Axioms for fuzzy bases of Hsueh fuzzy matroids

1 Introduction

The concept of matroid was coined by Whitney in 1935 to study an abstract theory of independence. Matroids also play an important role in combinatorial optimization. It is well-known that for an independent set systems of a finite set, the problem of finding a maximum weight set can be solved by a greedy algorithm if and only if the system is a matroid. Generalizations of matroids interest many scholars, and when we generalize matroids and related optimization problems to fuzzy setting, there usually exist two natural ways. One is using interval weights or something like to instead of precise values in a matroid-based optimization problems; the other is to fuzzify matroids.

Goetschel and Voxman [1] first introduced fuzzy matroids (named G-V fuzzy matroids) in 1988. The greedy algorithm and other related topics concerning G-V fuzzy matroids were discussed subsequently (see [2, 3]). Note if a fuzzy matroid on X is to be defined as a pair (X, Ψ), where Ψ is a nonempty family of fuzzy subsets of X satisfying some conditions, then this fuzzy matroid in fact is some structure on [0, 1]  ^X . Since [0, 1]  ^X is an infinite set, fuzzy matroids on X should be members of the class of infinite matroids. Based on this idea, Hsueh [4] in 1993 proposed his fuzzy matroids. [6] compares Hsueh fuzzy matroids with other four kinds of fuzzy matroids and studies fuzzy bases of Hsueh fuzzy matroids. As a continuation of these results, we studies axioms for fuzzy bases of Hsueh fuzzy matroids in this paper. When generalizing some axiom system of matroids to fuzzy setting, the corresponding axiom system usually becomes complicated and only determines some very special kind of fuzzy matroids (see, e.g. [5, 7, 10, 5, 7, 10]), while the main result in this paper shows that the fuzzy analog of base axioms for crisp matroids can determine general Hsueh fuzzy matroids. In addition, properties of fuzzy matroids are studied and H fuzzy matroids on two-element sets are characterized.

The arrangement of this paper is as follows. In the next section, we present some fundamental results which will be needed in this paper. Section 3 studies properties of H fuzzy matroids. we generalize the concept of submatroid to fuzzy setting and then two equivalent descriptions of H fuzzy matroids are given. After these preliminaries, fuzzy base axioms for H fuzzy matroids are obtained in Section Hsueh. Finally, a concluding remark is given in the last section.

2 Preliminaries

Now we present some fundamental notions and results on matroids, fuzzy sets and fuzzy matroids.

Definition 2.1. (Oxley[9]). A matroid M is a pair ( X , I ) consisting of a finite set X and a nonempty family I of subsets of X which satisfies the following two conditions:

(I1) If Y ∈ I and Z ⊆ Y, then Z ∈ I (hereditary property).(I2) If Y , Z ∈ I and |Y| < |Z| (where |Y| denotes the cardinality of Y), then there exists W ∈ I such that Y ⊂ W ⊆ Y ∪ Z (augmentation property).

Let M=(X, I) be a matroid. The members of I are the independent sets of M, maximal independent sets are bases of M. Matroids have many equivalent descriptions. For a more detailed discussion of axiom systems of matroids, see [9].

Let X be a finite set. A fuzzy set μ on X is a mapping μ : X ⟶ [0, 1]. We use [a] to denote the fuzzy set taking constant value a on X (a∈ [0, 1]) and χ _Y the characteristic function of Y (Y ⊆ X) . The fuzzy set [a] ∧ χ _{x}   (a > 0) (usually denoted by x _a ) is called a fuzzy point. We denote the family of all fuzzy sets on X by [0, 1]  ^X , and denote the set of all fuzzy points on X by J ([0, 1]  ^X ). For any μ, ν ∈ [0, 1]  ^X , we write μ ≤ ν if μ (x) ≤ ν (x) for each x ∈ X, and μ < ν if μ ≤ ν and μ ≠ ν. μ ∨ ν and μ ∧ ν are fuzzy sets which satisfy (μ ∨ ν) (x) = max {μ (x) , ν (x)} and (μ ∧ ν) (x) = min {μ (x) , ν (x)} (∀ x ∈ X). For each μ ∈ [0, 1]  ^X , the non-negative real number ∑_x∈X μ (x) (written as |μ|) is called the cardinality of μ; we write supp μ = {x ∈ X ∣ μ (x) >0}, m (μ)= min{μ (x)∣ x ∈ supp μ}, μ _[a] = {x ∈ X ∣ μ (x) ≥ a} (where 0 ≤ a ≤ 1),, R ⁺ (μ) = {μ (x) ∣ μ (x) >0, x ∈ X} and ↓μ = {ν ∣ μ ≥ ν, ν ∈ [0, 1]  ^X }.

Goetschel and Voxman generalized Definition 2.1 to fuzzy setting and introduced the concept of fuzzy matroids as follows.

Definition 2.2. (Goetschel and Voxman [1]). Let X be a finite set and Ψ ⊆ [0, 1]  ^X , then we call (X, Ψ) a G-V fuzzy matroid if it satisfies

(FI1) If μ ∈ Ψ, ν ∈ [0, 1]  ^X and ν < μ, then ν ∈ Ψ.(GVFI2) If μ, ν ∈ Ψ and 0 < |supp μ| < |supp ν|, then there exists ω ∈ Ψ such that

 μ < ω ≤ μ ∨ ν .

Definition 2.3. (Hsueh [4]). Suppose that X is a finite set and that Ψ ⊆ [0, 1]  ^X is a nonempty family of fuzzy sets satisfying:

(FI1) If μ ∈ Ψ, ν ∈ [0, 1]  ^X and ν < μ, then ν ∈ Ψ.(HFI2) If μ, ν ∈ Ψ and |μ| < |ν|, then there exists an element x ∈ X and a fuzzy subset ω ∈ Ψ such that ω (y) = μ (y) (∀ y ∈ X - {x}) and μ (x) < ω (x) ≤ ν (x) . (FI3) Let μ be a fuzzy subset of X. If {ν ∈ [0, 1]  ^X  ∣ ν < μ} ⊆ Ψ, then μ ∈ Ψ.

Then we call M = ( X, Ψ) an H fuzzy matroid.

Remark 2.4. (1) Let (X, Ψ) be a fuzzy independent set system (i.e. Ψ ⊆ [0, 1]  ^X and satisfies (FI1)), then the numbers of Ψ are called fuzzy independent sets and maximal fuzzy independent sets are named fuzzy bases.

(2) G-V fuzzy matroids and H fuzzy matroids are all special kinds of fuzzy independent set systems. That is to say, the difference between G-V fuzzy matroids and H fuzzy matroids just lies in using different ways to fuzzify (I2).

3 Properties of H fuzzy matroids

In this section, we begin with a result corresponding to submatroids in crisp matroids.

Proposition 3.1. Suppose that (X, Ψ) is an H fuzzy matroid and μ ∈ [0, 1]  ^X . Let Ψ| _μ  = {ν ∧ μ ∣ ν ∈ Ψ}, then (X, Ψ| _μ ) is an H fuzzy matroid.

Proof. Note that Ψ| _μ  = ↓ μ ∩ Ψ, it is easy to check that Ψ| _μ satisfies (FI1),(FI2) and (FI3).

Example 3.2. Let X = {x, y}, Φ = { μ ∈ [ 0 , 1 ] X ∣ μ ( x ) + μ ( y ) = 1 , 1 2 ≤ μ ( x ) ≤ 5 6 } and Ψ = ⋃ {↓ μ ∣ μ ∈ Φ}, then we can check that (X, Ψ) is an H fuzzy matroid. Again, let μ = x 2 3 ∨ y 1, then Ψ| _μ = { μ ∈ [ 0 , 1 ] X ∣ μ ( x ) + μ ( y ) = 1 , 1 2 ≤ μ ( x ) ≤ 2 3 }. Note that Ψ and Ψ| _μ have the same shape, we shall prove in Section Hsueh that all H fuzzy matroid on two-element sets have the same structure (see Proposition 4.2).

Remark 3.3. (1) Proposition 3.1 also holds for a G-V fuzzy matroid. That is, if (X, Ψ) is a G-V fuzzy matroid, then (X, Ψ| _μ ) is also a G-V fuzzy matroid (μ ∈ [0, 1]  ^X ).

(2) Fuzzy bases of the H fuzzy matroids (X, Ψ) and (X, Ψ| _μ ) in Example 3.2 have the same cardinality respectively. Actually, all fuzzy bases of an H fuzzy matroid have the same cardinality and we shall prove this result in the following. For a G-V fuzzy matroid (X, Ψ), fuzzy bases need not exist and even if (X, Ψ) has fuzzy bases and its fuzzy bases have the same cardinality, (X, Ψ| _μ ) does not necessarily have fuzzy bases with the same cardinality (see Example 3.4).

Example 3.4. Let X = {x, y}, Ψ = { ν ∈ [ 0 , 1 ] X ∣ ν ≤ x 1 2 ∨ y 1 or ν ≤ x 1 ∨ y 1 2 }, then we can check that (X, Ψ) is a G-V fuzzy matroid. Again, let μ = x 3 4 ∨ y 7 8, then (X, Ψ| _μ ) is a G-V fuzzy matroid, but the two fuzzy bases of (X, Ψ| _μ ) (i.e. x 1 2 ∨ y 7 8 and x 3 4 ∨ y 1 2 ) have different cardinalities.

The following result gives an equivalent description of (HFI2).

Proposition 3.5. Let (X, Ψ) be a fuzzy independent set system, then (HFI2) is equivalent to

(HFI2 ^∗) If μ, ν ∈ Ψ and |μ| < |ν|, then there exists ω ∈ Ψ such that μ < ω ≤ μ ∨ ν .

Proof. (HFI2)⇒ (HFI2 ^∗) is trivial, and for the converse, we suppose that ω is a fuzzy independent set and satisfies μ < ω ≤ μ ∨ ν . {x ∈ X|μ (x) < ω (x)} is obviously nonempty, then we choose x ₀ ∈ {x ∈ X|μ (x) < ω (x)} and define ω _{x 0}  ∈ [0, 1]  ^X as follows: ω x 0 ( x ) = { ω ( x 0 ) , x = x 0 , μ ( x ) , x ∈ X - { x 0 } .

By (FI1), ω _{x 0}  ∈ Ψ, and then (HFI2) holds obviously.

Proposition 3.6. Suppose that (X, Ψ) is an H fuzzy matroid, then (X, Ψ) has fuzzy bases.

Proof. We prove existence of fuzzy bases by induction on |X|. When |X|=1 or 2, it can be easily verified that (X, Ψ) has fuzzy bases. Assume that the assertion holds when |X| = k. Now let X = {x ₁, x ₂, . . . , x _k+1} and we shall prove that the H fuzzy matroid (X, Ψ) has fuzzy bases in the following. Again, let X _k = {x ₁, x ₂, . . . , x _k }, then it holds from Proposition 3.1 that (X _k , Ψ| _{χ X k} ) is an H fuzzy matroid. By the induction hypothesis,(X _k , Ψ| _{χ X k} ) has fuzzy bases. Assume that μ is a fuzzy base of (X _k , Ψ| _{χ X k} ), then define

b _μ = sup{a ∈ [0, 1] ∣ μ ∨ (x _k+1)  _a  ∈ Ψ}

If b _μ  = 0, then μ is obviously a fuzzy base of (X, Ψ). Thus we consider the case b _μ  > 0 in the following. Let

b ^∗ = sup{b _ν  ∣ ν is a fuzzy base of (X _k , Ψ| _{χ X k} )}.

Again, let

Ψ _k = {ν ∈ [0, 1]  ^{X _k}  ∣ ν ∨ (x _k+1)  _{b ^∗}  ∈ Ψ}

Note (x _k+1)  _{b ^∗}  ∈ Ψ, thus [0] ∈ Ψ _k . That is Ψ _k ≠ ∅.

We can routinely check that Ψ _k satisfies (FI1),(FI2) and (FI3), thus (X _k , Ψ _k ) is an H fuzzy matroid. Use the induction hypothesis again, there is a fuzzy base μ k ∗ of (X _k , Ψ _k ). Note Ψ _k  ⊆ Ψ| _{χ X k} , by Proposition 3.5, we can easily verify that | μ k ∗ | ≤ | μ |.

We assume | μ k ∗ | < | μ | and derive a contradiction. Let c = min{ | μ | - | μ k ∗ | , b ∗ }. By the definition of b ^∗, we can find a fuzzy base μ ^★ of (X _k , Ψ| _{χ X k} ) such that b ^∗ - b _{μ ^★}  < c/4 . Let d = b _{μ ^★}  - c/4, then μ ^★ ∨ (x _k+1)  _d  ∈ Ψ holds from the definition of b _{μ ^★} . Note μ ^★ and μ are fuzzy bases of (X _k , Ψ| _{χ X k} ), Proposition 3.5 can easily imply |μ ^★| = |μ|. Because

| μ ★ ∨ ( x k + 1 ) d | = | μ ★ | + d > | μ k ∗ | + c + d

= | μ k ∗ | + b μ ★ + 3 c / 4 > | μ k ∗ | + b ∗ + c / 2

> | μ k ∗ ∨ ( x k + 1 ) b ∗ | ,

we have ω ∈ Ψ satisfying μ k ∗ ∨ ( x k + 1 ) b ∗ < ω ≤ ( μ k ∗ ∨ ( x k + 1 ) b ∗ ) ∨ ( μ ★ ∨ ( x k + 1 ) d ). Obviously, ω (x _k+1) = b ^∗, thus there exists x ₁ ∈ X _k such that ω ( x 1 ) > μ k ∗ ( x 1 ). Therefore, ω ∧ χ _{X k}  ∈ Ψ and ω ∧ χ X k > μ k ∗, contrary to the fact that μ k ∗ is a fuzzy base of (X _k , Ψ _k ).

Finally, we show that μ k ∗ ∨ ( x k + 1 ) b ∗ is a fuzzy base of (X, Ψ) and our lemma is proved. Since μ k ∗ ∨ ( x k + 1 ) b ∗∈Ψ, we need only to show that for any υ ∈ [0, 1]  ^X and υ > μ k ∗ ∨ ( x k + 1 ) b ∗ , υ ∉ Ψ holds. Two cases are considered in the following.

Case 1: υ (x _k+1) = b ^∗. | μ k ∗ | = | μ | has been proved, note that μ is a fuzzy base of (X _k , Ψ| _{χ X k} ), it can be verified easily from Proposition 3.5 that μ k ∗ is also a fuzzy base of (X _k , Ψ| _{χ X k} ). Because υ ∧ χ X k > μ k ∗ and μ k ∗ is a fuzzy base of (X _k , Ψ| _{χ X k} ), we have υ ∧ χ _{X k}  ∉ Ψ. So does υ.

Case 2: υ (x _k+1) > b ^∗. Since υ ≥ μ k ∗ ∨ ( x k + 1 ) υ ( x k + 1 ), we need only to show that μ k ∗ ∨ ( x k + 1 ) υ ( x k + 1 ) ∉ Ψ, while this assertion clearly holds from the definition of b ^∗.

Remark 3.7. (1) ([8])Suppose that (X, Ψ) is a fuzzy independent set system and I _a = {μ _[a] ∣ μ ∈ Ψ} (a∈ (0,1]). Since X is finite, there is at most a finite number of families of subsets of X. Thus there is a finite sequence {a ₀, a ₁, . . . , a _n } such that

0 = a ₀ < a ₁ < . . . < a _n  = 1 (such a sequence will be denoted by 〈a ₀, a ₁, . . . , a _n 〉).

(2) As stated above, a G-V fuzzy matroid need not have fuzzy bases. The condition “closed" is proposed to ensure the existence of fuzzy bases for G-V fuzzy matroids (see Theorem 1.10 of [2]). The fuzzy independent set system ( { x , y } , { μ ∈ [ 0 , 1 ] X ∣ μ < x 1 ∨ y 1 2 } ) with the property (FI2) is closed, but does not satisfies (FI3). Conversely, we assert here that an H fuzzy matroid is a closed fuzzy independent set system. The relations of “closed", (FI3) and other conditions concerning the existence of fuzzy bases for fuzzy independent set systems will be discussed thoroughly in our future work.

The following result concerns with the level structure of H fuzzy matroids.

Proposition 3.8. Suppose (X, Ψ) is an H fuzzy matroid with the fundamental sequence 〈a ₀, a ₁, . . . , a _n 〉, then there is Y ⊆ X such that I _a  = 2 ^Y for each a ∈ (0, a ₁).

Proof. For a ∈ (0, a ₁), choose a maximal element A of I _a . To prove I _a  = 2 ^A , we assume that there is B ∈ I _a  - 2 ^A and derive a contradiction. Since B ∉ 2 ^A , we can choose x ∈ B - A. By the definition of I _a , we have x _a , χ _A  ∧ [a] ∈ Ψ. Obivously, there is b ∈ (0, a) such that

|χ _A  ∧ [b] | < |x _a | = a.

It follows from Proposition 3.5 that there existsω ∈ Ψ such that

(χ _A  ∧ [b]) < ω ≤ (χ _A  ∧ [b]) ∨ x _a  .

Note ω _[m(ω)] = A ∪ {x}, thus A ∪ { x } ∈ I _m (ω) . Since 0 < m (ω) < a ≤ a ₁, it holds from Remark 3.8 that I _m (ω)  = I _a . Thus we have A ∪ { x } ∈ I _a , it contradicts the fact that A is a maximal element of I _a .

Remark 3.9. A fuzzy independent set system (X, Ψ) is perfect if (X, I _a ) is a matroid for each a ∈ (0, 1]. Proposition 3.8 shows that (X, I _a ) is a matroid for each a ∈ (0, a ₁), while it is still not known if ( X , I _a ) is a matroid for each a ∈ [a ₁, 1]. That is, we do not know if an H fuzzy matroid is perfect.

At last of this section, we give another equivalent description of (HFI2).

Proposition 3.10. Let be an H fuzzy matroid, then (HFI2) is equivalent to

(HFI2 ^∗∗) For every μ ∈ [0, 1]  ^X , all maximal fuzzy independent sets of μ (i.e., elements in Max{ν ∈ Ψ|ν ≤ μ}) have the same cardinality.

Proof. Suppose μ ∈ [0, 1]  ^X and ν ₁, ν ₂∈ Max{ν ∈ Ψ|ν ≤ μ} . If |ν ₁| ≠ |ν ₂|, without loss of generality, we assume |ν ₁| < |ν ₂|. By Proposition 3.1, ν ₁ and ν ₂ are fuzzy bases of the H fuzzy matroid (X, Ψ| _μ ). It follows from Proposition 3.5 that there exists ω ∈ Ψ such that ν ₁ < ω ≤ ν ₁ ∨ ν ₂ ≤ μ, which contradicts the fact that ν ₁ is a fuzzy base of (X, Ψ| _μ ). Conversely, let μ, ν ∈ Ψ and |μ| < |ν| . Proposition 3.1 implies that μ and ν are fuzzy independent sets of the H fuzzy matroid (X, Ψ|_μ∨ν). By Proposition 3.6, there exist a fuzzy base of (X, Ψ|_μ∨ν) such that μ < ω ≤ μ ∨ ν . That is ω∈ Max{σ ∈ Ψ|σ ≤ μ ∨ ν} Choose x ₀ ∈ X such that μ (x ₀) < ω (x ₀). Define ω ∗ ( x ) = { ω ( x 0 ) , x = x 0 , μ ( x ) , x ∈ X - { x 0 } .

Then ω ^∗ ≤ ω implies ω ^∗ ∈ Ψ, and so (HFI2) holds.

Proposition 3.10 implies

Corollary 3.11. Fuzzy bases of an H fuzzy matroid have the same cardinality.

4 Fuzzy base axioms of H fuzzy matroids

After some preliminaries of Section 3, we come to the major goal of this paper, a theorem that characterize H fuzzy matroids by means of fuzzy bases.

Theorem 4.1. Let X be a nonempty finite set and Φ ⊆ [0, 1]  ^X . Then Φ is the set of fuzzy bases of an H fuzzy matroid on [0, 1]  ^X if and only if the following two conditions hold:

(FB1) All the members of Φ have the same cardinality.

(FB2) If μ, ν ∈ Φ and ω ∈ [0, 1]  ^X satisfying μ ∧ ν < ω < ν, then there is ω ₁ ∈ [0, 1]  ^X satisfying μ ∧ ν < ω ₁ ≤ μ such that ω ∨ ω ₁ ∈ Φ.

Proof. Suppose that Φ is the set of fuzzy bases of an H fuzzy matroid. (FB1) holds from Corollary 3.11. Let μ, ν ∈ Φ and ω is a fuzzy set on X such that μ ∧ ν < ω < ν . We shall show that (FB2) holds. Define

X μ ν μ = { x ∈ X | μ ( x ) > ν ( x ) } ,

X _μν  = {x ∈ X|μ (x) = ν (x)} ,

X μ ν ν = { x ∈ X | μ ( x ) < ν ( x ) } .

It is easy to check that μ ≠ ν, thus X μ ν μ ≠ ∅ and X μ ν ν ≠ ∅ . Since μ ∧ ν < ω < ν, it holds that ω (x) = ν (x) for x ∈ X μ ν μ ∪ X μ ν . And ω < ν implies that there is x 0 ∈ X μ ν ν such that μ (x ₀) < ω (x ₀) . Define ω ^∗ ∈ [0, 1]  ^X as follows: ω ∗ ( x ) = { ω ( x ) , x ∈ X μ ν ν , μ ( x ) , x ∈ X - X μ ν ν .

Obviously, μ < ω ^∗ and ω < ω ^∗ . Note that μ ∈ Φ and ω is a fuzzy independent set, it follows from Proposition 3.10 that there exists μ ^∗ ∈ Φ such that ω ≤ μ ^∗ ≤ ω ^∗ . Let

ω 1 = [ μ ∗ ∧ χ X - X μ ν ν ] ∨ [ ( μ ∧ ν ) ∧ χ X μ ν ν ] .

We can check that μ ∗ ( x ) = ω 1 ( x ) ≥ ω ( x ) ( x ∈ X - X μ ν ν ) and ω ₁ (x) = ( μ ∧ ν ) ( x ) ≤ ω ( x ) = μ ∗ ( x ) ( ∀ x ∈ X μ ν ν ) , thus μ ^∗ = ω ∨ ω ₁ ∈ Φ . Finally, μ ∧ ν < ω ₁ ≤ μ can be easily verified, as desired.

Conversely, suppose that Φ ⊆ [0, 1]  ^X satisfies (FB1) and (FB2), we shall prove that Φ is the set of fuzzy bases of an H fuzzy matroid on [0, 1]  ^X  . Let ↓Φ = {μ ∈ [0, 1]  ^X | there exists ω _μ  ∈ Φ such that μ ≤ ω _μ } . By (FB1), we have Max↓Φ = Φ . Now we need only to show that (X, ↓ Φ) is an H fuzzy matroid. Since (FI1) holds trivially for (X, ↓ Φ), we shall show that (X, ↓ Φ) has the property of (HFI2 ^∗) in the following, and then by Proposition 3.5, we finish the proof.

Suppose μ, ν ∈ ↓ Φ and |μ| < |ν| . By the definition of ↓Φ, we can choose ω _μ  ∈ Φ such that μ < ω _μ  . If there is x 0 ∈ X μ ν ν such that ω _μ  (x ₀) > μ (x ₀), then define ω μ ∗ ( x ) = { min { ν ( x 0 ) + μ ( x 0 ) 2 , ω μ ( x 0 ) + μ ( x 0 ) 2 } , x = x 0 , μ ( x ) , x ∈ X - { x 0 } .

Obviously, ω μ ∗ < ω μ, thus ω μ ∗ ∈ ↓ Φ. It is easy to check that ω μ ∗ satisfies μ < ω μ ∗ ≤ μ ∨ ν , as desired. Now we assume that ω _μ  (x) = μ (x) < ν (x) for all x ∈ X μ ν ν. Choose ω _ν  ∈ Φ such that ω _ν  ≥ ν, and then for x ∈ X μ ν ν, ω _μ  < ω _ν holds obviously. If for every x ∈ X - X μ ν ν, we have μ (x) ≤ ω _ν  (x), then define ω ν ∗ ( x ) = { ( ν ( x ) + μ ( x ) ) / 2 , x ∈ X μ ν ν , μ ( x ) , x ∈ X - X μ ν ν .

Then ω ν ∗ < ω ν implies ω ν ∗ ∈ ↓ Φ, and it follows from the definition of ω ν ∗ that μ < ω ν ∗ ≤ μ ∨ ν , as desired. Now we consider the case that there exists y 0 ∈ X - X μ ν ν such that ν (y ₀)≤ ω _ν  (y ₀) < μ (y ₀) . Obviously, y 0 ∈ X μ ω ν μ, so X μ ω ν μ ≠ ∅ . For every n ∈ ℕ, we define ω n ∗ ∈ [ 0 , 1 ] X as follows: ω n ∗ ( x ) = { μ ( x ) - μ ( x ) - ω ν ( x ) 2 n , x ∈ X μ ω ν μ , ω μ ( x ) , x ∈ X - X μ ω ν μ .

Observe that ω μ ∧ ω ν < ω n ∗ < ω μ, thus (FB2) implies that there exists ω n ★ with ω μ ∧ ω ν < ω n ★ ≤ ω ν such that ω n ∗ ∨ ω n ★ ∈ Φ (∀n ∈ {1, 2, 3, . . .}). Let ω n = ω n ∗ ∨ ω n ★ and the following properties are satisfied trivially:

ω _ν  (x) ≥ ω _n  (x) ≥ μ (x) = ω _μ  (x), if x ∈ X μ ν ν;

ω _ν  (x) < ω _n  (x) < μ (x) , if x ∈ X μ ω ν μ ;

ω _ν  (x) ≥ ω _n  (x) ≥ μ (x), if x ∈ X μ ω ν ω ν ∩ ( X - X μ ν μ );

ω _n  (x) = μ (x), if x ∈ X _{μω ν} .

Note that the above four sets (i.e., X μ ν ν , X μ ω ν μ , X μ ω ν ω ν ∩ ( X - X μ ν μ ) and X _{μω ν} ) are pairwise disjoint and their union is X.

Since { ω n } n ∈ ℕ is a sequence of points of the compact set Φ, { ω n } n ∈ ℕ has a limit point which belongs to Φ and we denote the limit point by ω° . When x ∈ X μ ν μ ∪ ( X μ ω ν ω ν ∩ ( X - X μ ω ν μ ) ), it can be easily verified that ω° (x) ≥ μ (x). Moreover, ω° (x) = μ (x) holds apparently when x ∈ X _{μω ν} . For every x ∈ X μ ω ν μ, we have

lim n → ∞ ω n ( x ) = μ ( x ) ,

so it follows that ω° (x) = μ (x) ( ∀ x ∈ X μ ω ν μ ).

We now assert that there exists z 0 ∈ X μ ν μ satisfying ω° (z ₀) > μ (z ₀). To prove this assertion, we assume that ω° (x) = μ (x) for all x ∈ X μ ν μ and derive a contradiction. In the following, we consider some inequalities concerning cardinalities of fuzzy sets. By the relation of ω° and μ preceding discussed, we have

| ω ∘ | - | μ | = ∑ x ∈ X μ ν μ ( ω ∘ ( x ) - μ ( x ) ) +∑ x ∈ X μ ω ν μ ( ω ∘ ( x ) - μ ( x ) )

   +∑_x∈Y (ω° (x) - μ (x)) +∑_{x∈X μω ν}  (ω° (x) - μ (x))

  =∑_x∈Y (ω° (x) - μ (x))

≤∑_x∈Y (ω _ν  (x) - μ (x)) ,

where Y = X μ ω ν ω ν ∩ ( X - X μ ν μ ).

Since |μ| < |ν|, it holds that

∑ x ∈ X μ ν ν ( ν ( x ) - μ ( x ) ) > ∑ x ∈ X μ ν μ ( μ ( x ) - ν ( x ) )

                                   ≥ ∑ x ∈ X μ ω ν μ ( μ ( x ) - ω ν ( x ) ) .

That is

∑ x ∈ X μ ν ν ( ν ( x ) - μ ( x ) ) - ∑ x ∈ X μ ω ν μ ( μ ( x ) - ω ν ( x ) ) = ∑ x ∈ X μ ν ν ( ν ( x ) - μ ( x ) ) + ∑ x ∈ X μ ω ν μ ( ω ν ( x ) - μ ( x ) ) > 0 .

Note that

∑ x ∈ X μ ν ν ( ω ν ( x ) - μ ( x ) ) + ∑ x ∈ X μ ω ν μ ( ω ν ( x ) - μ ( x ) ) ≥ ∑ x ∈ X μ ν ν ( ν ( x ) - μ ( x ) ) + ∑ x ∈ X μ ω ν μ ( ω ν ( x ) - μ ( x ) ) > 0 ,

and

∑_{x∈X μω ν}  (ω _ν  (x) - μ (x)) =0 .

Thus, we have

| ω ν | - | μ | = ∑ x ∈ X μ ν ν ( ω ν ( x ) - μ ( x ) )

+∑ x ∈ X μ ω ν μ ( ω ν ( x ) - μ ( x ) ) +∑_x∈Y (ω _ν  (x) - μ (x))

+∑_{x∈X μω ν}  (ω _ν  (x) - μ (x))

>∑_x∈Y (ω _ν  (x) - μ (x))

≥|ω°| - |μ|,

contrary to the fact that all the members of Φ have the same cardinality.

Finally, we define ω ( x ) = { min { ν ( z 0 ) + μ ( z 0 ) 2 , ω ∘ ( z 0 ) + μ ( z 0 ) 2 } , x = z 0 , μ ( x ) , x ∈ X - { z 0 } .

Then ω < ω° implies ω ∈ ↓ Φ, and it follow from the definition of ω that μ < ω ≤ ν ∨ μ, this finishes the proof.

Finally, we show a simple application of Theorem 4.1.

Proposition 4.2. Let X = {x, y} and Φ ⊆ [0, 1]  ^X , then Φ is a set of fuzzy bases of an H fuzzy matroid if and only if for some a ∈ (0, 2], Φ is a closed interval in the straight line x + y = a.

Remark 4.3. A closed interval of x + y = a means a set {μ ∈ [0, 1]  ^X |μ (x) + μ (y) = a, b ≤ μ (x) ≤ c} (∀  0 ≤ b ≤ c ≤ 1) . Note that the closed interval defined here contains one-element set as its special case.

Proof of Proposition 4.2. Let Φ be the set of fuzzy bases of an H fuzzy matroid (X, Ψ). By (FB1) in Theorem 4.1, all elements of Ψ have the same cardinality. That is, there exists a ∈ (0, 2] such that Φ ⊆ {μ ∈ [0, 1]  ^X |μ (x) + μ (y) = a} . To prove that Φ is a closed interval of x + y = a, we need only to show that for every pair of different elements μ, ν ∈ Φ and each d ∈ (μ (x) , ν (x)) (without loss of generality, we assume μ (x) < ν (x)), x _d  ∨ y _a-d ∈ Φ. Let ω = x _d  ∨ y _ν(y). Obviously, μ ∧ ν < ω < ν, then it holds from Theorem 4.1 that there is ω ₁ ∈ [0, 1]  ^X satisfying μ ∧ ν < ω ₁ < μ such that ω ∨ ω ₁ ∈ Φ . We can easily check that ω ∨ ω ₁ = x _d  ∨ y _a-d, thus x _d  ∨ y _a-d ∈ Φ, as desired. Conversely, it follows straightly from Proposition 3.10.

5 Conclusion

In general, when we generalize axiom systems of crisp matroids to fuzzy setting, two problems may arise. One is that the system of fuzzy analogs of crisp axioms does not determine fuzzy matroids. That is to say, we need add additional conditions to systems of fuzzification of crisp axioms. Another problem is that complicated axiom systems often suit very special kinds of fuzzy matroids. Theorem 4.1 gives a group of two axioms by which an H fuzzy matroid can be determined. The two axioms correspond nicely to base axioms of crisp matroids and contain no extra conditions. More importantly, our result holds for arbitrary H fuzzy matroids.

Acknowledgments

The authors would like to thank very sincerely Prof. Tofigh Allahviranloo and the unknown reviewers for their valuable comments and helpful suggestions. This work is supported by the National Natural Science Foundation of China (No. 61202178) and the Fundamental Research Funds for the Central Universities (No. JB150709).