#### Abstract

A new approach to the fuzzification of convex structures is introduced. It is also called an -fuzzifying convex structure. In the definition of -fuzzifying convex structure, each subset can be regarded as a convex set to some degree. An -fuzzifying convex structure can be characterized by means of its -fuzzifying closure operator. An -fuzzifying convex structure and its -fuzzifying closure operator are one-to-one corresponding. The concepts of -fuzzifying convexity preserving functions, substructures, disjoint sums, bases, subbases, joins, product, and quotient structures are presented and their fundamental properties are obtained in -fuzzifying convex structure.

#### 1. Introduction and Preliminaries

Convexity theory has been accepted to be of increasing importance in recent years in the study of extremum problems in many areas of applied mathematics. The concept of convexity which was mainly defined and studied in in the pioneering works of Newton, Minkowski, and others as described in [1] now finds a place in several other mathematical structures such as vector spaces, posets, lattices, metric spaces, graphs, and median algebras. This development is motivated not only by the need for an abstract theory of convexity generalizing the classical theorems in due to Helly, Caratheodory, and so forth but also by the necessity to unify geometric aspects of all these mathematical structures. Some more details can be found in [2].

In 1994, Rosa presented the notion of fuzzy convex structures in [3, 4]. In 2009, Maruyama generalized it to -fuzzy setting in [5]. A fuzzy convex structure is a pair of in which is a crisp subset of the set of -fuzzy subsets of a nonempty set satisfying certain set of axioms.

In this paper, from a completely different point of view, we introduce the notion of -fuzzifying convexity on a nonempty set by means of a mapping satisfying three axioms, where is a complete lattice and is the set of all subsets of . Thus, each subset of can be regarded as a convex set to some degree.

Throughout this paper, denotes a complete lattice with an order-reversing . The smallest element (or zero element) and the largest element (or unit element) in are denoted by and , respectively. , resp, , denotes the collection of all subsets, respectively, all finite subsets of a nonempty set .

The binary relation in is defined as follows: for , if and only if for every subset , the relation always implies the existence of with [6]. is called the greatest minimal family of in the sense of [7], denoted by . Moreover, for , we define .

In a completely distributive lattice with an order-reversing , there exist and for each , , and (see [7]).

Theorem 1 (see [7]). *If is completely distributive, then for a subfamily of , one has*(1)*; that is, is an mapping.*(2)*; that is, is a union-preserving mapping.*

For a complete lattice , , and , we use the following notation:

Suppose that is completely distributive and we define

Some properties of these cut sets can be found in [8â€“13].

Theorem 2 (see [8, 10, 13]). *If is completely distributive, then for each -fuzzy set in , one has *(1)*.*(2)*, .*(3)*, .*

Lemma 3. *Let be a completely distributive lattice and let . Then the following conditions are equivalent:*(1)*.*(2)*; if then .*(3)*; if then .*(4)*; if then .*

*Proof. *It is easy to know that , , and hold. Next we prove and .

; if , then . Hence, , . By , . Then .

; if then . ; if then . â€‰By , . Hence, and thus . So .

*Definition 4 (see [2]). *A subset of is called a convexity if it satisfies the following three conditions: (C1), ;(C2)if is nonempty, then ;(C3)if is nonempty and totally ordered by inclusion, then .

The fuzzy sets in are called convex sets, and the pair is called a convex structure.

If satisfies (C1) and (C2), then is called a closure structure.

Theorem 5 (see [2, 14]). *Let be a convex structure and . For , , where is the hull operator of .*

*Definition 6 (see [2]). *A closure operator on is domain finite (or algebraic) provided for each and for each there is a finite set with .

Theorem 7 (see [2]). *Let be a closure structure. Then the following conditions are equivalent:*(1)* is a convexity on ;*(2)*the closure operator of is domain finite;*(3)* is stable for updirected union; that is, if is an updirected set then .*

In 1994, Rosa presented the notion of fuzzy convex structures in [3, 4]. In 2009, Maruyama generalized it to -fuzzy setting in [5] as follows.

*Definition 8 (see [5]). *For a nonempty set and a subset of , is called a fuzzy convex structure if and only if satisfies the following conditions:(MC1), ;(MC2)if is nonempty, then ;(MC3)if is nonempty and totally ordered by inclusion, then .

Based on papers [15, 16], we can obtain the following definitions and theorems.

*Definition 9 (see [15, 16]). *A mapping is called an -fuzzifying closure system if it satisfies the following conditions:(MYC);(MYC2)if is nonempty, then .

A pair is called an -fuzzifying closure system space provided that is an -fuzzifying closure system on .

*Definition 10 (see [15, 16]). *An -fuzzifying closure operator on is a mapping satisfying the following conditions: (CL1) for every ;(CL2);(CL3).

Theorem 11 (see [15, 16]). *Let be an -fuzzifying closure operator on . Define a mapping by
**
Then is an -fuzzifying closure system.*

Theorem 12 (see [15, 16]). *Let be an -fuzzifying closure system space. Define by
**
Then is an -fuzzifying closure operator on .*

Theorem 13 (see [15, 16]). *Let be an -fuzzifying closure system space and let be an -fuzzifying closure operator on . Then and .*

*Definition 14 (see [17]). *A fuzzy vector space is a pair , where is a vector space on a totally ordered field and is a mapping satisfying for any , . If is a fuzzy vector space, then for each , is a subspace of .

#### 2. -Fuzzifying Convex Structures

In this section, from a completely different point of view, we introduce the notion of -fuzzifying convexity on a nonempty set by means of a mapping . Thus, each subset of can be regarded as a convex set to some degree.

*Definition 15. *A mapping is called an -fuzzifying convexity on if it satisfies the following conditions: (MYC1) ;(MYC2) if is nonempty, then ;(MYC3) if is nonempty and totally ordered by inclusion, then .

If is an -fuzzifying convexity on , then the pair is called an -fuzzifying convex structure. For , can be regarded as the degree to which is a convex set. When , an -fuzzifying convex structure is called a fuzzifying convex structure for short.

If satisfies (MYC1) and (MYC2), then is called an -fuzzifying closure structure.

*Remark 16. *(1) We can see that an -fuzzifying closure structure is exactly an -fuzzifying closure system space with .

(2) By Theorems 11 and 12, we can verify that there is a one-to-one correspondence between -fuzzifying closure structures and -fuzzifying closure operators with (CL0), where(CL0) for every .

Therefore, we can obtain the -fuzzifying closure operator induced by an -fuzzifying closure structure that satisfies (CL0)â€“(CL3).

*Example 17 (see [16, 18]). *Let be a universe of discourse. A mapping is called a fuzzifying topology on if it satisfies the following conditions:(FOA1);(FOA2), ;(FOA3), .

And is called a fuzzifying topological space. Furthermore, if a fuzzifying topology satisfies (SOA2), , then is called a saturated fuzzifying topology and is called an Alexandroff fuzzifying topological space.

We can see that an Alexandroff fuzzifying topological space is a fuzzifying convex structure.

*Example 18. *Let be a fuzzy vector space. Define a mapping by
where is the standard convexity of for each ; that is, if and only if for all and for each with , . Then is a fuzzifying convex structure. It is easy to see that satisfies (MYC1). Now we prove that satisfies (MYC2) and (MYC3).

(MYC2) Suppose that is nonempty and take any . Then for each , . By the definition of , we know that, for each , there exists such that . Since is the standard convexity of , . This shows that . By the arbitrariness of , we obtain that .

(MYC3) Suppose that is nonempty and totally ordered by inclusion, and take any . Then for each , . By the definition of , we know that, for each , there exists such that . Since is the standard convexity of , . This shows that . By the arbitrariness of , we obtain that .

*Example 19. *Let be a crisp convex structure. Define by
Then it is easy to prove that is a fuzzifying convex structure.

*Example 20. *Let be a nonempty set and let be a mapping defined by
Then it is easy to prove that is a fuzzifying convex structure. If with , then 0.5 is the degree to which is a convex set.

Theorem 21. *Let be a mapping. Then is an -fuzzifying convex structure if and only if, for each , is a convex structure.*

*Proof. *
This is straightforward.

Theorem 22. *If is completely distributive, then a mapping is an -fuzzifying convexity if and only if, for each , is a convexity.*

*Proof. **Sufficiency*. (MYC1) For each , . We have .

(MYC2) Let be nonempty and for , . Thus, . We know that and then for each . Since for each , is a convexity, ; that is, . Therefore, by the arbitrariness of and Lemma 3, .

(MYC3) Let be nonempty and totally ordered by inclusion and let for . Thus, . We know that and then for each . Since for each , is a convexity, ; that is, . Therefore, by the arbitrariness of and Lemma 3, .*Necessity*. Suppose that is an -fuzzifying convexity and . Now we prove that is a convexity.

(C1) By and , we know that and . This implies that , .

(C2) If , then for all , . Hence, . By , we know that
This shows . Hence, .

(C3) If is nonempty and totally ordered by inclusion, then for each , . Hence, . By we know that
This shows . Therefore, . The proof is completed.

Now we consider the conditions that a family of convexities forms an -fuzzifying convexity. By Theorem 2, we can obtain the following result.

Corollary 23. *Let be completely distributive and let be an -fuzzifying convexity. Then *(1)* for any with ;*(2)* for any with .*

Theorem 24. *Let be completely distributive and let be a family of convexities. If , for all , then there exists an -fuzzifying convexity such that .*

*Proof. *Suppose that for all . Define by
Next we show that for all . On one hand, if , then . There exists such that and . By , we know that . Hence, . On the other hand, if , then there exists such that and . So ; that is, . Then . Therefore, for all . By Theorem 22, we can obtain that is an -fuzzifying convexity and for all .

Theorem 25. *Let be completely distributive and let be a family of convexities. If , for all , then there exists an -fuzzifying convexity such that .*

*Proof. *
This is straightforward.

*Definition 26. *Let , be -fuzzifying convexities on . If , for all , which is denoted by , then is said to be coarser than and is said to be finer than .

Theorem 27. *Let be a family of -fuzzifying convexities on . Then is an -fuzzifying convexity on , where is defined by
**
for each . Obviously, is coarser than for all .*

*Proof. *This is straightforward.

#### 3. Characterizations of -Fuzzifying Convex Structures by -Fuzzifying Closure Operators

In this section, we always suppose that is a completely distributive lattice with an order-reversing .

*Definition 28. *An -fuzzifying closure operator on is called domain finite (or algebraic) if it satisfies the following condition: for each and ,(MDF).

Theorem 29. *Let be an -fuzzifying closure structure. Then the following conditions are equivalent:*(1)* is an -fuzzifying convexity on ;*(2)*the closure operator of is domain finite.*(3)* is stable for updirected union; that is, if is an updirected set, then .*

*Proof. * Let be the -fuzzifying closure operator of . By Remark 16, we know that satisfies (CL0)â€“(CL3). Next we need to prove that satisfies (MDF). Since is an -fuzzifying convex structure, by Theorem 22, for each , is a convex structure. Let be the closure operator of for each . It is obvious that . Conversely, let be any element in with the property of . Then by Theorem 12, we have
By the arbitrariness of and Lemma 3, we have . Therefore, .

For any nonempty , which is an updirected set, we need to prove .

Let be any element in with the property of . Then by Theorem 11, we have
Hence,
Therefore, .

By Theorems 7 and 22, it is trivial.

By Remark 16 and Theorems 13 and 29, we can obtain the following theorem, which shows that an -fuzzifying convex structure can be characterized by means of an -fuzzifying closure operator, which satisfies (CL0)â€“(CL3) and (MDF).

Theorem 30. *There is a one-to-one correspondence between -fuzzifying convex structures and -fuzzifying closure operators in Definition 10 satisfying (CL0) and (MDF).*

#### 4. -Fuzzifying Convexity Preserving Functions

In this section, still denotes a completely distributive lattice. We will generalize the notion of convexity preserving functions to -fuzzy setting.

Theorem 31. *Let be an -fuzzifying convex structure and a surjective function. Define a mapping by
**
Then is an -fuzzifying convex structure.*

*Proof. *
(MYC1) holds from the following equalities:

For (MYC2), for any nonempty set , let be any element in with the property of . Then ,
For each , there exists such that and . Note that and . Finally, we have
This implies that .

For (MYC3), for any nonempty set , which is totally ordered by inclusion, let be any element in with the property of . Then , . For each , there exists such that and . Since is surjective and is totally ordered by inclusion, we have which is totally ordered by inclusion and then . Note that . Finally, we have
This implies that .

*Definition 32. *Let and be -fuzzifying convex structures. A function is called an -fuzzifying convexity preserving function if for all .

*Definition 33. *Let and be -fuzzifying convex structures. A function is called an -fuzzifying convex-to-convex function if for all .

*Definition 34. *Let and be -fuzzifying convex structures. A function is an -fuzzifying isomorphism if is a bijection, an -fuzzifying convexity preserving function, and an -fuzzifying convex-to-convex function.

The following theorem gives a characterization of -fuzzifying convexity preserving functions.

Theorem 35. *Let and be two -fuzzifying convex structures. A surjective function is an -fuzzifying convexity preserving function if and only if for all .*

*Proof. **Necessity*. If is an -fuzzifying convexity preserving function, then for all . Hence, for all , we have
*Sufficiency*. If for all , then
for all . This shows that is an -fuzzifying convexity preserving function.

The next three theorems are trivial.

Theorem 36. *If and are -fuzzifying convexity preserving functions, then is an -fuzzifying convexity preserving function.*

Theorem 37. *Let and be -fuzzifying convex structures. Then a function is an -fuzzifying convexity preserving function if and only if is a convexity preserving function for any .*

Theorem 38. *Let and be -fuzzifying convex structures. Then a function is an -fuzzifying convexity preserving function if and only if is a convexity preserving function for any .*

#### 5. Quotient -Fuzzifying Convex Structures

In this section, the notions of quotient structures and quotient functions are generalized to -fuzzy setting.

Theorem 39. *Let be an -fuzzifying convex structure and a surjective function. Define a mapping by
**
Then is an -fuzzifying convex structure and we call a quotient -fuzzifying convexity of with respect to and . Moreover, it is easy to see that is an -fuzzifying convexity preserving function from to .*

*Proof. *(MYC1) holds from the following equalities:

(MYC2) can be shown from the following fact: for any nonempty set ,

For (MYC3), if is nonempty and totally ordered by inclusion, then

Theorem 40. *Let be an -fuzzifying convex structure and a surjective function. Then is the finest convexity on such that is an -fuzzifying convexity preserving function.*

*Proof. *Let be an -fuzzifying convexity on such that is an -fuzzifying convexity preserving function from to ; then , . We have , . Therefore, .

*Definition 41. *Let and be -fuzzifying convex structures. A function is called an -fuzzifying quotient function if is surjective and is a quotient -fuzzifying convexity with respect to and .

Theorem 42. *If is an -fuzzifying quotient function, then is an -fuzzifying convexity preserving function if and only if is an -fuzzifying convexity preserving function.*

*Proof. *
Since is an -fuzzifying quotient function, we know that is surjective and , .*Necessity*. Since is an -fuzzifying convexity preserving function, , . Thus, ,
*Sufficiency*. Since is an -fuzzifying convexity preserving function, ,

Theorem 43. *If is a surjective -fuzzifying convexity preserving function and an -fuzzifying convex-to-convex function, then is a quotient -fuzzifying convexity. Moreover, is an -fuzzifying quotient function.*

*Proof. *Since is a surjective -fuzzifying convexity preserving function and an -fuzzifying convex-to-convex function, we have , and , . Since is surjective, for all , . Hence,
So for each and then is a quotient -fuzzifying convexity.

Based on Theorem 39, we can obtain the following result.

Theorem 44. *Let be an -fuzzifying convex structure and let be an equivalence relation defined on . Let be the usual quotient set and let be the projection mapping from to . Define by
**
Then is an -fuzzifying convexity on and is a quotient -fuzzifying convex structure of .*

#### 6. Substructures and Disjoint Sums of -Fuzzifying Convex Structures

In this section, still denotes a completely distributive lattice. We will give the substructures and disjoint sums of -fuzzifying convex structures and discuss some of their fundamental properties.

Theorem 45. *Let be an -fuzzifying convex structure, . Then is an -fuzzifying convex structure on , where , . One calls an -fuzzifying substructure of .*

*Proof. *(1) Clearly, .

(2) For any , we have
where . Since , we have .

(3) For any , which is nonempty and totally ordered by inclusion, let be any element in with the property of ; that is, . Then for each , there exists such that and ; that is, . By Theorem 21, for each , is a convex structure. Let denote the hull operator of for each . Then for all . Since is nonempty and totally ordered by inclusion, is nonempty and totally ordered by inclusion. Hence, ; that is, . By Theorem 5, . So we have . This implies that .

Theorem 46. *Let be a family of pairwise disjoint -fuzzifying convex structures; that is, for . Consider the set and , is the usual inclusion mapping (i.e., , ). Define a mapping by , . Then is an -fuzzifying convexity on . is called an -fuzzifying sum convexity of and is denoted by . The -fuzzifying convex structure is called the -fuzzifying sum convex structure of , written as .*

*Proof. *(1) Clearly, .

(2) For any nonempty set , we have

(3) For any , which is nonempty and totally ordered by inclusion, we have

*Remark 47. *Let be a family of -fuzzifying convex structures. Consider , . Then for . For each , the usual mapping , is one-to-one and onto and it naturally induces the mapping from to , still denoted by . Define such that for each . Then one can easily verify that it is an -fuzzifying convexity on and is an -fuzzifying isomorphism. Thus, we may identify with . In the sequel, we will assume that any family of -fuzzifying convex structures has an -fuzzifying sum convex structure (up to an -fuzzifying isomorphism), but in the proof we still tacitly assume that the discussed family consists of pairwise disjoint -fuzzifying convex structures. This is because there is no difference between two families of -fuzzifying convex structures from a point of isomorphism.

Theorem 48. *Let . Then is the finest -fuzzifying convexity on such that are -fuzzifying convexity preserving functions.*

*Proof. *If there is an -fuzzifying convexity on such that , is an -fuzzifying convexity preserving function, then for each and , and thus . Therefore, .

Theorem 49. *Let and let be an -fuzzifying convex structure. Then is an -fuzzifying convexity preserving function from to if and only if the composition is an -fuzzifying convexity preserving function for each .*

* Proof. **Necessity*. It is easy to obtain the necessity by Theorem 42.*Sufficiency*. Since the composition is an -fuzzifying convexity preserving function for each , we have for each , . And thus . It implies that is an -fuzzifying convexity preserving function from to .

Theorem 50. *Let and for each . Then , where .*

*Proof. *Let () be the usual inclusion mapping and () be the restriction of to . Then by Theorems 45 and 46, we obtain, ,
We also have ,
Note that for any with , we know . Hence, ,
Therefore,
for every . Conversely, suppose that for . For any , there exists such that and . Let . Then (where ), and , . Hence, , where . We obtain . So