bases axioms and circuits axioms for fuzzifying matroids

Fuzzy Sets and Systems 161 (2010) 3155–3165www.elsevier.com/locate/fss

Bases axioms and circuits axioms for fuzzifying matroids�

Wei Yaoa,b,∗, Fu-Gui Shic

a Department of Mathematics, Shaanxi Normal University, Xi’an 710062, PR Chinab Department of Mathematics, Hebei University of Science and Technology, 050018 Shijiazhuang, PR China

c Department of Mathematics, Beijing Institute of Technology, 100081 Beijing, PR China

Received 22 June 2008; received in revised form 7 May 2010; accepted 10 July 2010Available online 15 July 2010

Abstract

This paper shows that weighted graphs are examples of fuzzifying matroids. Axioms of bases and that of circuits are establishedfor fuzzifying matroids.© 2010 Elsevier B.V. All rights reserved.

Keywords: Algebra; Weighted graph; Weighted matroid; Fuzzifying matroid; Base-map; Circuit-map

1. Introduction

Matroids were introduced by Whitney in 1935 as a generalization of both graphs and linear independence in vectorspaces. It is well-known that matroids play an important role in mathematics, especially in applied mathematics, whichare precisely the structures for which the very simple and efficient greedy algorithm works [5,6,17]. In 1988, matroidswere generalized to fuzzy fields by Goetschel and Voxman [7]. Their approach to the fuzzification of matroids preservesmany basic properties of crisp matroids. From then on, fuzzy bases, fuzzy circuits, fuzzy rank functions and fuzzyclosure operators are widely studied [8–11,14–16], while none of the corresponding axioms has been established. In[19], based on the ideal of a fuzzifying topology [23], Shi introduced a new approach to fuzzification of matroids,namely a fuzzifying matroid. Indeed, a fuzzifying matroid is equivalent to a closed fuzzy pre-matroid in [15,16](cf. [19, Remark 11]). Fuzzy rank functions (which are different from that in [10]) are defined and studied. A one-to-one correspondence is established between all fuzzifying matroids and all fuzzy rank functions. Recently, Shi andWang [20], Wang and Wei [21] defined a concept of fuzzifying closure operators, by which fuzzifying matroids arealso characterized. The results in [19–21] indicate the reasonability of the definition of fuzzifying matroids.

Roughly speaking, Shi defined the concept of fuzzifying matroids firstly and temporarily from viewpoint of theoreticaspects. Since a graph is one of the motivation and basic examples of crisp matroids, it is natural to ask that:

Does there exist some kinds of graphs (or fuzzy graphs) which can be supplied as examples of fuzzifying matroids?

� This paper is supported by the NNSF of China (10971242, 10926055), the Foundation of Hebei Province (A2010000826,09276158) and theFoundation of HEBUST (XL200821, QD200957).

∗ Corresponding author at: Department of Mathematics, Shaanxi Normal University, Xi’an 710062, PR China.E-mail address: [email protected] (W. Yao).

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3156 W. Yao, F.-G. Shi / Fuzzy Sets and Systems 161 (2010) 3155–3165

In this paper, we will show that weighted graphs are examples of fuzzifying matroids. Our main aim is to establishaxioms of bases and that of circuits for fuzzifying matroids. This paper is organized as follows. In Section 2, we willrecall some basic notions and results related to fuzzy sets, (fuzzy) graphs and (fuzzifying) matroids. In Section 3, weshow that weighted graphs are examples of fuzzifying matroids. We establish axioms of bases and that of circuits forfuzzifying matroids in Sections 4 and 5, respectively. Conclusions and remarks are made in the last section.

2. Preliminaries

2.1. Fuzzy set theory

Fuzzy set theory, introduced by Zadeh [24], has been widely applied to handle uncertainties like vagueness, ambiguity,and imprecision in linguistic variables.

For X a set, a map A : X −→ [0, 1] is called a fuzzy subset of X. Denote by F(X ) the family of all fuzzy subsetsof X. For any A ∈ F(X ) and any a ∈ [0, 1], define A[a] = {x ∈ X | A(x)≥a}, called the a-cut set of A. For a familyS ⊆ [0, 1]X and a ∈ [0, 1], denote Sa = {A[a]| A ∈ S}.

A weighted function on a set X is a map f : X −→ R, where R is the set of all real numbers. If X is finite, then everyweighted function f can be formulated as a map f : X −→ [0, 1], which is exactly a fuzzy subset of X. In this paper, aweighted function on a finite set always means a fuzzy subset of the given set.

2.2. Graphs and matroids

For graph and matroid theory, we refer to [5,17].Let E be a set and A ⊆ 2E . Define

MaxA= {A ∈ A| A ⊆ B ∈ A implies A = B},MinA= {A ∈ A| A ⊇ B ∈ A implies A = B},UppA= {X ∈ 2E | there exists A ∈ A such that A ⊆ X},LowA= {X ∈ 2E | there exists A ∈ A such that X ⊆ A},OppA= {X ∈ 2E | X /∈ A}.

A (crisp) matroid is a pair (E, I), where E is a finite set and I ⊆ 2E satisfies

(I1) ∅ ∈ I;(I2) A ⊆ B ∈ I implies A ∈ I;(I3) for A, B ∈ I, if |A| < |B|, then there exists e ∈ B − A such that A ∪ e ∈ I.

A weighted matroid (E, I, w) is a matroid (E, I) equipped with a weighted function w : E −→ [0, 1].For a matroid (E, I), the family BI = Max I is called the family of bases, a member of BI is called a base of the

matroid (E, I), in other words, a base is a maximal element of I. The family CI = Min(Opp(I)) is called the familyof circuits, a member of CI is called a circuit of the matroid (E, I), in other words, a circuit is a maximal element ofOpp(I). Every matroid can be characterized by axioms of bases and that of circuits [17].

Two matroids (Ei , Ii ) (i ∈ I ) are called isomorphic if there exists a bijection f : E1 −→ E2 such that A ∈ I1 ifff (A) ∈ I2. It is well-known that, for a given graph a matroid can be naturally constructed. For explicit, let E be theedge set of a graph G and let I be the family of subsets of E that do not contain any of the edges of cycle of G. Then(E, I) is a matroid, which is called the circuit matroid or the cycle matroid of G. A matroid is called graphic if it isisomorphic to the circuit matroid of a graph, which is very important in matroid theory [17]. Of course, each weightedgraph can induce a weighted matroid with the weighted function unchanging.

The idea of a fuzzy graph was introduced by Rosenfeld [18] and has been applied in a lot of fields, such as the theoryof fuzzy clustering [22] and some optimization problems [4,12]. A fuzzy graph is a quadruple G = (V, E, �, �), whereE ⊆ V × V and � ∈ F(E), � ∈ F(V ) satisfying �(e)≤ min{�(x), �(y)}(∀e = (x, y) ∈ E). If � ≡ 1, then G is calleda graph of crisp vertex set and fuzzy edge set [1], which is indeed a weighted graph.

W. Yao, F.-G. Shi / Fuzzy Sets and Systems 161 (2010) 3155–3165 3157

2.3. Fuzzy matroids and fuzzy pre-matroids

A Goetschel–Voxman fuzzy matroid [7] is a pair (E, �), where E is a set and � ⊆ [0, 1]E is a nonempty family thatsatisfies

(FM1) if � ∈ �, � ∈ [0, 1]E and �≤�, then � ∈ �;(FM2) if �, � ∈ � and |supp �| < |supp �|, then there exists � ∈ � such that

(a) � < �≤� ∨ �;(b) m(�)≥ min{m(�), m(�)}.

where supp � = {x ∈ E | �(x) > 0}, m(�) = min{�(x)| x ∈ supp �} and � < � means �≤� but � � �.A fuzzy subset � ∈ [0, 1]E is called a base [8] (resp., circuit [9]) of (E, �) if � ∈ � (resp., � /∈ �) and it is maximal

(resp., minimal) in � (resp., [0, 1]E − �).For a fuzzy matroid (E, �), the pair (E, �r ) is a crisp matroid for each r ∈ (0, 1] [7].A Novak fuzzy pre-matroid [15] is a pair (E,I), where I ⊆ [0, 1]E satisfies (FM1) and for each r ∈ (0, 1], (E,Ir )

is a crisp matroid ((E,Ir ) is called the r-level of (E,I)). Obviously, a Novak fuzzy pre-matroid is an extension of aGoetschel–Voxman fuzzy matroid.

1 Since there are only a finite number of distinct crisp matroids that can be defined on a finite set E, there are onlya finite number of distinct r-levels of (E,I). According to [15], the family of levels of a fuzzy pre-matroid is totallyordered with respect to crisp inclusion. More precisely, suppose (E,I) is a fuzzy pre-matroid whose levels are (E,Ir )for r ∈ (0, 1]. Then the following holds: if r ′, r ′′ ∈ (0, 1] and r ′ < r ′′ then Ir ′ ⊇ Ir ′′ and consequently everyfuzzy pre-matroid introduces a partition of the interval (0, 1] into a finite number of subintervals, every subintervalcorresponding to a same level matroid. This fact has been already pointed out by Goetschel and Voxman [7] for fuzzymatroids. Denote the set of all end points of these subintervals by F(I), which is finite and uniquely associated witha fuzzy pre-matroid (E,I). If every subinterval defined by two consecutive members of F(I) is right-closed, thenfuzzy pre-matroid (E,I) is said to be closed (this concept was introduced for the first time in [7] for fuzzy matroids).

2.4. Cut matroids and strong cut matroids of fuzzifying matroids

Definition 2.1 (Shi [19]). Let E be a finite set and I : 2E −→ [0, 1] be a map satisfying:

(FI1) I(∅) = 1;(FI2) if A ⊆ B, then I(A)≥I(B);(FI3) if |A| < |B|, then there exists e ∈ B − A such that I(A ∪ e)≥ min{I(A),I(B)}.The pair (E,I) is called a fuzzifying matroid on E.

In Definition 2.1, if we replace [0, 1] by {0, 1}, then a fuzzifying matroid reduces to a crisp matroid. We call afuzzifying matroid (E,I) trivial if I(E) = 1 (that is I ≡ 1).

Remark 2.2. 2 A fuzzy measure or a non-additive measure [2,3] on set X is a function � : 2E −→ [0, 1] satisfying�(∅) = 0, �(X ) = 1 and �(A)≤�(B)(∀A ⊆ B ⊆ X ). Let (E,I) be a nontrivial fuzzifying matroid, define �I = 1 −I,that is �I(A) = 1 − I(A)(∀A ⊆ E). Then �I is a special fuzzy measure on E.

Let I : 2E −→ [0, 1] be an arbitrary map. For r ∈ (0, 1], define I[r ] = {A ⊆ E | I(A)≥r} and for s ∈ [0, 1),define I(s) = {A ⊆ E | I(A) > s}. If (E,I) be a fuzzifying matroid, then (E,I[r ]) (resp., (E,I(s))) is a crisp matroidon E (a detailed proof will be given in Theorem 2.8 below), called the r-cut matroid (resp., s-strong cut matroid)of (E,I) [19].

Example 2.3. Let E = {a, b, c} and define I : 2E −→ [0, 1] by

∅�1; {b}, {c}, {b, c}�0.7; {a}, {a, b}�0.3; {a, c}, E�0.

1 This paragraph is partly cited from [15,16].2 This is told by one reviewer, thanks.


Then (E,I) is a fuzzifying matroid [19, Example 3]. Clearly, the r-cut matroids are

J[r ] =

⎧⎪⎨⎪⎩

{∅, {a}, {b}, {c}, {a, b}, {b, c}}, r ∈ (0, 0.3],

{∅, {b}, {c}, {b, c}}, r ∈ (0.3, 0.7],

{∅}, r ∈ (0.7, 1].

The corresponding closed fuzzy pre-matroid is

� = {�, � ∈ [0, 1]E | �(a) = 0, �(b)≤0.7, �(c)≤0.7; �(a)≤0.3, �(b)≤0.7, �(c) = 0},which exactly has two bases 0/a + 0.7/b + 0.7/c and 0.3/a + 0.7/b + 0/c.

Theorem 2.4. Let (E,I) be a fuzzifying matroid and for r ∈ (0, 1], (E,I[r ]) be its r-cut matroid. Then there is a finitesequence 0 = r0 < r1 < · · · < rn = 1 such that

(1) if ri < a, b≤ri+1, then I[a] = I[b], 0≤i≤n − 1;(2) if ri < a≤ri+1 < b≤ri+2, then I[a]�I[b], 0≤i≤n − 2.

Analogously, we have:

Theorem 2.5. Let (E,I) be a fuzzifying matroid and for s ∈ [0, 1), (E,I(s)) be its s-strong cut matroid. Then there isa finite sequence 0 = s0 < s1 < · · · < sn < 1 such that

(1) if si≤a, b < si+1, then I(a) = I(b), 0≤i≤n − 1;(2) if si≤a < ri+1≤b < si+2, then I(a)�I(b), 0≤i≤n − 2.

In order to prove Theorems 2.4 and 2.5, we need some lemmas first.Now, we shall define an equivalence relation ∼ on (0,1] by a ∼ b ⇔ I[a] = I[b](∀a, b ∈ (0, 1]). Since E is a

finite set, the number of crisp matroids on E is also finite. That is, there exist at most finitely many equivalence classes,respectively, denoted by A1, A2, . . . , An .

Lemma 2.6.(1) For any a, b ∈ (0, 1], if a≤b, then I[a] ⊇ I[b].(2) Each Ai (i = 1, 2, . . . , n) is an interval.

Proof. Trivial. �

Lemma 2.7. For any i ∈ {1, 2, . . . , n}, sup Ai ∈ Ai . Thus each Ai is a left open and right closed interval.

Proof. Suppose ri = sup Ai and (E, I) is the corresponding cut matroid with respect to Ai . Then I ⊇ I[ri ] holdsobviously. For all A ∈ I, A ∈ I[r ] and I(A)≥r for all r ∈ Ai . It follows that I(A)≥ri and A ∈ I[ri ]. Thus, I ⊆ I[ri ].Hence I = I[ri ] and ri ∈ Ai . �

By Lemma 2.7, we obtain a sequence r1, r2, . . . , rn ∈ (0, 1] which separates different cut matroids from each other.Suppose that r1 < r2 < · · · < rn . Then rn must be equal to 1. Thus Theorem 2.4 can be easily proved and Theorem2.5 can be proved dually.

Theorem 2.8 (Shi [19, Corollary 7]). Let I : 2E −→ [0, 1] be a map. Then the following statements are equivalent:

(1) (E,I) is a fuzzifying matroid.(2) For any r ∈ (0, 1], (E,I[r ]) is a crisp matroid.(3) For any s ∈ [0, 1), (E,I(s)) is a crisp matroid.

Proof. (2) �⇒ (1): (FI1) Clearly, I(∅) = 1 since ∅ ∈ I[1]. (FI2) For A, B ⊆ E with A ⊆ B, suppose that I(B) = r .If r = 0, then I(A)≥0 = I(B). If r � 0, then B ∈ I[r ], which implies A ∈ I[r ] and I(A)≥r = I(B). (FI3)


Suppose that A, B ⊆ E and |A| < |B|. Let r = min{I(A),I(B)}. If r = 0, then for any e ∈ B − A, we haveI(A ∪ e)≥0 = min{I(A),I(B)}. If r � 0, then A, B ∈ I[r ] and there exists e ∈ B − A such that A ∪ e ∈ I[r ] andI(A ∪ e)≥r = min{I(A),I(B)}.

(1) �⇒ (3): Suppose that (E,I) is a fuzzifying matroid and s ∈ [0, 1). (I1) Clearly, ∅ ∈ I(s). (I2) For A ⊆ B ∈ I(s),we have I(A)≥I(B) > s, then A ∈ I(s). (I3) Suppose that A, B ∈ I(s) and |A| < |B|, then there exists e ∈ B − Asuch that I(A ∪ e)≥ min{I(A),I(B)} > s and then A ∪ e ∈ I(s).

(3) �⇒ (2): For r ∈ (0, 1], we only need to show that there exists s ∈ [0, 1) such that I[r ] = I(s). If there is noA ⊆ E such that I(A) = r , then r � 1 and I[r ] = I(r ). If there is no A ⊆ E such that I(A) < r , put s = r/2, thens ∈ [0, 1) and I[r ] = I(s) = 2E . Otherwise, there is a biggest r1 ∈ [0, 1) such that r1 < r and there exist some A ⊆ Ewith I(A) = r1, put s = (r1 + r )/2, then s ∈ [0, 1) and I[r ] = I(s). �

3. Weighted matroids as fuzzifying matroids

In this section, we will show that weighted graphs are examples of fuzzifying matroids.Let (E, I) be a weighted matroid [5] with the weighted function w : E −→ [0, 1]. Define II : 2E −→ [0, 1] for

A ⊆ E ,

II (A) ={

mine∈A

w(e), A ∈ I,

0 otherwise.

By Theorem 2.8, it is easy to verify that (E,II ) is a fuzzifying matroid.

Remark 3.1.(1) Since a weighted matroid can be naturally constructed from a weighted graph, namely the circuit matroid with the

weighted function unchanging, weighted graphs also are examples of fuzzifying matroids.(2) If (E, I, w) is a loop-free weighted matroid (i.e., a weighted matroid with no loop), then for all e ∈ E, {e} ∈ I,

and then for all e ∈ E, w(e) = II ({e}). Thus both independent sets and weighted function of a loop-free weightedmatroid (E, I, w) can be nicely formulated by the corresponding fuzzifying matroid (E,II ) with w = II |{{e}|e∈E},the map II restricted on {{e}|e ∈ E}.

(3) The loop in a weighted graph in many practical examples maybe has no use. For example, the cities in a map andthe roads between them consists a weighted graph with the length of roads as the weight. When one want to travelfrom one city to another as quickly as possible, loops will never be walked through. Thus loops in these cases areof no use.

Let G be a weighted graph. Denote by M[G] = (E, I[G]) the corresponding weighted matroid of G andM[G] =(E,II[G]).

Two fuzzifying matroidsM1 = (E1,I1) andM2 = (E2,I2) are called isomorphic, in symbolsM1�M2, if thereexists a bijection f : E1 −→ E2 such that for all A ⊆ E1, I1(A) = I2( f (A)).

Definition 3.2. We call a fuzzifying matroidM = (E,I) to be graphic ifM�M[G] for a weighted graph G.

Example 3.3. Let E = {x, y, z} and define I : 2E −→ [0, 1] by E�0; {x}, {x, y}, {x, z}� 14 ; {y}, {y, z}� 1

2 ; {z},∅�1. Then (E,I) is a graphic fuzzifying matroid. One corresponding weighted graph is

��

��

��

��

�

�

x 14

y 12

z1

where 14 , 1

2 , 1 are the weights of x, y, z, respectively.


4. Bases axioms for fuzzifying matroids

For some fuzzy matroids, as the following example shows, maybe there is no base or circuit.

Example 4.1. (1) (A fuzzy matroid with no base) Let E be a finite set and � = {A ∈ [0, 1]E | ∀e ∈ E, A(e) < 0.5}.Then (E, �) is a Goetschel–Voxman fuzzy matroid, which has no base.

(2) (A fuzzy matroid with no circuit) Let E be a finite set and � = {A ∈ [0, 1]E | ∀e ∈ E, A(e)≤0.5}. Then (E, �)is a closed Goetschel–Voxman fuzzy matroid, which has no circuit.

In this section (resp., the following section), we will establish axioms of bases (resp., circuits) for fuzzifying matroids.These results show a fuzzifying matroid is better than a fuzzy matroid in these aspects.

Recall in a matroid (E, I), BI = Max I is called the family of bases of I, which has the following property:

(B) LowBI = I.

It is easy to see if B ⊆ 2E has the property (B), then we have BI ⊆ B. That is to say, BI is the least family of subsetsof E with the property (B). Based on which, we give the following definition.

Definition 4.2. Suppose that (E,I) is a fuzzifying matroid. We call a map P : 2E −→ [0, 1] a quasi-base-map of(E,I) if for all A ⊆ X, I(A) = ∨

A⊆B P(B).

Clearly, if (E,I) is a fuzzifying matroid, then I itself is a quasi-base-map.Let (E,I) be a fuzzifying matroid. Define PI : 2E −→ [0, 1] by

PI(B) =∨

{a ∈ (0, 1]| B ∈ B(I[a])} (∀B ⊆ E).

By Theorem 2.4, we have for any B ∈ 2E ,

PI(B) ={

max{a ∈ (0, 1]| B ∈ B(I[a])} if B ∈ B(I[a]) for some a ∈ (0, 1],

0 otherwise.

Theorem 4.3. Let (E,I) be a fuzzifying matroid. Then PI is the minimal quasi-base-map of (E,I), called thebase-map.

Proof. (1) PI is a quasi-base-map. Let A ⊆ E . For any B ⊆ E with A ⊆ B, if B ∈ B(I[a]) for some a � 0, thenB ∈ I[a] and a≤I(B)≤I(A). Thus PI(B)≤I(A) and by the arbitrariness of B, I(A)≥∨

A⊆B PI(B). Conversely,suppose that I(A) = a � 0. Then A ∈ I[a] and there exists B ∈ B(I[a]) such that A ⊆ B. Thus PI(B)≥a = I(A) andI(A)≤ ∨

A⊆B PI(B).(2) Minimality ofPI. Suppose that F is a quasi-base-map of (E,I), i.e., for any A ⊆ E,I(A) = ∨

A⊆B F(B). Forany A ⊆ E , if A ∈ B(I[a]) ⊆ I[a] for some a � 0, then I(A)≥a. Then there exists B ⊇ A such that F(B) = I(A)≥a.It is easy to see that I(B)≥F(B)≥a and B ∈ I[a]. Since A is a base of (E,I[a]), we have B = A. Therefore F(A)≥aand then F(A)≥PI(A). Hence F≥PI. �

Theorem 4.4. Let (E,I) be a fuzzifying matroid. Then

(1) for any a ∈ (0, 1], Max(PI)[a] = B(I[a]);(2) for any B ∈ 2E , if PI(B) = a � 0, then B ∈ Max(PI)[a].

Proof. (1) Suppose that a > 0. For any B ∈ Max(PI)[a], we have PI(B)≥a and then there exists b≥a such thatB ∈ B(I[b]) ⊆ I[b] ⊆ I[a]. There exists A ∈ B(I[a]) such that B ⊆ A. Then by I(A) = ∨

A⊆C PI(C), there existsC ⊇ A such that PI(C)≥a and C ∈ (PI)[a]. By the maximality of B in (PI)[a], we have B = C ∈ B(I[a]). ThusMax(PI)[a] ⊆ B(I[a]). Conversely, for any B ∈ B(I[a]), by the definition of PI, we have PI(B)≥a and B ∈ (PI)[a].Suppose that B ⊆ A ∈ (PI)[a]. Then PI(A)≥a and there exists b≥a such that A ∈ B(I[b]) ⊆ I[b] ⊆ I[a]. By themaximality of B in I[a], we have B = A. Thus B ∈ Max(PI)[a]. Therefore Max(PI)[a] ⊇ B(I[a]).

(2) Suppose that PI(B) = a � 0. Then B ∈ B(I[a]) = Max(PI)[a]. �


Remark 4.5. In Theorem 4.4, the conclusion (1) cannot be replaced by

(PI)[a] = B(I[a]) (∀a ∈ (0, 1]).

That is, Max(PI)[a] = (PI)[a] does not hold generally for a given fuzzifying matroid (E,I). Also the conclusion (2)cannot be rewritten as

(PI)[a] = {B ⊆ E | BI(B) = a} (∀a ∈ (0, 1]).

For example, for the fuzzifying matroid (E,I) in Example 3.3. It is easy to verify that I[1/4] = 2E\{E}, I[1/2] =2{y,z},I[1] = {∅, {y}}. Then PI({x, y}) = PI({x, z}) = 1

4 ,PI({y, z}) = 12 ,PI({y}) = 1 and 0 otherwise. Thus

(PI)[1/4] = {{x, y}, {y, z}, {x, z}} and (PI)[1/2] = {{y}, {y, z}}.

Theorem 4.6. Let P : 2E −→ [0, 1] be a map satisfying

(FB1) for any a ∈ (0, 1], MaxP[a] is a family of bases for a crisp matroid on E;(FB2) for any B ∈ 2E , if P(B) = a � 0, then B ∈ MaxP[a].

Then there is a unique fuzzifying matroid with P as its base-map.

Proof. Define IP : 2E −→ [0, 1] by

IP(A) =∨A⊆B

P(B) (∀A ∈ 2E ).

Step 1: (E,IP) is a fuzzifying matroid. (FI2) is obvious and (FI1) holds since MaxP[1] is a family of bases forsome matroids on E. To show (FI3), for all A, B ∈ 2E with |A| < |B|. Put a = IP(A) ∧IP(B). If a = 0, then for anye ∈ B − A,IP(A ∪ e)≥0 = IP(A) ∧ IP(B). In the following, we suppose that a > 0. Then IP(A)≥a,IP(B)≥a

and then there exist A1, B1 ∈ 2E with A ⊆ A1, B ⊆ B1 such thatP(A1)≥a,P(B1)≥a. Put I = Low(MaxP[a]), wehave A1, B1 ∈ I and then A, B ∈ I by (I2). Thus there exists e ∈ B − A such that A ∪ e ∈ I. Then A ∪ e ⊆ C forsome C ∈ MaxP[a] ⊆ P[a]. Hence IP(A ∪ e)≥P(C)≥a = IP(A) ∧ IP(B).

Step 2: For any a ∈ (0, 1], MaxP[a] ⊆ B((IP)[a]). In fact, for any B ∈ MaxP[a], we have IP(B)≥P(B)≥a andB ∈ (IP)[a]. Suppose that B ⊆ A ∈ (IP)[a]. Then IP(A)≥a and then there exists C ⊇ A such that P(C)≥a. ThusC ∈ P[a]. By the maximality of B in P[a], B = C = A. It follows that B ∈ B((IP)[a]).

Step 3:P = PIP . By the definition of IP,P is a quasi-base-map of (E,IP) and by Theorem 4.3, we only need toshow that P≤PIP . For any B ∈ 2E , suppose that P(B) = a � 0. By Step 2 and (FB2), B ∈ B((IP)[a]) and then by thedefinition of PIP , PIP (B)≥a = P(B). �

Corollary 4.7. (1) P : 2E −→ [0, 1] is the base-map for a fuzzifying matroid if and only if P satisfies (FB1) and(FB2).

(2) Let (E,I) be a fuzzifying matroid. Then IPI = I.(3) If P : 2E −→ [0, 1] satisfies (FB1) and (FB2), then PIP = P.

Theorem 4.8. Let (E,I) be a fuzzifying matroid and B ∈ 2E . If PI(B) � 0, then PI(B) = I(B).

Proof. Obviously, PI(B)≤I(B). Suppose that PI(B) = a � 0. Then B ∈ B(I[a]). We only need to prove that for anyB ⊆ A, PI(A)≤a. If PI(A)≥a, then A ∈ (PI)[a]. And there exists C ∈ Max(PI)[a] = B(I[a]) such that A ⊆ C . Itfollows that B = C = A. This completes the proof. �

Corollary 4.9. Let (E,I) be a fuzzifying matroid. Then for any B ∈ 2E ,

PI(B) ={I(B) if B ∈ B(I[a]) for some a ∈ (0, 1],

0 otherwise.


Example 4.10. In Example 2.3, the base-map PI of (E,I) is computed as

∅�1; {b, c}�0.7; {a, b}�0.3; {a}, {b}, {c}, {a, c}, E�0.

In Example 3.3, the base-map PI of (E,I) is computed as

{z}�1; {y, z}� 12 ; {x, y}, {x, z}� 1

4 ; ∅, {x}, {y}, E�0.

5. Circuits axioms for fuzzifying matroids

In this section, we will establish axioms of circuits for fuzzifying matroids. Recall in a matroid (E, I), CI =Min(Opp(I)) is called the family of circuits of I, which has the following property:

(C) Upp CI = Opp(I).

It is easy to see if C ⊆ 2E has the property (C), then we have CI ⊆ C. That is to say, CI is the least family of subsetsof E with the property (C). Based on which, we define:

Definition 5.1. Suppose that (E,I) is a fuzzifying matroid. We call a map C : 2E −→ [0, 1] is a quasi-circuit-map of(E,I) if for all A ⊆ X, I(A) = 1 − ∨

B⊆A C(B).

It is easy to see that if (E,I) is a fuzzifying matroid, then 1−I (this means for any A ⊆ E, (1−I)(A) = 1−I(A))is a quasi-circuit-map.

Let (E,I) be a fuzzifying matroid. Define CI : 2E −→ [0, 1] by

∀C ∈ 2E , CI(C) = 1 −∧

{a ∈ [0, 1)| C ∈ C(I(a))}.

Then by Theorem 2.5, we have for all C ∈ 2E ,

CI(C) ={

1 − min{a ∈ [0, 1)|C ∈ C(I(a))} if C ∈ C(I(a)) for some a ∈ [0, 1),

0 otherwise.

Theorem 5.2. Let (E,I) be a fuzzifying matroid. Then CI is the minimal quasi-circuit-map of (E,I), called thecircuit-map.

Proof. Analogize to Theorem 4.3. �

Theorem 5.3. Let (E,I) be a fuzzifying matroid. Then

(1) for any a ∈ (0, 1], Min(CI)[a] = C(I(1−a));(2) for any C ∈ 2E , if CI(C) = a � 0, then C ∈ Min(CI)[a].

Proof. (1) Suppose that a � 0. For any C ∈ Min(CI)[a], we have CI(C)≥a and then there exists b≤1 − a such thatC ∈ C(I(b)) and C /∈ I(b). Thus C /∈ I(1−a) and there exists A ∈ C(I(1−a)) such that A ⊆ C . Then I(A)≤1 − a and byI(A) = 1−∨

B⊆A CI(B), there exists B ⊆ A such that CI(B)≥a and B ∈ (CI)[a]. By the minimality of C in (CI)[a],we have C = B = A ∈ C(I(1−a)). Thus Min(CI)[a] ⊆ C(I(1−a)). Conversely, for any C ∈ C(I(1−a)), by the definition ofCI, we have CI(C)≥a and C ∈ (CI)[a]. Suppose that C ⊇ B ∈ (CI)[a]. Then CI(B)≥a and there exists b≤1 − asuch that B ∈ C(I(b)). Then B /∈ I(b) and B /∈ I(1−a). By the minimality of C in Opp(I(1−a)), we have C = B andthen C ∈ Min(CI)[a]. Thus Min(CI)[a] ⊇ C(I(1−a)).

(2) is similar to Theorem 4.4(2). �

Remark 5.4. Analogizing to Remark 4.5, in Theorem 5.3, the conclusion (1) cannot be replaced by

(CI)[a] = CI(1−a) (∀a ∈ [0, 1)).


That is, Min(CI)[a] = (CI)[a] does not hold generally for a given fuzzifying matroid (E,I). Also the conclusion (2)cannot be rewritten as

(CI)[a] = {C ⊆ E | CI(C) = a} (∀a ∈ [0, 1)).

We still can take the example in Remark 4.5 to check the two claims. Since it analogizes to that in Remark 4.5, we omitit here.

Theorem 5.5. Let C : 2E −→ [0, 1] be a map satisfying

(FC1) for any a ∈ [0, 1), MinC[a] is a family of circuits for a crisp matroid on E;(FC2) for any C ∈ 2E , if C(C) = a � 0, then C ∈ MinC[a].

Then there is a unique fuzzifying matroid with C as its circuit-map.

Proof. Define IC : 2E −→ [0, 1] by

IC(A) = 1 −∨B⊆A

C(B) (∀A ∈ 2E ).

Analogizing to Theorem 4.6, the proof is separated into three steps: Step 1, (E,IC) is a fuzzifying matroid; Step 2,for any a � 0, MinC[a] ⊆ C((IC)(1−a)); and Step 3, C = CIC . Here we only prove Step 1 since the other two are similarto that in Theorem 4.6.

We only need to show that for any a ∈ [0, 1), (IC)(a) is a crisp independence system on E by Theorem 2.8. We willshow that for any a ∈ [0, 1), Opp((IC)(a)) = Upp(MinC[1−a]). Indeed, A ∈ Opp((IC)(a)) iff IC(A)�a iff IC(A)≤aiff

∨B⊆A C(B)≥1 − a iff ∃B ⊆ A such that C(B)≥1 − a iff ∃B ⊆ A such that B ∈ C[1−a] iff ∃C ⊆ A such that C ∈

MinC[1−a] iff A ∈ Upp(MinC[1−a]). Hence Opp((IC)(a)) = Upp(MinC[1−a]) and (IC)(a) = Opp(Upp(MinC[1−a]))is a crisp independence system on E. �

Corollary 5.6. (1) C : 2E −→ [0, 1] is the circuit-map for a fuzzifying matroid if and only if P satisfies (FC1) and(FC2).

(2) Let (E,I) be a fuzzifying matroid. Then ICI = I.(3) C : 2E −→ [0, 1] satisfies (FC1) and (FC2), then CIC = C.

Theorem 5.7. Let (E,I) be a fuzzifying matroid and C ∈ 2E . If CI(C) � 0, then CI(C) = 1 − I(C).

Proof. Analogize to Theorem 4.8. �

Corollary 5.8. Let (E,I) be a fuzzifying matroid. Then for any C ∈ 2E ,

CI(C) ={

1 − I(C) if C ∈ C(I(a)) for some a ∈ [0, 1),

0 otherwise.

Example 5.9. In Example 2.3, the circuit-map CI of (E,I) is computed as

{a, c}�1; {a}�0.7; {b}, {c}�0.3; ∅, {a, b}, {b, c}, E�0.

In Example 3.3, the circuit-map CI of (E,I) is computed as

E�1; {x}� 34 ; {y}� 1

2 ; A�0 (|A| = 2 or A = {y}).

6. Conclusions and remarks

This paper shows that weighted graphs (and even weighted matroids) are examples of fuzzifying matroids. Sinceweighted matroids are important in application in some combinatorial optimization problems, as has been claimed in


[19], it is possible that fuzzifying matroids also would be applied to some combinatorial optimization problems. Thispaper also gives the axioms of base-maps and that of circuit-maps for fuzzifying matroids. That is, each fuzzifyingmatroids can be uniquely determined by its base-map or its circuit-map. The conclusions in this paper and that in[19,20] indicate that the definition of fuzzifying matroids is reasonable. While for some fuzzy matroids, as Example4.1 has shown, maybe there is no base or circuit. Therefore, if we want to study a fuzzy matroid, certainly first is a crispsubfamily of fuzzy subsets of a finite set which satisfies several axioms, we should try hard to find a more reasonabledefinition than that in the sense of Goetschel and Voxman.

Remark 6.1. In this paper, we only study fuzzifying matroids instead of L-fuzzifying matroids in [19]. It does notindicate that we cannot transplant the results in this paper onto a completely distributive lattice. Indeed, we confirm thatthe results in this paper can even be transplanted onto some more general lattices than completely distributive lattices.The reason why we only study the case L = [0, 1] is that, since weighted graphs are examples of fuzzifying matroids,when the results are put into practice, the unit interval [0, 1] is more effective than any other abstract lattices.

Remark 6.2. Following one reviewer’s comments and also as is stated in Remark 2.2, if (E,I) is a fuzzifying matroid,then �I = 1 − I is a fuzzy measure on E. The base-map (resp., circuit-map) for (E,I) is somewhat like to form ofmaximal sets (resp., minimal sets) [2,3] for �I.

Remark 6.3. Every crisp matroid (E, I) can be regarded as a fuzzifying matroid (E, �I ). Conversely, for a fuzzifyingmatroid (E,I), we can make many crisp matroids, besides which, (E,I[1]) and (E,I(0)) are the simplest two. For acrisp matroid (E, I), it is easy to check that (�I )[1] = (�I )(0) = I. For a fuzzifying matroid (E,I), it is also easy tocheck that �(I[1])≤I and �(I(0))≥I. Categorically speaking, crisp matroids can be embedded in fuzzifying matroids asa simultaneously reflective and coreflective subcategory [13].

Acknowledgements

The authors are thankful to the anonymous reviewers for their constructive comments and suggestions.

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