuzzy 𝑡1-filter⇒ 𝐿/𝜇 ∈ B[𝑡

1] ⇒ 𝐿/𝜇 ∈ B[𝑡

2] ⇒

𝜇 is a fuzzy 𝑡2-filter.

Theorem 17. Suppose there are fuzzy 𝑡1-filter and fuzzy 𝑡

2-

filter on 𝐿. If B[𝑡1] = B[𝑡

2], then 𝜇 is a fuzzy 𝑡

1-filter if and

only if 𝜇 is a fuzzy 𝑡2-filter.

Proof. 𝜇 is a fuzzy 𝑡1-filter⇔ 𝐿/𝜇 ∈ B[𝑡

1] ⇔ 𝐿/𝜇 ∈ B[𝑡

2] ⇔

𝜇 is a fuzzy 𝑡2-filter.

Remark 18. The above results give the general principle ofthe relations among fuzzy 𝑡-filters. If we want to judge therelations among fuzzy 𝑡-filter, we only resort to the relationsamong 𝑡-algebras. Since the relations among 𝑡-algebras areknown to us, we can easily obtain the relations among fuzzy𝑡-filters.

Theorem 19. Let 𝐿 be a residuated lattice. If 𝜇 is a fuzzyimplicative filter, then 𝜇 is a fuzzy 𝑛-contractive filter.

Proof. It is obvious that B[𝑥 → 𝑥2] ⊆ B[𝑥𝑛 → 𝑥𝑛+1]. ByTheorem 16, the result is clear.

Lemma 20 (see [5, 27]). Let 𝐿 be a residuated lattice. If 𝐿 is aHeyting algebra, then 𝐿 is an Rl-monoid.

Lemma 21 (see [11]). Let 𝐿 be a residuated lattice. Then thefollowing are equivalent:

(1) 𝐿 is an MV-algebra;(2) (𝑥 → 𝑦) → 𝑦 = (𝑦 → 𝑥) → 𝑥, ∀𝑥, 𝑦 ∈ 𝐿.

Lemma 22 (see [25]). Let 𝐿 be a residuated lattice. Then 𝐿 isan MV-algebra if and only if 𝐿 is a regular BL-algebra.

Lemma 23. Let 𝐿 be a residuated lattice. Then the followingare equivalent:

(1) 𝐿 is a Boolean algebra;(2) 𝑥 ∨ 𝑥∗ = 1, ∀𝑥 ∈ 𝐿;(3) 𝐿 is regular and idempotent.

Proof. (1)⇔(2) Reference [11], Proposition 2.10.(1)⇒(3) If 𝐿 is a Boolean algebra, then 𝐿 is an idempotent

MV-algebra. Thus, 𝐿 is regular and idempotent.(3)⇒(1) If 𝐿 is regular and idempotent, then 𝐿 is a regular

Rl-monoid. Thus, 𝐿 is an MV-algebra. Also, 𝐿 is idempotent;therefore, 𝐿 is a Boolean algebra.

Lemma 24. Let 𝐿 be a residuated lattice. Then the followingare equivalent:

(1) 𝐿 is a Boolean algebra;(2) 𝐿 is an idempotent R0-algebra.

Proof. (1)⇒(2) If𝐿 is a Boolean algebra, then𝐿 is a R0-algebra([30], Example 8.5.1) and 𝐿 is idempotent.

(2)⇒(1) Suppose 𝐿 is an idempotent R0-algebra. Since 𝐿is a R0-algebra, then 𝐿 is regular. By Lemma 23, we have that𝐿 is a Boolean algebra.

Lemma25. Let𝐿 be a residuated lattice. 𝑡1and 𝑡2are arbitrary

terms on 𝐿. Then, B[𝑡1⊗ 𝑡2] = B[𝑡

1] ∩ B[𝑡

2].

Proof. 𝐿 ∈ B[𝑡1⊗ 𝑡2] ⇔ 𝐿 ∈ B and 𝑡

1⊗ 𝑡2= 1 ⇔ 𝐿 ∈ B;

𝑡1= 1 and 𝑡

2= 1 ⇔ 𝐿 ∈ B[𝑡

1] ∩ B[𝑡

2].

Theorem 26. Let 𝐿 be a residuated lattice. Then, 𝜇 is a fuzzyBoolean filter if and only if 𝜇 is both a fuzzy regular and a fuzzyimplicative filter.

Proof. 𝜇 is a fuzzy Boolean filter⇔ 𝐿/𝜇 ∈ B[𝑥∨𝑥∗] ⇔ 𝐿/𝜇 ∈B[(𝑥∗∗ → 𝑥)⊗ (𝑥 → 𝑥2)] ⇔ 𝐿/𝜇 ∈ B[𝑥∗∗ → 𝑥]∩B[𝑥 →

𝑥2

] ⇔ 𝐿/𝜇 ∈ B[𝑥∗∗ → 𝑥] and 𝐿/𝜇 ∈ B[𝑥 → 𝑥2] ⇔ 𝜇 isboth a fuzzy regular and a fuzzy implicative filter.

Remark 27. Using the same method, we can easily obtain thefollowing results.

Theorem 28. Let 𝐿 be a residuated lattice. Then

(1) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is a fuzzyfantastic and fuzzy implicative filter;

(2) 𝜇 is a fuzzy fantastic filter if and only if 𝜇 is a fuzzyregular and fuzzy divisible filter;

(3) every fuzzy implicative filter is a fuzzy divisible one;(4) if 𝜇 is a fuzzy prelinear filter, then 𝜇 is a fuzzy fantastic

filter if and only if 𝜇 is both a fuzzy regular and a fuzzydivisible filter;

(5) if 𝜇 is a fuzzy Boolean filter, then 𝜇 is a fuzzy 𝑛-contractive filter.

Remark 29. The notion of fuzzy 𝑡-filter and the general prin-ciple are not only applicable on residuated lattices, but alsoeven transferable to all their subvarieties. Taking advantageof the relations among 𝑡-algebras, we can easily obtain thefollowing results.

Theorem 30. Let 𝐿 be a Boolean-algebra. Then the fuzzyprelinear filter, fuzzy fantastic filter, fuzzy divisible filter, fuzzyregular filter, and fuzzy 𝑛-contractive and fuzzy Boolean filtercoincide.

Theorem 31. Let 𝐿 be an MV-algebra. Then

(1) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is a fuzzyimplicative filter;

(2) the fuzzy prelinear filter, fuzzy fantastic filter, fuzzydivisible filter, and fuzzy regular filter coincide.

Theorem 32. Let 𝐿 be a Gödel-algebra. Then

(1) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is a fuzzyregular filter if and only if 𝜇 is a fuzzy fantastic filter;

(2) the fuzzy prelinear filter, fuzzy divisible filter, fuzzy 𝑛-contractive filter, and fuzzy implicative filter coincide.

Theorem 33. Let 𝐿 be a BL-algebra; then

(1) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is both a fuzzyimplicative and a fuzzy regular filter;


(2) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is both a fuzzyimplicative and a fuzzy fantastic filter;

(3) 𝜇 is a fuzzy fantastic filter if and only if 𝜇 is a fuzzyregular filter;

(4) the fuzzy prelinear filter and fuzzy divisible filtercoincide.

Theorem 34. Let 𝐿 be an MTL-algebra. Then,

(1) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is both a fuzzyimplicative and a fuzzy regular filter;

(2) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is both a fuzzyimplicative and a fuzzy fantastic filter;

(3) 𝜇 is a fantastic filter if and only if 𝜇 is a regular anddivisible filter;

(4) if𝜇 is a fuzzy implicative filter, then𝜇 is a fuzzy divisiblefilter.

Theorem 35. Let 𝐿 be a Heyting-algebra. Then

(1) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is a fuzzyregular filter if and only if 𝜇 is a fuzzy fantastic filter;

(2) the fuzzy implicative filter, fuzzy divisible filter, andfuzzy 𝑛-contractive filter coincide.

Theorem 36. Let 𝐿 be a R0-algebra; then


(2) every fuzzy Boolean filter is a fuzzy fantastic filter;(3) 𝜇 is a fuzzy fantastic filter if and only if 𝜇 is a fuzzy

divisible filter;(4) the fuzzy prelinear filter and fuzzy regular filter coin-

cide;(5) every fuzzy implicative filter is a fuzzy divisible one.

Theorem 37. Let 𝐿 be a regular residuated lattice. Then


(2) 𝜇 is a fuzzy fantastic filter if and only if 𝜇 is a fuzzydivisible filter;

(3) every fuzzy Boolean filter is a fuzzy fantastic filter;(4) every fuzzy implicative filter is a fuzzy divisible one.

Theorem 38. Let 𝐿 be a Rl-monoid. Then

(1) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is a fuzzyimplicative and fuzzy fantastic filter;

(2) 𝜇 is a fuzzy Boolean filter if and only if 𝜇 is a fuzzyimplicative and fuzzy regular filter;

(3) 𝜇 is a fuzzy fantastic filter if and only if 𝜇 is a fuzzyregular filter;

(4) every fuzzy implicative filter is a fuzzy divisible one.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China, Grant nos. 11371130 and 61273018, andby the Research Fund for the Doctoral Program of HigherEducation of China, Grant no. 20120161110017.

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Research ArticleOn Power Idealization Filter Topologies ofLattice Implication Algebras

Shi-Zhong Bai1,2 and Xiu-Yun Wu1,3

1 School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China2 School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China3Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, China

Correspondence should be addressed to Xiu-Yun Wu; [email protected]

Received 24 June 2014; Accepted 2 August 2014; Published 28 August 2014


Copyright © 2014 S.-Z. Bai and X.-Y. Wu. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The aim of this paper is to introduce power idealization filter topologies with respect to filter topologies and power ideals of latticeimplication algebras, and to investigate some properties of power idealization filter topological spaces and their quotient spaces.

1. Introduction and Preliminaries

By generalizing Boolean algebras and Lukasiewicz implica-tion algebras [1], Xu [2] defined the concept of lattice impli-cation algebra which is regarded as an efficient approach todeal with lattice valued logical systems. Later, Xu and Qin [3]defined the concept of the filer topology of a lattice impli-cation algebra which takes the set of all filters of the latticeimplication algebra as a base. Based on these definitions andsome results in [4], we introduce and study power idealiza-tion topologies with respect to filter topologies and powerideals of lattice implication algebras.

Now we recall some definitions and notions of latticeimplication algebras and topological spaces.

Let (𝐿, ∧, ∨, 0, 1) be a bounded lattice with the greatest 1and the smallest 0. A system (𝐿, ∧, ∨, , → , 0, 1) is called aquasi-lattice implication algebra if : 𝐿 → 𝐿 is an order-reserving involution and →: 𝐿 × 𝐿 → 𝐿 is a map (called animplication operator) satisfying the following conditions forany 𝑥, 𝑦, 𝑧 ∈ 𝐿:

(1) 𝑥 → (𝑦 → 𝑧) = 𝑦 → (𝑥 → 𝑧),(2) 𝑥 → 𝑥 = 1,(3) 𝑥 → 𝑦 = 𝑦 → 𝑥,(4) 𝑥 → 𝑦 = 𝑦 → 𝑥 = 1 implies 𝑥 = 𝑦,(5) (𝑥 → 𝑦) → 𝑦 = (𝑦 → 𝑥) → 𝑥.

A quasi-lattice implication algebra (𝐿, ∧, ∨, , → , 0, 1) iscalled a lattice implication algebra if the implication operator→ further fulfils the following conditions:

(6) (𝑥 ∨ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∧ (𝑦 → 𝑧),(7) (𝑥 ∧ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∨ (𝑦 → 𝑧).

A lattice implication algebra (𝐿, ∧, ∨, , → , 0, 1) will besimply denoted by 𝐿.

Let 𝐿 be a lattice implication algebra and let 𝜑 be a subsetof 2𝐿. We use 𝜑𝑐 to denote the complement {𝐿 \ 𝐴 : 𝐴 ∈ 𝜑},where 𝐿 \ 𝐴 = {𝑥 ∈ 𝐿 : 𝑥 ∉ 𝐴}. A subset 𝜏 ⊆ 2𝐿 is called atopology on 𝐿, if 𝜏 satisfies the following:

(1) 0, 𝐿 ∈ 𝜏,(2) 𝐴, 𝐵 ∈ 𝜏 implies 𝐴 ∩ 𝐵 ∈ 𝜏,(3) {𝐴

𝑡∈ 𝜏 : 𝑡 ∈ 𝑇} ⊆ 𝜏 implies ∪

𝑡∈𝑇𝐴𝑡∈ 𝜏.

Elements of 𝜏 are called 𝜏-open sets and the complementsof them are called 𝜏-closed. The pair (𝐿, 𝜏) is called atopological space. A subsetB of 𝜏 is called a base of 𝜏, if foreach 𝐴 ∈ 𝜏 and each 𝑥 ∈ 𝐴, there exists 𝐵 ∈ B such that𝑥 ∈ 𝐵 ⊆ 𝐴.

Let 𝐿 be an implication algebra. A subset 𝐹 of 𝐿 is calleda filer, if 𝐹 satisfies the following: (1) 1 ∈ 𝐹; (2) 𝑥, 𝑥 → 𝑦 ∈ 𝐹implies 𝑦 ∈ 𝐹. The collection of all filters in 𝐿 is denotedby F(𝐿), or F briefly. Clearly, F consists a base of some


http://dx.doi.org/10.1155/2014/812145


topology 𝑇F(𝐿), briefly 𝑇F. Usually, 𝑇F is called the filtertopology generated by F. And the pair (𝐿, 𝑇F) is called thefilter topological space.A subset𝑈 ⊆ 𝐿 is called𝑇F-neighbor-hood of 𝑥 ∈ 𝐿, or neighborhood of 𝑥 in 𝑇F if 𝑥 ∈ 𝑈 ∈ 𝑇F.The set of all 𝑇F-neighborhoods of 𝑥 is denoted by N𝑇F(𝑥).Since F ⊆ 𝑇F and [𝑥) = ∩{𝐹 : 𝑥 ∈ 𝐹 ∈ F} ∈ F, [𝑥) is thesmallest element ofN

𝑇F(𝑥).

The closure operator and interior operator of 𝑇F aredenoted by 𝑐 and 𝑖. Clearly, for every𝐴 ⊆ 𝐿, 𝑐(𝐴) = ∩{𝐿\[𝑥) :𝑥 ∈ 𝐿, [𝑥) ∩ 𝐴 = 0} and 𝑖(𝐴) = ∪{[𝑥) : 𝑥 ∈ 𝐿, [𝑥) ⊆ 𝐴}. Thefollowing proposition describes 𝑐(𝐴).

Proposition 1. Let (𝐿, 𝑇F) be the filer topology generated byF(𝐿). Then for 𝐴 ⊆ 𝐿, 𝑐(𝐴) = {𝑥 ∈ 𝐿 : [𝑥) ∩ 𝐴 ̸= 0}.

Proof. The proof is trivial since [𝑥) is the smallest 𝑇F-neighborhood of 𝑥.

Let 𝐿 be a lattice implication algebra and let 2𝐿 be thepower set of 𝐿. A nonempty subsetI of 2𝐿 is called a powerideal of 𝐿 if I satisfies the following: (1) 𝐴, 𝐵 ∈ 2𝐿 and 𝐴 ⊆𝐵 ∈ I imply 𝐴 ∈ I; (2) 𝐴, 𝐵 ∈ I implies 𝐴 ∪ 𝐵 ∈ I. Thecollection of all power ideals in 2𝐿 is denoted by I(𝐿), orbrieflyI. Note thatI

0= {0} is the smallest power ideal and

I𝐿= 2𝐿 is the greatest power ideal. Moreover, ifI,J ∈ I,

then (1)I∩J ∈ I; (2)I∨J = {𝐼∪ 𝐽 : 𝐼 ∈ I, 𝐽 ∈ J} ∈ I.

2. Local Functions and Power IdealizationFilter Topologies

Let 𝐿 be a lattice implication algebra, let 𝑇F be the filter topo-logy, and let I be a power ideal. An operator ∗ on 2𝐿 isdefined as follows:

𝐴∗(I, 𝑇F) = {𝑥 ∈ 𝐿 : ∀𝑈 ∈ N𝑇F (𝑥) , 𝐴 ∩ 𝑈 ∉ I} (1)

for every 𝐴 ⊆ 𝐿.The operator ∗ is called the local function with respect to

𝑇F andI. 𝐴∗ is called local function of 𝐴. We usually write

𝐴∗(I) or 𝐴∗ instead of 𝐴∗(I, 𝑇F).Clearly, 𝑥 ∈ 𝐴∗ if and only if [𝑥) ∩ 𝐴 ∉ I. Thus 𝐴∗ =

{𝑥 ∈ 𝐿 : [𝑥) ∩ 𝐴 ∉ I}. The following proposition gives somefurther details of 𝐴∗.

Proposition 2. Let (𝐿, 𝑇F) be the filter topological space andI,J ∈ I. Then

(1) 𝐴∗(I0) = 𝑐(𝐴) and 𝐴∗(I

𝐿) = 0;

(2) if 𝐴 ⊆ 𝐵, then 𝐴∗(I) ⊆ 𝐵∗(I);(3) ifI ⊆ J, then 𝐴∗(J) ⊆ 𝐴∗(I);(4) 𝐴∗(I) = 𝑐(𝐴∗(I)) ⊆ 𝑐(𝐴);(5) (𝐴∗)∗(I) ⊆ 𝐴∗(I);(6) if 𝐴 ⊆ I, then 𝐴∗(I) = 0;(7) if 𝐴 ∈ 𝑇𝑐F, then 𝐴

∗(I) ⊆ 𝐴;

(8) if 𝐵 ∈ I, then (𝐴 ∪ 𝐵)∗(I) = 𝐴∗(I) = (𝐴 \ 𝐵)∗(I);(9) (𝐴 ∪ 𝐴∗(I))∗(I) = 𝐴∗(I);

(10) (𝐴 ∪ 𝐵)∗(I) = 𝐴∗(I) ∪ 𝐵∗(I);(11) 𝐴∗(I)\𝐵∗(I) = (𝐴\𝐵)∗(I)\𝐵∗(I) ⊆ (𝐴\𝐵)∗(I);(12) if {1} ∉ I, then [𝑥)∗(I) = 𝐿 for each 𝑥 ∈ 𝐿;(13) if {1} ∉ I and 1 ∈ 𝐴 ⊆ 𝐿, then [𝐴)∗(I) = [𝐴∗(I)) =

𝐿.

Proof. (1) By Proposition 1, 𝑥 ∈ 𝐴∗(I0) if and only if [𝑥) ∩

𝐴 ̸= 0 if and only if 𝑥 ∈ 𝑐(𝐴). Thus 𝐴∗(I0) = 𝑐(𝐴). Since

[𝑥) ∩ 𝐴 ∈ 2𝐿 = I𝐿for each 𝑥 ∈ 𝐿, 𝐴∗(𝐿) = 0.

(2) Let 𝐴 ⊆ 𝐵 and 𝑥 ∈ 𝐴∗(I). Then [𝑥) ∩ 𝐴 ∉ I. SinceI is a power ideal and [𝑥) ∩ 𝐴 ⊆ [𝑥) ∩ 𝐵, [𝑥) ∩ 𝐵 ∉ I and so𝑥 ∈ 𝐵

∗(I). Thus 𝐴∗(I) ⊆ 𝐵∗(I).(3) Let I ⊆ J and 𝑥 ∈ 𝐴∗(I). Then [𝑥) ∩ 𝐴 ∉ J. It

follows that [𝑥) ∩ 𝐴 ∉ I and so 𝑥 ∈ 𝐴∗(I). Thus 𝐴∗(J) ⊆𝐴∗(I).(4) If 𝑥 ∉ 𝑐(𝐴), then 𝑥 ∈ 𝐿 \ 𝑐(𝐴) ∈ 𝑇F and so [𝑥) ⊆

𝐿 \ 𝑐(𝐴). Thus [𝑥) ∩ 𝐴 ⊆ (𝐿 \ 𝑐(𝐴)) ∩ 𝐴 = 0 ∈ I. Thisimplies 𝑥 ∉ 𝐴∗(I) and so 𝐴∗(I) ⊆ 𝑐(𝐴). Then 𝑐(𝐴∗(I)) ⊆𝑐(𝑐(𝐴)) = 𝑐(𝐴).

It is clear that 𝐴∗(I) ⊆ 𝑐(𝐴∗(I)). Next, we prove𝑐(𝐴∗(I)) ⊆ 𝐴∗(I).

Let 𝑥 ∈ 𝑐(𝐴∗(I)). By Proposition 1, [𝑥) ∩ 𝐴∗(I) ̸= 0.Then there exists 𝑦 ∈ [𝑥) ∩ 𝐴∗(I). By 𝑦 ∈ 𝐴∗(I), [𝑦) ∩ 𝐴 ∉I. By𝑦 ∈ [𝑥), [𝑦) ⊆ [𝑥).Thus [𝑥)∩𝐴 ∉ I and so𝑥 ∈ 𝐴∗(I).Therefore 𝑐(𝐴∗(I)) ⊆ 𝐴∗(I).

(5) By (4), (𝐴∗(I))∗(I) ⊆ 𝑐(𝐴∗(I)) = 𝐴∗(I).(6) Since [𝑥) ∩ 𝐴 ⊆ 𝐴 ∈ I for each 𝑥 ∈ 𝐿, 𝐴∗(I) = 0.(7) Suppose that 𝑥 ∈ 𝐴∗(I) \ 𝐴. Then 𝑥 ∈ 𝐿 \ 𝐴 ∈ 𝑇F.

Thus [𝑥) ⊆ 𝐿 \ 𝐴 and so [𝑥) ∩ 𝐴 ⊆ (𝐿 \ 𝐴) = 0 ∈ I. Hence𝑥 ∉ 𝐴

∗(I) which is a contradiction. Therefore 𝐴∗(I) ⊆ 𝐴.(8) By (2), (𝐴 \ 𝐵)∗(I) ⊆ 𝐴∗(I) ⊆ (𝐴 ∪ 𝐵)∗(I). Next,

we prove the inverse inclusions.If 𝑥 ∉ (𝐴\𝐵)∗(I), then ([𝑥)∩𝐴) \𝐵 = [𝑥)∩ (𝐴\𝐵) ∈ I.

Thus [𝑥) ∩ 𝐴 ⊆ 𝐼 ∪ 𝐵 ∈ I which follows fromI is a powerideal. This implies 𝑥 ∉ 𝐴∗(I). Thus 𝐴∗(I) ⊆ (𝐴 \ 𝐵)∗(I)and so 𝐴∗(I) = (𝐴 \ 𝐵)∗(I).

If 𝑥 ∉ 𝐴∗(I), then [𝑥) ∩ 𝐴 ∈ I. Since 𝐵 ∈ I,

[𝑥) ∩ (𝐴 ∪ 𝐵) ⊆ ([𝑥) ∩ 𝐴) ∪ 𝐵 ∈ I. (2)

Thus 𝑥 ∉ (𝐴 ∪ 𝐵)∗(I). This implies (𝐴 ∪ 𝐵)∗(I) ⊆ 𝐴∗(I)and so (𝐴 ∪ 𝐵)∗(I) = 𝐴∗(I).

(9) Clearly, 𝐴∗(I) ⊆ (𝐴 ∪ 𝐴∗(I))∗(I). Conversely, if𝑥 ∉ 𝐴∗(I), then [𝑥) ∩ 𝐴 ∈ I. Let [𝑥) ∩ 𝐴 = 𝐼. Then 𝐴 ⊆𝐼 ∪ (𝐿 \ [𝑥)). By (2), (7), (8), and [𝑥) ∈ 𝑇F,

𝐴∗

(I) ⊆ (𝐼 ∪ (𝐿 \ [𝑥)))∗

(I) = (𝐿 \ [𝑥))∗

(I) ⊆ 𝐿 \ [𝑥) .

(3)

Thus 𝐴 ∪ 𝐴∗(I) ⊆ (𝐿 \ [𝑥)) ∪ 𝐴 and so

[𝑥) ∩ (𝐴 ∪ 𝐴∗

(I)) ⊆ ((𝐿 \ [𝑥)) ∪ 𝐴) ∩ [𝑥)

= 𝐴 ∩ [𝑥) = 𝐼 ∈ I.(4)

This implies𝑥 ∉ (𝐴∪𝐴∗(I))∗(I) and so (𝐴∪𝐴∗(I))∗(I) ⊆𝐴∗(I).

(10) 𝐴∗(I) ∪ 𝐵∗(I) ⊆ (𝐴 ∪ 𝐵)∗(I) is clear. Conversely,if 𝑥 ∉ 𝐴∗(I) ∪ 𝐵∗(I), then [𝑥) ∩ 𝐴, [𝑥) ∩ 𝐵 ∈ I. Thus


[𝑥) ∩ (𝐴 ∪ 𝐵) = ([𝑥) ∩ 𝐴) ∪ ([𝑥) ∪ 𝐵) ∈ I. This implies𝑥 ∉ (𝐴 ∪ 𝐵)

∗(I). Therefore (𝐴 ∪ 𝐵)∗(I) ⊆ 𝐴∗(I) ∪ 𝐵∗(I).

(11) We firstly prove 𝐴∗(I) \ 𝐵∗(I) ⊆ (𝐴 \ 𝐵)∗(I)

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