The Study on the ( L , M )-Fuzzy Independent Set Systems

Independent sets play an important role in matroid theory. In this paper, the definitions of pre-independent fuzzy set system and independent fuzzy set system in L-fuzzy setting are presented. Independent M-fuzzifying set system is introduced and some of its properties are discussed. Further independent (L,M)-fuzzy set system is given and some of its properties are obtained. The relations of these independent set systems in the setting of fuzzy vector spaces and fuzzy graphs are showed.


Introduction
As a generalization of both graphs and matrices, matroids were introduced by Whitney in 1935.It plays an important role in mathematics, especially in applied mathematics.
Matroids are precisely the structures for which the simple and efficient greedy algorithm works.
In 1988, the concept of fuzzy matroids was introduced by R. Goetschel and W.
The approach to the fuzzification of matroids preserves many basic properties of crisp matroids, and an M-fuzzifying matroid and its fuzzy rank function are one-to-one corresponding.Further the concept of (L,M)-fuzzy matroid was presented by Shi [7], it is a wider generalization of M-fuzzifying matroids.
Independent set systems play an important role in matroids theory.In this paper, firstly, the pre-independent fuzzy set system, independent fuzzy set system to L-fuzzy setting, independent M-fuzzifying set system, and independent (L,M)-fuzzy set system are presented.Secondly, the properties of these independent set systems are discussed.Finally, the relevance of these independent set systems in the setting of fuzzy vector spaces and fuzzy graphs are given.

Preliminaries
Let E be a non-empty finite set.We denote the power set of E by 2 E and the cardinality of X by X for any [ ] 0,1 -fuzzy set system is called a fuzzy set system for short.
A set system ( ) Use fuzzy sets on E instead of crisp sets, Novak [4] obtained the definition of independent fuzzy set systems as follows.
Definition 2.1.Let E be a finite set and F be a fuzzy subset family on E .If F satisfies the following condition: then the pairs ( ) , E F is called an independent fuzzy set system.
Throughout this paper, let E be a finite set, both L and M denote completely distributive lattices.The smallest element and the largest element in L are denoted by ⊥ and Τ , respectively.We often do not distinguish a crisp subset of E and its characteristic function.

Independent L-Fuzzy Set Systems and Theirs Properties
There is not a method such that we immediately believe which way of fuzzification of a crisp structure is more natural than others.Nevertheless, it seems to be widely accepted that any fuzzifying structures have an analogous crisp structures as theirs levels.
Consequently, an L-fuzzy set system ( ) is an independent set system for each pre-independent [ ] 0,1 -fuzzy set system is an fuzzy pre-independent set system [4].In this section, we introduce the concept of independent L-fuzzy set system and discuss theirs properties.
Definition 3.1.Let E be a finite set.If a mapping then the pair ( ) , E  is called an independent L -fuzzy set system.An independent [ ] 0,1 -fuzzy set system is precise a fuzzy independent set system [4].
For each is an independent set system.By Theorem 3.2, it is easy to obtain the following.
Corollary 3.3.Let ( ) , E  be an independent L-fuzzy set system.Then ( ) , E  be a pre-independent L-fuzzy set system.
Conversely, given a family of independent set systems, we can obtain an independent L-fuzzy set system.Theorem 3.4.Let E be a finite set and is an independent set system for each ⊥ , we have χ ∅ ∈  .We show that  satisfies the property (LH) as follows. , ) By Theorem 3.2, we get a family of independent set systems by an independent Lfuzzy set system ( ) , E  .Subsequently, Theorem 3.4 tells us the family of independent set systems can induce an independent L-fuzzy set system ( ) is not true.
In the following, we will prove when  satisfies the condition ( ) s (which will be given in Theorem 3.5), we have =   .Theorem 3.5.Let E be a finite set and ( ) , E  be an independent L-fuzzy set system.We suppose that  satisfies the statement: (s) Proof.Obviously ⊆   .We show that ⊆   as follows.
A ∈ ∀ , this implies ⊥ .Since  is an independent L-fuzzy set system, then We call  is strong if it satisfies the condition (LH) and ( ) s , then the pair ( ) , E  is called a strong independent L-fuzzy set system.Fuzzy matroids which are introduced by Goerschel and Voxman [1] are a subclass of strong independent L-fuzzy set systems.
For a strong independent L-fuzzy set system, we can obtain an equivalent description as follows.
Theorem 3.6.Let ( ) , E  be an independent L-fuzzy set system.Then  is strong if and only if Proof.For each is an independent set system as follows. , , there is G ∈  such that ( ) r G B = .By Lemma 3.7, there exists a L ∈ such that ( ) is an independent set system.

Independent M-Fuzzifying Set Systems
In crisp independent set system , E J , we can regard J as a mapping satisfies the property (H).Use fuzzy sets instead of crisp sets, Novak [4] presented an approach to the fuzzification of independent set systems, which is called fuzzy independent set system.In fact, we may consider such a mapping : and the following statement: then the pair ( ) , E  is called an independent M-fuzzifying set system.Specially, when an independent [ ] 0,1 -fuzzifying set system is also called an independent fuzzifying set system for short.

Theorem 4.2. Let
: 2 E M →  be a mapping.Then the following statements are equivalent: (i) ( )  is an independent set system; (iii) For each ( ) is an independent set system; (vi) For each ( ) . Thus  is an independent M-fuzzifying set system on E.
( ) ( ) satisfies the condition (H), we have for any ( ) is an independent set system we have We can similarly prove the remainder statements are also equivalent.
be a mapping.Then the following statements are equivalent:  is an independent set system; (iii) For each  is an independent set system.
Remark 4.4.In Proposition 2 of [4], Novak has illuminated that when , a closed fuzzy independent set system is equivalent with an independent fuzzifying set system.M-fuzzifying matroids [13] are precise a subclass of the independent Mfuzzifying set system.

Independent (L,M)-Fuzzy Set Systems
In this section, we obtain the definition of independent M-fuzzifying set systems and discuss theirs properties.Definition 5.1.Let E be a finite set and L,M be lattices.A mapping : E L M →  satisfies the following statement: then the pair ( ) , E  is called an independent (L,M)-fuzzy set system.
Obviously, an independent (2,M)-fuzzy set system can be viewed as an independent M-fuzzifying set system, where { } 2 , = ⊥ Τ .Moreover, an independent (L,2)-fuzzy set system is called an independent L-fuzzy set system.An crisp independent set system can be regarded as an independent (2,2)-fuzzy set system.Theorem 5.2.Let E be a finite set and be a mapping.Then the following statements are equivalent: , r E  is an independent L-fuzzy set system; (v) For each ( ) , r E  is an independent L-fuzzy set system.
The prove is trivial and omitted.Corollary 5.3.Let E be a finite set and [ ] [ ] : 0,1 0,1 E →  be a mapping.Then the following conditions are equivalent:  is an independent fuzzy set system; (ii)  is an independent fuzzy set system.

Some Examples of Independent (L,M)-Fuzzy Set Systems
be a fuzzy graph, where ( ) then the pair ( ) , E  is an independent fuzzifying set system.

Conclusion
In this paper, pre-independent fuzzy set system and independent fuzzy set system to L-fuzzy setting are defined.Independent M-fuzzifying set system is introduced and obtained its some properties.Further the definition of independent M-fuzzifying set system is generalized to independent (L,M)-fuzzy set system, and its some properties are proved.Finally, the relevance of generalized independent set systems are presented in the setting of fuzzy vector spaces and fuzzy graphs.
greatest minimal family of b in the sense of[10], denoted by ( )

,Theorem 3 . 8 .
the strong independent [ ] 0,1 -fuzzy set systems are the perfect independent set systems which are defined by Novak [4].Lemma 3.7 ([13]).If G is a finite L-fuzzy set and ( ) a G ≠ ∅ for ( ) Let E be a finite set and ( ) , E  be an independent L-fuzzy set system.For each vector space.If E is a subset of V , we define a subfamily of [ ] the pair ( ) , E  is an independent fuzzy set system.

Funds
The project is supported by the Science & Technology Program of Beijing Municipal Commission of Education (KM201611417007, KM201511417012), the NNSF of China (11371002), the academic youth backbone project of Heilongjiang Education Department (1251G3036), and the foundation of Heilongjiang Province (A201209).
The set of non-unit prime elements in L is denoted by ( ) E  is an independent fuzzy set system.The prove is trivial and omitted.Similarly, we can obtain easily the followings.
, = be a fuzzy vector space.If E is a subset of V , we define a mapping