M-FUZZIFYING INTERVAL SPACES 1. Introduction Convexity theory

Iranian Journal of Fuzzy Systems Vol. 14, No. 1, (2017) pp. 145-162

145

M -FUZZIFYING INTERVAL SPACES Z. Y. XIU AND F. G. SHI

Abstract. In this paper, we introduce the notion of M -fuzzifying interval spaces, and discuss the relationship between M -fuzzifying interval spaces and M -fuzzifying convex structures. It is proved that the category MYCSA2 can be embedded in the category MYIS as a reflective subcategory, where MYCSA2 and MYIS denote the category of M -fuzzifying convex structures of M -fuzzifying arity ≤ 2 and the category of M -fuzzifying interval spaces, respectively. Under the framework of M -fuzzifying interval spaces, subspaces and product spaces are presented and some of their fundamental properties are obtained.

1. Introduction 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 Rn 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 Rn due to Helly, Caratheodory etc; but also by the necessity to unify geometric aspects of all these mathematical structures. Some more details can be found in [9, 24]. In 1994, Rosa presented the notion of fuzzy convex structures in [15, 16]. In 2009, Maruyama generalized it to M -fuzzy setting in [11]. A fuzzy convex structure is a pair of (X, C) in which C is a crisp subset of the set of M -fuzzy subsets of a nonempty set X satisfying certain set of axioms. Recently, a new approach to the fuzzification of convex structures is introduced in [21]. It is called an M -fuzzifying convex structure, in which each subset of X can be regarded as a convex set to some degree. In classical theory of convex structures, interval operators provide a natural and frequent method of describing or constructing convex structures. In [13], fuzzy interval operators were defined to describe fuzzy convex structures. The present paper is a continuation of investigations on M -fuzzifying convex structures, started in [21]. Now we are concerned with the so-called M -fuzzifying interval spaces, Received: September 2015; Revised: February 2016; Accepted: June 2016 Key words and phrases: M -fuzzifying interval spaces, M -fuzzifying convex structures, M fuzzifying interval preserving functions, Subspaces, Product spaces.

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Z. Y. Xiu and F. G. Shi

and discuss the relationship between M -fuzzifying interval spaces and M -fuzzifying convex structures from the category theory point view . We also define the notions of a subspace and a product space of M -fuzzifying interval spaces and study some of their fundamental properties. 2. Preliminaries W V0 Throughout this paper, (M, , , ) denotes a completely distributive lattice 0 with an order-reversing involution . The smallest element (or zero element) and the largest element (or unit element) in M are denoted by ⊥ and >, respectively. 2X , 2X f in , denotes the collection of all subsets, all finite subsets of a nonempty set X respectively. An element a in M is called co-prime if a 6 b ∨ c implies a 6 b or a 6 c [5]. The set of non-zero co-prime elements in M is denoted by J(M ). The binary relation ≺ in M is defined as follows: for a, b ∈ M , a ≺ b if and only if for every subset D ⊆ M , the relation b ≤ sup D always implies the existence of d ∈ D with a ≤ d [2]. {a ∈ M : a ≺ b} is called the greatest minimal family of b in the sense of [25], denoted by β(b). Moreover, for b ∈ M , we define α(b) = {a ∈ M : b ≺op a}. 0 In a completely distributive lattice M with involution , there W an order-reversing V 0 exist α(b) and β(b) for each b ∈ M , b = β(b) = α(b) and a ≺ b ⇔ b ≺op 0 a (see[25]). For A ∈ M X and a ∈ M , we use the following notations: A[a] = {x ∈ X : A(x) ≥ a}, A(a) = {x ∈ X : a ∈ β(A(x))}, A[a] = {x ∈ X : a 6∈ α(A(x))}, A(a) = {x ∈ X : A(x) a}. Some properties of these cut sets can be found in [7, 12, 17, 18, 19, 20]. ← → : M Y −→ M X : M X −→ M Y and fM Let f : X −→ W Y be a mapping. Define fM → X ← by fM (A)(y) = f (x)=y A(x) for A ∈ M and y ∈ Y , and fM (B) = B ◦ f for B ∈ M Y , respectively. → → Here we point out that fM and fM are the usual Zadeh image and preimage operators, respectively. For more details we refer the reader to [14]. In [8, 6, 22], the concept of M -fuzzy non-negative real numbers were introduced as follows: An M -fuzzy non-negative real number W is an equivalence class V[λ] of antitone maps λ : R −→ M satisfying λ(0−) = t, λ(+∞) = t∈R λ(t) = ⊥, where [λ] = {µ : ∀t > 0, λ(t−) = µ(t−)}. We do not distinguish between an M -fuzzy real number [λ] and a left continuous fuzzy function λ representing it. The set of all non-negative M -fuzzy real numbers is denoted by [0, +∞)(M ). Theorem 2.1. [25] For a subfamily {ai : i ∈ Ω} of M , we have S V V S (1) α i∈Ω ai = i∈Ω α(ai ), i.e., α is a − mapping. W S W S (2) β − mapping. i∈Ω ai = i∈Ω β(ai ), i.e., β is a Theorem 2.2. [17] For each M -fuzzy set A in M X and a ∈ M , S i∈Ω (Ai )(a) .

W

i∈Ω

Ai

(a)

=

M -fuzzifying Interval Spaces

147

Based on [10] and [22], we can give the following definition: Definition 2.3. A map d : X × X −→ [0, +∞)(M ) is called an M -fuzzifying pseudo-quasi-metric on X if it satisfies the following conditions. For all x, y, z ∈ X and for all s, t > 0, (Md1) x = y =⇒ d(x, y)(0+) = ⊥; (Md2) d(x, z)(r + s) ≤ d(x, y)(r) ∨ d(y, z)(s); (Md3) d(x, y) = d(y, x). Definition 2.4. [24] Let I : X × X −→ 2X be a mapping satisfying the following conditions: for all x, y ∈ X, (I1) x, y ∈ I(x, y); (I2) I(x, y) = I(y, x). Then I is called an interval operator on X, and I(x, y) is the interval between x and y. The resulting pair (X, I) is called an interval space. Definition 2.5. [24] A convex structure (X, C) is of arity ≤ n provided its convex sets are precisely the sets with the property that co(F ) ⊆ C for each subset F with |F | ≤ n. Theorem 2.6. [24] A convex structure is induced by an interval operator if and only if it is of arity ≤ 2. Theorem 2.7. [24] Let (X, CX ) and (Y, CY ) be convex structures and coX , coY denote the hull operators, respectively. (1) f : X −→ Y is a convexity preserving function if and only if f (coX (F )) ⊆ coY (f (F )) for any F ∈ 2X f in . If (X, CX ) is of arity ≤ n, then it suffices to consider sets |F | ≤ n. (2) f : X −→ Y is a convex-to-convex function if and only if f (coX (F )) ⊇ coY (f (F )) for any F ∈ 2X f in . If (X, CX ) is of arity ≤ n, then it suffices to consider sets |F | ≤ n. Definition 2.8. [4] Let A ∈ M X , ∅ 6= Y ⊆ X. We define an M -fuzzy set A|Y ∈ M Y as follows: ∀y ∈ Y , (A|Y )(y) = A(y). A|Y is called the restriction of A to Y . Definition 2.9. [13] Let X be a set and let I : X × X −→ [0, 1]X be a mapping with the following properties: for all x, y ∈ X, (FI1) I(x, y)(x) > 0, I(x, y)(y) > 0; (FI2) I(x, y) = I(y, x). Then I is called a fuzzy interval operator on X. The resulting pair (X, I) is called a fuzzy interval space and I(x, y) is called the fuzzy interval between x and y for each x, y ∈ X. Definition 2.10. [21] A mapping C : 2X −→ M is called an M -fuzzifying convexity on X if it satisfies the following conditions: (MYC1) C(∅) = C(X) = >; T V (MYC2) if {Ai : i ∈ Ω} ⊆ 2X is nonempty, then C( i∈Ω Ai ) ≥ i∈Ω C(Ai );

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X (MYC3) ifS{Ai : i ∈ Ω} V ⊆ 2 is nonempty and totally ordered by inclusion, then C( i∈Ω Ai ) ≥ i∈Ω C(Ai ). If C is an M -fuzzifying convexity on X, then the pair (X, C) is called an M -fuzzifying convex structure.

Theorem 2.11. [21] Let C : 2X −→ M be a mapping. Then the following are equivalent. (1) (X, C) is an M -fuzzifying convex structure. (2) For each a ∈ M \{⊥}, (X, C[a] ) is a convex structure. (3) For each a ∈ α(⊥), (X, C [a] ) is a convex structure. Definition 2.12. [21] Let (X, CX ) and (Y, CY ) be M -fuzzifying convex structures. A function f : X −→ Y is called an M -fuzzifying convex-to-convex function if CY (f (A)) ≥ CX (A) for all A ∈ 2X . f : X −→ Y is called an M -fuzzifying convexity preserving function if CX f −1 (B) ≥ CY (B) for all B ∈ 2Y . Theorem 2.13. [21] Let (X, CX ) and (Y, CY ) be M -fuzzifying convex structures. Then a function f : (X, CX ) −→ (Y, CY ) is an M -fuzzifying convexity preserving [a] [a] function if and only if f : (X, CX ) −→ (Y, CY ) is a convexity preserving function for any a ∈ α(⊥). Theorem 2.14. [21] Let (X, C) be an M -fuzzifying convex structure. Define a mapping coC : 2X −→ M by ^ ∀x ∈ X, ∀A ⊆ X, coC (A)(x) = C(B)0 . x∈B⊇A /

Then coC satisfies the following conditions: (C0) coC (∅)(x) = ⊥ for every x ∈ X. (C1) coC (A)(x) = > for every x ∈ A. (C2) A ⊆ B =⇒ coV C (A) ≤ co WC (B). (C3) coC (A)(x) = x∈B⊇A y ∈B / coC (B)(y). W / (MDF) coC (A)(x) = {coC (F )(x) : F ∈ 2A f in }. Conversely, let a mapping co : 2X −→ M X satisfy (C0)−(C3) and (M DF ). Define a mapping Cco : 2X −→ M by ^ ∀A ⊆ X, Cco (A) = (co(A)(x))0 . x∈A /

Then (X, Cco ) is an M -fuzzifying convex structure. Moreover, coCco = co. Theorem 2.15. [21] Let (X, C) be an M -fuzzifying convex structure. Then CcoC = C. Remark 2.16. If (X, C) is an M -fuzzifying convex structure, then we call coC an M -fuzzifying hull operator for (X, C). For each x, y ∈ X, we call coC {x, y} the M -fuzzifying segment joining x and y. Theorem 2.17. [23] Let C be the M -fuzzifying convex structure on X. If β(a∧b) = β(a) ∩ β(b) for any a, b ∈ M, then for any a ∈ α(⊥) and A ⊆ X, co(C [a] ) (A) = coC (A)(a0 ) .


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Theorem 2.18. [21] Let (X, C) be an M -fuzzifying convex structure, ∅ = 6 Y ⊆ X. Then (Y, C|Y ) is an M -fuzzifying convex structure on Y , where (C|Y )(A) = W T {C(B) : B ∈ 2X , B Y = A} for every A ∈ 2Y . We call (Y, C|Y ) an M fuzzifying substructure of (X, C). Definition 2.19. [21] Let {(Xt , Ct )}t∈T be a family of M -fuzzifying convex structures. Let X be the product of the sets Xt for t ∈ T , and let πt : X −→ Xt denote the for each t ∈ T . Define a mapping ϕ : 2X −→ M by W projection W ϕ(A) = t∈T (πt )−1 (B)=A Ct (B) for each A ∈ 2X . Then the product convexity Q Q C of X, denoted by t∈T Ct , is the one generated by ϕ. That is t∈T Ct is the finest convexity containing ϕ. The resulting M -fuzzifying Q convex structure (X, C) is called the product of {(Xt , Ct )}t∈T and is denoted by t∈T (Xt , Ct ). Theorem 2.20. [21] Let (Y, D) be an M -fuzzifying convex structure and f : X → Y a surjective function. Define a mapping f −1 (D) : 2X → M by _ ∀A ∈ 2X , f −1 (D)(A) = D(B) : f −1 (B) = A . Then (X, f −1 (D)) is an M -fuzzifying convex structure. Definition 2.21. [21] Let {(X, Ct )}t∈T be a family of M -fuzzifying convex strucW tures on X. Define a maping ϕ : 2X −→ M by ϕ(A) = t∈T Ct (A) for each F A ∈ 2X . FThen the join of {Ct }t∈T , denoted by t∈T Ct , is the one generated by ϕ. That is t∈T Ct is the F finest convexity containing ϕ. The resulting M -fuzzifying convex structure (X, t∈T Ct ) is called the join of {(X, Ct )}t∈T and is denoted by F (X, C ). t t∈T Theorem 2.22. [21] Let {(Xt , Ct )}t∈T be a family of M -fuzzifying Q convex structures. Let X be the product of the sets Xt for t ∈ T . Then t∈T (Xt , Ct ) = F −1 (Ct )). t∈T (X, (πt ) 3. M -fuzzifying Interval Spaces In this section, we generalize the notion of interval spaces to M -fuzzy setting and discuss the relationship between M -fuzzifying interval spaces and M -fuzzifying convex structures. Definition 3.1. A mapping I : X × X −→ M X is called an M -fuzzifying interval operator on X if it satisfies the following conditions: for all x, y ∈ X, (MYI1) I(x, y)(x) = I(x, y)(y) = >; (MYI2) I(x, y) = I(y, x). If I is an M -fuzzifying interval operator on X, then the pair (X, I) is called an M fuzzifying interval space. For x, y ∈ X, I(x, y) is the M -fuzzifying interval between x and y. Remark 3.2. Definitions 2.9 and 3.1 are both generalizations of interval spaces. However, Definition 2.9 is related to fuzzy convex structures in [15, 16], and an M fuzzifying interval space is defined for M -fuzzifying convex structures introduced in [21].

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Example 3.3. Let d be an M -fuzzifying pseudo-quasi-metric on X. Define an operator Id : X × X −→ M X by ^ _ ∀x, y, z ∈ X, Id (x, y)(z) = (d(p, z)(r))0 . r>0 p∈{x,y}

It is easy to verify that Id is an M -fuzzifying interval operator on X. Theorem 3.4. Let I : X ×X −→ M X be a mapping. Then the following conditions are equivalent: (1) (X, I) is an M -fuzzifying interval space. (2) For each a ∈ M \{>}, (X, I a ) is an interval space, where I a (x, y) = I(x, y)(a) for all x, y ∈ X. (3) For each a ∈ α(⊥), (X, Ia ) is an interval space, where Ia (x, y) = I(x, y)(a) for all x, y ∈ X. Proof. (2) ⇒ (1) (MYI1) For each a ∈ M \{>}, let x, y ∈ I a . Then we have x, y ∈ I a (x, y) = I(x, y)(a) , that is, I(x, y)(x) a and I(x, y)(y) a. Hence I(x, y)(x) = I(x, y)(y) = >. (MYI2) For any x, y, z ∈ X and a ∈ M \{>}, I(x, y)(z) a

⇔ ⇔ ⇔ ⇔ ⇔

z ∈ I(x, y)(a) z ∈ I a (x, y) z ∈ I a (y, x) z ∈ I(y, x)(a) I(y, x)(z) a.

This implies I(x, y) = I(y, x). (1) ⇒ (2) Suppose that I : X × X −→ M X is an M -fuzzifying interval operator and a ∈ M \{>}. Now we prove that (X, I a ) is an interval space. (I1) For all x, y ∈ X, by I(x, y)(x) = I(x, y)(y) = >, we know that x, y ∈ I(x, y)(a) = I a (x, y). (I2) For all x, y ∈ X,, by I(x, y) = I(y, x), we know that I a (y, x) = I(y, x)(a) = I(x, y)(a) = I a (x, y). The proof of (1) ⇔ (3) is similar to that of (1) ⇔ (2) and is omitted.

The next theorem shows that an M -fuzzifying convex structure induces an M fuzzifying interval space in a natural way. Theorem 3.5. Let (X, C) be an M -fuzzifying convex structure. Define a mapping SegC : X × X −→ M X by ^ ∀x, y, z ∈ X, SegC (x, y)(z) = coC {x, y}(z) = C(B)0 . z ∈B⊇{x,y} /

Then SegC is an M -fuzzifying interval operator. We call SegC the M -fuzzifying segment operator for (X, C). Proof. It is obvious that SegC satisfies (MYI1) and (MYI2).


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From [24], we know that if (X, I) is an interval space, then the convex structure (X, C), introduced by (X, I), is obtained as follows: C = {A ⊆ X : ∀x, y ∈ S A, I(x, y) ⊆ A}, that is, A ∈ C ⇔ A ⊇ {I(x, y) : ∀x, y ∈ A} ⇔ ∀z ∈ / A, ∀x, y ∈ A, z ∈ / I(x, y). Based on the above fact, we can get the following theorem, which shows that an M -fuzzifying interval space induces an M -fuzzifying convex structure. Theorem 3.6. Let (X, I) be an M -fuzzifying interval space. Define a mapping CI : 2X −→ M by ^ ^ (I(x, y)(z))0 .

∀A ⊆ X, CI (A) =

z ∈A / x,y∈A

Then (X, CI ) is an M -fuzzifying convex structure. Moreover, I ≤ SegCI . Proof. Since (X, I) is an M -fuzzifying interval space, by Theorem 3.4, for each a ∈ M \{>}, (X, I a ) is an interval space. Let (X, C a ) denote the convex strucure induced by (X, I a ). By Theorem 2.6, (X, C a ) is of arity ≤ 2. Then for each a ∈ M \{⊥}, we have A ∈ (CI )[a]

⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

C ≥a VI (A)V

0 z ∈A / x,y∈A (I(x, y)(z)) ≥ a ∀z ∈ / A, ∀x, y ∈ A, (I(x, y)(z))0 ≥ a ∀z ∈ / A, ∀x, y ∈ A, I(x, y)(z) ≤ a0 0 ∀z ∈ / A, ∀x, y ∈ A, z ∈ / I(x, y)(a ) a0 ∀z ∈ / A, ∀x, y ∈ A, z ∈ / I (x, y) 0 A ∈ Ca .

0

This implies (CI )[a] = C a for each a ∈ M \{⊥}. By Theorem 2.11, (X, CI ) is an M -fuzzifying convex structure. To show the last inequality, let p, q ∈ X. Then for every h ∈ X, SegCI (p, q)(h)

= = = ≥ =

V CI (B)0 / Vh∈B⊇{p,q} V V ( z∈B (I(x, y)(z))0 )0 / Vh∈B⊇{p,q} W / Wx,y∈B ( z∈B / / x,y∈B I(x, y)(z)) Vh∈B⊇{p,q} I(p, q)(h) h∈B⊇{p,q} / I(p, q)(h).

This implies I ≤ SegCI .

Lemma 3.7. Let (X, I) be an M -fuzzifying interval space and CI be the M 0 fuzzifying convexity induced by (X, I). Then for each a ∈ α(⊥), (CI )[a ] = CIa . Proof. For each A ⊆ X and a ∈ α(⊥), 0

A ∈ (CI )[a

]

⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

a0 ∈ / α(C VI (A))V 0 a0 ∈ / α( z∈A / x,y∈A (I(x, y)(z)) ) S S 0 a∈ / z∈A α((I(x, y)(z)) ) / x,y∈A ∀z ∈ / A, ∀x, y ∈ A, a0 ∈ / α((I(x, y)(z))0 ) ∀z ∈ / A, ∀x, y ∈ A, a ∈ / β(I(x, y)(z)) ∀z ∈ / A, ∀x, y ∈ A, z ∈ / I(x, y)(a) ∀z ∈ / A, ∀x, y ∈ A, z ∈ / Ia (x, y) A ∈ C Ia . 0

This implies for each a ∈ α(⊥), that (CI )[a ] = CIa .

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Next we give the formula of the M -fuzzifying hull operator of an M -fuzzifying convex structure induced by an M -fuzzifying interval space. Theorem 3.8. Let (X, I) be an M -fuzzifying interval W space and CI be the M ∞ fuzzifying convexity induced by (X, I). Then co (A) = CI k=0 Ak , where A0 = A, W Ak+1 (z) = x,y∈X I(x, y)(z) ∧ Ak (x) ∧ Ak (y). Proof. For each a ∈ α(⊥), z ∈ (Ak+1 )(a)

⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

a ∈ β(A Wk+1 (z)) a ∈ β( x,y∈X I(x, y)(z) ∧ Ak (x) ∧ Ak (y)) S a ∈ x,y∈X β(I(x, y)(z) ∧ Ak (x) ∧ Ak (y)) S a ∈ x,y∈X [β(I(x, y)(z)) ∩ β(Ak (x)) ∩ β(Ak (y))] ∃x, y ∈ X, s.t., a ∈ β(I(x, y)(z)), a ∈ β(Ak (x)), a ∈ β(Ak (y)) ∃x, y ∈ X, s.t., z ∈ I(x, y)(a) , x, y ∈ (Ak )(a) ∃x, y ∈ X, s.t., z ∈ Ia (x, y), x, y ∈ (Ak )(a) . S This implies (Ak+1 )(a) = {Ia (x, y)|x, y ∈ (Ak )(a) }, for each a ∈ α(⊥). Since (X, Ia ) is anSinterval space for each a ∈ α(⊥), by Theorem 3.4, we can obtain S ∞ coCIa (A) = n=0 (Ak )(a) , where A0 = A, (Ak+1 )(a) = {Ia (x, y)|x, y ∈ (Ak )(a) }. W∞ Furthermore, coCIa (A) = ( n=0 Ak )(a) . By Lemma 3.7, for each a ∈ α(⊥), W∞ 0 (CI )[a ] = CIa . So we have co(CI )[a0 ] = coCIa and then co(CI )[a0 ] (A) = ( n=0 Ak )(a) . By Theorem 2.17, for each a ∈ α(⊥), co(CI )[a] (A) = coCI (A)(a0 ) . Hence for each W∞ W∞ a ∈ α(⊥), coCI (A)(a) W = ( n=0 Ak )(a) . Therefore co(CI ) (A) = n=0 Ak , where A0 = A, Ak+1 (z) = x,y∈X I(x, y)(z) ∧ Ak (x) ∧ Ak (y). From [24], we know that if (X, C) is a convex structure and coC its hull operator, then the following holds: A ∈ C ⇔ A ⊇ coC (A) ⇔ ∀x ∈ / A, x ∈ / coC (A) = S {coC (F ) : F ∈ 2A / A, ∀F ∈ 2A / coC (F ). f in } ⇔ ∀x ∈ f in , x ∈ Based on the above fact, we can give the following theorem, which gives a relation between an M -fuzzifying convex structure (X, C) and its hull operator coC . Theorem 3.9. Let (X, C) be an M -fuzzifying convex structure and coC be the M V V 0 fuzzifying hull operator. Then ∀A ⊆ X, C(A) = x∈A (co A C (F )(x)) . / F ∈2 f in

X

Proof. Since coC satisfies (MDF), for all A ∈ 2 , V 0 C(A) = Vx∈A / (co W C (A)(x)) A 0 = / (V {coC (F )(x) : F ∈ 2f in }) Vz∈A 0 = x∈A / F ∈2A , (coC (F )(x)) . f in

Based on Definition 2.5 and Theorem 3.9, we can give the following definition. Definition 3.10. Let (X, C) be an M -fuzzifying convex structure. (X, C) is of M -fuzzifying arity ≤ n if it satisfies ^ ^ ∀A ⊆ X, C(A) = (coC (F )(x))0 . x∈A / F ∈2A f in ,|F |≤n


153

Theorem 3.11. An M -fuzzifying convex structure is induced by an M -fuzzifying interval operator if and only if it is of M -fuzzifying arity ≤ 2. Proof. Necessity. Let (X, CI ) be the M -fuzzifying convex structure induced by an M -fuzzifying interval operator (X, I) and SegCI denote the M -fuzzifying segment operator V of (X,VCI ). Next we prove (X, CI ) is of M -fuzzifying arity ≤ 2, that is, 0 CI (A) = z∈A / {x,y}⊆A (coCI {x, y}(z)) . Since I ≤ SegCI , we know that CI (A) ≥ V V 0 z ∈A / {x,y}⊆A (coCI {x, y}(z)) . For each a ∈ M \{⊥}, let CI (A) ≥ a and coa denote the hull operator of (X, (CI )[a] ). Then CI (A) ≥ a

⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ V

A ∈ (C SI )[a] A = {coa {x, y} : {x, y} ⊆ A} ∀z ∈ / A, ∀{x, y} ⊆ A, z ∈ / coa ({x, y}) z∈ / coa {x, y} ⊇ {x, y}, coa {x, y} ∈ (CI )[a] zV∈ / coV a {x, y} ⊇ {x, W y}, CI (coa {x, y}) ≥ a ( CI (B)) ≥ a / V{x,y}⊆A Vz ∈B⊇{x,y} / Vz∈A CI (B)0 )0 ≥ a / V{x,y}⊆A ( z ∈B⊇{x,y} / Vz∈A 0 z ∈A / {x,y}⊆A (coCI {x, y}(z)) ≥ a. V 0 z ∈A / {x,y}⊆A (coCI {x, y}(z)) . Therefore the proof of ne-

This implies CI (A) ≤ cessity is completed. Sufficiency. If (X, C) is of M -fuzzifying arity ≤ 2, then ^ ^ ^ ^ C(A) = (coC {x, y}(z))0 = (SegC (x, y)(z))0 . z ∈A / {x,y}⊆A

z ∈A / x,y∈A

Let IC = SegC . Obviously, IC is an M -fuzzifying interval operator on X. So (X, C) can be induced by the M -fuzzifying interval operator IC . Theorem 3.12. Let (X, C) be an M -fuzzifying convex structure of M -fuzzifying arity ≤ 2. Then C = CIC . Proof. By Theorems 3.6 and 3.11, for all A ∈ 2X , ^ ^ ^ ^ CIC (A) = (IC (x, y)(z))0 = z ∈A / x,y∈A

(coC {x, y}(z))0 = C(A).

z ∈A / {x,y}⊆A

Therefore C = CIC .

4. MYCSA2 as a Subcategory of MYIS In this section, we intend to investigate relations between M -fuzzifying interval spaces and M -fuzzifying convex structures of M -fuzzifying arity ≤ 2 from a categorical aspect. The category of M -fuzzifying convex structures of M -fuzzifying arity ≤ 2 and M -fuzzifying convexity preserving functions is denoted by MYCSA2. Definition 4.1. Let (X, IX ) and (Y, IY ) be M -fuzzifying interval spaces. (1) If a function f : X −→ Y satisfies the following condition: for x, y ∈ X, → fM (IX (x, y)) ≤ IY (f (x), f (y)),

then f is called an M -fuzzifying interval preserving function.

154


(2) If a function f : X −→ Y satisfies the following condition: for x, y ∈ X, → fM (IX (x, y)) = IY (f (x), f (y)),

then f is called an M -fuzzifying interval-to-interval function. The category of M -fuzzifying interval spaces and M -fuzzifying interval preserving functions is denoted by MYIS. Theorem 4.2. Let (X, CX ) and (Y, CY ) be M -fuzzifying convex structures and coX , coY denote their M -fuzzifying hull operators, respectively. Then f : X −→ Y is an → M -fuzzifying convexity preserving function if and only if fM (coX (F )) ≤ coY (f (F )) X for any F ∈ 2f in . If (X, CX ) is of M -fuzzifying arity ≤ n, then it suffices to consider sets |F | ≤ n. W Proof. Necessity. We need to prove f (x)=y coX (F )(x) ≤ coY (f (F ))(y) for any F ∈ 2X f in and for any y ∈ Y , that is, for y ∈ Y , ∀x ∈ X, if f (x) = y, then coX (F )(x) ≤ coY (f (F ))(y) for any F ∈ 2X . For each D ∈ 2Yfin , if y ∈ / D ⊇ f (F ), f in −1 −1 then x ∈ / f (D) ⊇ F . Since CX f (D) ≥ CY (D), we know that V coX (F )(x) = CX (B)0 / Vx∈B⊇F −1 ≤ (D))0 / −1 (D)⊇F CX (f Vx∈f 0 ≤ y ∈D⊇f / (F ) CY (D) = coY (f (F ))(y). W This implies f (x)=y coX (F )(x) ≤ coY (f (F ))(y) for any F ∈ 2X f in and for any y ∈Y. Sufficiency. We need to prove CX f −1 (B) ≥ CY (B) for all B ∈ 2Y , that is, V V V V 0 0 ≥ f −1 (B) (coX (F )(p)) p∈f / −1 (B) q ∈B / G∈2B (coY (G)(q)) . F ∈2f in

f in

f −1 (B)

∀p ∈ / f −1 (B) and F ∈ 2f in , let y = f (p). Then y = f (p) ∈ / B and f (F ) ∈ B → 2f in . By fM (coX (F )) ≤ coY (f (F )) for any F ∈ 2X , we know that coX (F )(p) ≤ f in coY (f (F ))(y) and then (coX (F )(p))0 ≥ (coY (f (F ))(y))0 . So V V 0 CX f −1 (B) = f −1 (B) (coX (F )(p)) p∈f / −1 (B) V V F ∈2f in 0 ≥ B (coY (f (F ))(f (p))) / Vf (p)∈B V f (F )∈2f in ≥ (coY (G)(q))0 q ∈B / G∈2B f in = CY (B). By Definition 3.10, we can easily obtain that if (X, CX ) is of M -fuzzifying arity ≤ n, then in the above proof it suffices to consider sets |F | ≤ n. Theorem 4.3. Let (X, CX ) and (Y, CY ) be M -fuzzifying convex structures and coX , coY denote their M -fuzzifying hull operators, respectively. Then f : X −→ Y is an → M -fuzzifying convex-to-convex function if and only if fM (coX (F )) ≥ coY (f (F )) for X any F ∈ 2f in . If (Y, CY ) is of M -fuzzifying arity ≤ n, then it suffices to consider sets |F | ≤ n.


155

W Proof. Necessity. We need to prove f (x)=y coX (F )(x) ≥ coY (f (F ))(y) for any W V V 0 F ∈ 2X CX (B)0 ≥ y∈D⊇f f in and for any y ∈ Y , that is, f (x)=y x∈B⊇F / / (F ) CY (D) X a a for any F ∈ 2f in and for any y ∈ Y . For any a ∈ β(>), let coX and coY denote hull 0 0 operators of (X, (CX )[a ] ) and (Y, (CY )[a ] ), respactively. By Theorems 2.7 and 2.13, we can obtain coaY (f (F )) ⊆ f (coaX (F )) for any F ∈ 2X f in and for any a ∈ β(>). Let V 0 a be any element in β(>) such that y∈D⊇f C (D) a. Then we have X / (F ) V 0 / D, then a ≺ CY (D)0 y ∈D⊇f / (F ) CY (D) a ⇒ ∀D ⊇ f (F ), if y ∈ ⇒ ∀D ⊇ f (F ), if y ∈ / D, then CY (D) ≺op a0 ⇒ ∀D ⊇ f (F ), if y ∈ / D, then a0 ∈ α(CY (D)) 0 ⇒ ∀D ⊇ f (F ), if y ∈ / D, then D ∈ / (CY )[a ] ⇒ y ∈ coaY (f (F )) ⇒ y ∈ coaY (f (F )) ⊆ f (coaX (F )) ⇒ ∃x ∈ coaX (F ), s.t., f (x) = y 0 ⇒ ∀B ⊇ F, if x ∈ / B, then B ∈ / (CX )[a ] ⇒ ∀B ⊇ F, if x ∈ / B, then a0 ∈ α(CX (B)) ⇒ ∀B ⊇ F, if x ∈ / B, then CX (B) ≺op a0 ⇒ V ∀B ⊇ F, if x ∈ / B, then a ≺ CX (B)0 ⇒ C (B)0 ≥ a / Wx∈B⊇F VX ⇒ CX (B)0 ≥ a / Wf (x)=y x∈B⊇F ⇒ f (x)=y coX (F )(x) ≥ a. W This implies f (x)=y coX (F )(x) ≥ coY (f (F ))(y). Sufficiency. We need to prove CY (f (A)) ≥ CX (A) for all A ∈ 2X , that is, ^ ^ ^ ^ (coY (G)(z))0 ≥ (coX (F )(x))0 . x∈A / F ∈2A f in

z ∈f / (A) G∈2f (A) f in

→ Note that for any F ∈ 2X )) ≥ coY (f (F )), that f in and for any y ∈ Y , fM (coX (F W W is, f (x)=y coX (F )(x) ≥ coY (f (F ))(y). It follows that ( f (x)=y coX (F )(x))0 ≤ W (coY (f (F ))(y))0 and then α(( f (x)=y coX (F )(x))0 ) ⊇ α((coY (f (F ))(y))0 ). Hence S we obtain f (x)=y α((coX (F )(x))0 ) ⊇ α((coY (f (F ))(y))0 ). V V 0 Let a be any element in M such that a ∈ α( z∈f f (A) (coY (G)(z)) ). / (A) G∈2 f in

Then ⇒ ⇒

V V 0 a ∈ α( z∈f f (A) (coY (G)(z)) ) / (A) S VG∈2f in 0 a ∈ z∈f (A) (coY (G)(z)) ) / (A) α( G∈2ff in S S 0 a ∈ z∈f f (A) α((coY (G)(z)) ) / (A) G∈2 f in

f (A)

⇒ ∃z ∈ / f (A), G ∈ 2f in , s.t., a ∈ α((coY (G)(z))0 ) W f (A) ⇒ ∃z ∈ / f (A), G ∈ 2f in , s.t., a ∈ α(( f (x)=z (coX (H)(x))0 ), where f (H) = G and H is f inite S f (A) ⇒ ∃z ∈ / f (A), G ∈ 2f in , s.t., a ∈ f (x)=z α((coX (H)(x))0 ) ⇒ ∃x ∈ X, / A, s.t., a ∈ α((coX (H)(x))0 ) V f (x)V= z, x ∈ 0 ⇒ a ∈ α( x∈A (co X (F )(x)) ), / F ∈2A f in

156


This implies CY (f (A)) ≥ CX (A) for any A ∈ 2X . By Definition 3.10, we can easily obtain that if (Y, CY ) is of M -fuzzifying arity ≤ n, then in the above proof it suffices to consider sets |F | ≤ n. By Theorems 4.2 and 4.3, we can easily obtain the following three theorems. Theorem 4.4. Let (X, CX ) and (Y, CY ) be M -fuzzifying convex structures and let SegCX , SegCY be their M -fuzzifying segment operators, respectively. (1) If f : (X, CX ) −→ (Y, CY ) is M -fuzzifying convexity preserving, then f : (X, SegCX ) −→ (Y, SegCY ) is M -fuzzifying interval preserving. (2) If f : (X, CX ) −→ (Y, CY ) is M -fuzzifying convexity preserving and M fuzzifying convex-to-convex, then f : (X, SegCX ) −→ (Y, SegCY ) is M -fuzzifying interval-to-interval. Theorem 4.5. Let (X, CX ) and (Y, CY ) be M -fuzzifying convex structures of M fuzzifying arity ≤ 2. (1) If f : (X, SegCX ) −→ (Y, SegCY ) is M -fuzzifying convexity preserving, then f : (X, CX ) −→ (Y, CY ) is M -fuzzifying convexity preserving. (2) If f : (X, SegCX ) −→ (Y, SegCY ) is M -fuzzifying interval-to-interval, then f : (X, CX ) −→ (Y, CY ) is M -fuzzifying convexity preserving and M -fuzzifying convexto-convex. Theorem 4.6. Let (X, IX ) and (Y, IY ) be M -fuzzifying interval spaces. If f : (X, IX ) −→ (Y, IY ) is M -fuzzifying convexity preserving, then f : (X, CIX ) −→ (Y, CIY ) is M -fuzzifying convexity preserving. Applying Theorems 3.6, 3.11, 4.4, 4.5 and 4.6, we find a functor F from MYCSA2 to MYIS, where F is defined by F : MYCSA2 −→ MYIS such that for all (X, C) ∈ |MYCSA2|, F((X, C)) = (X, IC ), for all f : (X, CX ) −→ (Y, CY ), F(f ) = f . Meanwhile, there exists a functor G from MYIS to MYCSA2, where G is defined by G : MYIS −→ MYCSA2 such that for all (X, I) ∈ |MYIS|, F((X, I)) = (X, CI ), for all f : (X, IX ) −→ (Y, IY ), G(f ) = f . Moreover, we have G ◦ F((X, C)) = (X, C) and F ◦ G((X, I)) ≥ (X, I). To sum up, we get the following theorem. Theorem 4.7. The category MYCSA2 can be embedded in the category MYIS as a reflective subcategory. Next we give an example to show MYCSA2 is not isomorphic to MYIS with respect to F and G. Example 4.8. Let (X, ρ) be the finite metric space, presented in FIGURE 1 as a weighted graph. Let (X, I) be an interval space induced by (X, ρ), where I(x, y) = {z|ρ(x, z) + ρ(z, y) = ρ(x, y)} and then (X, χI ) is an M -fuzzifying interval space, where M = {⊥, >}. Let (X, CI ) be a convex structure induced by (X, I) and then (X, χCI ) is an M -fuzzifying convex structure. We can see that I(x, y) = {x, y, z, q} and I(z, q) = {x, y, z, p, q}. It is obvious that I(z, q) ⊆ I(x, y) does not hold. So I(x, y) is not a convex set in (X, CI ). Let Seg be the segment operator of (X, χCI ). Then χI (x, y) 6= Seg(x, y). Therefore F ◦ G((X, χI )) 6= (X, χI ).


157

z 1

1 x

1

1.1 p 1

1

y

1 1

q

Figure 1. Weighted Graph of (X, ρ) 5. Subspaces and Product Spaces Theorem 5.1. Let (X, IX ) be an M -fuzzifying interval space, ∅ = 6 Y ⊆ X. Then (Y, IY ) is an M -fuzzifying interval space on Y , where IY = IX |Y . We call (Y, IY ) a subspace of (X, IX ). Proof. It is easy to verify that IY satisfies (MYI1) and (MYI2).

Theorem 5.2. Let (X, IX ) be an M -fuzzifying interval space, ∅ = 6 Y ⊆ X, and let i : Y −→ X be the inclusion mapping. Then i is an M -fuzzifying interval preserving function from (Y, IY ) to (X, IX ). Proof. The proof is obvious and is omitted.

Next we can easily obtain the following result, which is the generalization of the theorem in [24]: the relative interval operator induces a convexity which is finer than or equal to the relative convexity. That is if (X, IX ) is an interval space and ∅= 6 Y ⊆ X, then CIX |Y ⊆ CIY . Theorem 5.3. Let (X, IX ) be an M -fuzzifying interval space, ∅ = 6 Y ⊆ X. Then CIX |Y ≤ CIY . Proof. For A ⊆ Y , by Theorems 2.18 and 3.6, we have W (CIX |Y )(A) = {CIX (B) : B ∈ 2X , B ∩ Y = A} =

W

B∈2X ,B∩Y =A

V

p∈B /

V

0 m,n∈B (IX (m, n)(p)) .

Let a be any element in M \{⊥} such that (CIX |Y )(A) a. Then CIX |Y (A) a

⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

∃B ∈ 2X , B ∩ Y = A, ∀p ∈ / B, ∀m, n ∈ B, (IX (m, n)(p))0 ≥ a ∃B ∈ 2X , B ∩ Y = A, B ∈ C(IX )a0 A ∈ C(IX )a0 |Y A ∈ C(IX )a0 |Y 0 ∀z ∈ / A, ∀x, y ∈ A, z ∈ / (IX )a |Y (x, y) 0 ∀z ∈ / A, / IY (x, y)(a ) V V ∀x, y ∈ A, z ∈ 0 z ∈A / x,y∈A (IY (x, y)(z)) ≥ a C(IY ) (A) ≥ a.

158


This implies CIX |Y ≤ CIY .

Theorem 5.4. Let {(Xt , It )}t∈T be a family of M -fuzzifying interval spaces. Let X be the productVof the sets Xt for t ∈ T . Define a mapping I : X × X −→ M X by I(x, y)(z) = t∈T It (xt , yt )(zt ), where x = (xt )t∈T , y = (yt )t∈T , z = (zt )t∈T . Then (X, I) is an M -fuzzifying interval space Q on X. We call (X, I) the product space of {(Xt , It )}t∈T and denote it by (X, t∈T It ). Proof. The proof is obvious and is omitted.

Theorem 5.5. Let (X, I) be the product space of a family of M -fuzzifying interval spaces {(Xt , It )}t∈T . Let πt : X −→ Xt denote the projection for each t ∈ T . Then for each t ∈ T , πt is an M -fuzzifying interval preserving function from (X, I) to (Xt , It ) . Proof. The proof is obvious and is omitted.

Theorem 5.6. Let (X, I) be the product space of a family of M -fuzzifying in0 terval spaces {(Xt , It )}t∈T , where T is finite. Then for each a ∈ J(M ), I (a ) = 0 Q (a ) . t∈T It Proof. For each a ∈ J(M ) and for all x, y, z ∈ X, 0

∀z ∈ / I (a ) (x, y) ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ 0

This implies for each a ∈ J(M ), I (a )

I(x, y)(z) ≤ a0 0 (I(x, V y)(z)) ≥ a (W t∈T It (xt , yt )(zt ))0 ≥ a 0 t∈T (It (xt , yt )(zt )) ≥ a ∃t ∈ T, (It (xt , yt )(zt ))0 ≥ a ∃t ∈ T, It (xt , yt )(zt ) ≤ a0 0 ∃t ∈ T, zt ∈ / It (xt , yt )(a ) 0 Q (a ) z∈ / t∈T It (x, y). Q (a0 ) = t∈T It .

Theorem 5.7. Let {(X, CF t )}t∈T be a family of M W-fuzzifying Vconvex structures, where T is finite and C = t∈T Ct . Then C(A) = T Bt =A t∈T Ct (Bt ) for all t∈T

A ∈ 2X . W V b Proof. Let C(A) = T Bt =A t∈T Ct (Bt ). Next we prove Cb is an M -fuzzifying t∈T convexity. b = C(X) b (MYC1) It is easy to verify that C(∅) = >. T V b b λ ) for any nonempty (MYC2) We need to prove that C( λ∈Λ Aλ ) ≥ λ∈Λ C(A X {Aλ : λ ∈ Λ} ⊆ 2 . Let a be any element in M \{⊥} such that ^ _ ^ ^ b λ ). a≺ Ct (Bt,λ ) = C(A λ∈Λ

T

t∈T

Bt,λ =Aλ t∈T

λ∈Λ

T Then for each λ ∈ Λ, there exists a set {Bt,λ : t ∈ T } ⊆ 2X such that T {Bt,λ : t ∈ T } = Aλ and ∀t ∈ T , Ct (Bt,λ ) ≥ a, that is Bt,λ ∈ (Ct )[a] . Hence {Bt,λ :


159

T T T T λ T ∈ Λ} T ∈ (Ct )[a] and then Ct ( λ∈Λ Bt,λ ) ≥ a. By λ∈Λ Aλ = λ∈Λ t∈T Bt,λ = t∈T λ∈Λ Bt,λ , we have C(

\

_

Aλ ) = T

λ∈Λ

t∈T

T

λ∈Λ

^

T Bt,λ = λ∈Λ Aλ t∈T

Ct (

\

Bt,λ ) ≥ a.

λ∈Λ

V bT b This implies C( λ∈Λ Aλ ) ≥ λ∈Λ C(Aλ ). bS (MYC3) We need to prove that C(

V b λ ) for any nonempty Aλ ) ≥ λ∈Λ C(A and up-directed {Aλ : λ ∈ Λ} ⊆ 2X . Let a be any element in M \{⊥} such V W V V b λ ). Then for each λ ∈ Λ, that a ≺ λ∈Λ T Bt,λ =Aλ t∈T Ct (Bt,λ ) = λ∈Λ C(A t∈T T there exists an up-directed set {Bt,λ : t ∈ T } ⊆ 2X such that {Bt,λ : t ∈ T } = Aλ and ∀t ∈ T , Ct (Bt,λ ) ≥ a, that is Bt,λ ∈ (Ct )[a] . LetTcoat denote the hull operator of (X, (Ct )[a] ). Then coat (Aλ ) ⊆ Bt,λ and Aλ = {coat (Aλ ) : a t ∈ T }.S Since {Aλ : λ ∈ Λ} is up-directed, {coS t (Aλ ) : λ ∈ Λ} is up-directed. a a (A ) : λ ∈ Λ} ∈ (C ) . Then C ( ≥ a. Let Dt = Hence {co λ t T t [a] t λ∈Λ cot (A S T S S λ )) T a a co (A ). Since D = co (A ) = coa (Aλ ) = t λ λ∈Λ Sλ∈Λ t λ Vt∈T t S t∈T W t∈T λ∈Λ t V b T S t∈T Ct (Dt ) ≥ a. t∈T Ct (Bt ) ≥ λ∈Λ Aλ , we obtain C( λ∈Λ Aλ ) = t∈T Bt = λ∈Λ Aλ λ∈Λ

V

S

b b Therefore C( λ∈Λ C(Aλ ). λ∈Λ Aλ ) ≥ W Let D be an M -fuzzifying convexity satisfying D(A) ≥ ϕ(A) = t∈T Ct (A) for T each A ∈ 2X . Then for any set {Bt : t ∈ T } ⊆ 2X with t∈T Bt = A, ^

Ct (Bt ) ≤

t∈T

^

D(Bt ) ≤ D(

t∈T

\

Bt ) = D(A).

t∈T

W V b Hence C(A) = T Bt =A t∈T Ct (Bt ) ≤ D(A). We know that ϕ is a subbase of t∈T Cb and thus Cb is the joint of {Ct : t ∈ T }. Therefore Cb = C. Theorem 5.8. Let {(Xt , Ct )}t∈T be a family of M -fuzzifying convex structures, where T is V finite. Let (X, C) be the product of {(Xt , Ct )}t∈T . Then C(A) = W X Q Bt =A t∈T Ct (Bt ) for all A ∈ 2 . t∈T

Proof. By Theorem 2.20, {πt−1W(Ct )|t ∈ T } V is a family of M -fuzzifying convexities on W X. By Theorem 5.7, C(A) = T Bt =A t∈T π−1 (Dt )=Bt Ct (Dt ) for all A ∈ 2X , t t∈T V W where Dt ⊆ Xt and Bt ⊆ X. Next we prove that C(A) = Q Gt =A t∈T Ct (Gt ). t∈T On one hand, let a be any element in M \{⊥} such that a ≺ C(A). Then there T exists {Bt |t ∈ T } ⊆ 2X such that t∈T Bt = A and for each t ∈ T , πt−1 (Dt ) = Bt Q Q T and Ct (Dt ) ≥ a. Let Gt = Dt . Then we W get t∈T Gt V = t∈T Dt = t∈T πt−1 (Dt ) = T Bt = A and Ct (Gt ) ≥ a. Hence Q Gt =A t∈T Ct (Gt ) ≥ a. This implies t∈T Wt∈T V Q C (G ) ≥ C(A). On the other hand, let a be any element in t t G =A t∈T t t∈T W V M \{⊥} such that a ≺ Q Gt =A t∈T Ct (Gt ). Then there exists {Gt |t ∈ T } t∈T Q such that t∈T Gt = A and for each t ∈ T , Ct (Gt ) ≥ a. Let Dt = Gt and T T Q Q −1 Bt =Wπt−1 (Dt ). Then we get W t∈T Bt = t∈T πt (Dt ) = t∈T Dt = t∈T Gt = A and π−1 (Dt )=Bt Ct (Dt ) = π−1 (Dt )=Bt Ct (Gt ) ≥ a. Hence t

t

_ T

t∈T

^

_

Bt =A t∈T π −1 (Dt )=Bt t

Ct (Dt ) ≥ a.

160


This implies C(A) ≥

W

Therefore C(A) =

W

Q

t∈T

Q

t∈T

Gt =A

Bt =A

V

V t∈T t∈T

Ct (Gt ). Ct (Bt ) for all A ∈ 2X .

Theorem 5.9. [24] Suppose that (X, I) is the product of a family of interval spaces {(Xt , It )}t∈T , where T is finite. Let CI be the Q convexity induced by I and ∀t ∈ T , CIt be the convexity induced by It . Then CI = t∈T CIt . Based on Theorem 5.9, we give the following analysis: A ∈ CI

⇔ ⇔ ⇔ ⇔ ⇔

∀z ∈ / A, ∀x, y ∈ A, z ∈ / I(x, y) ∀z ∈ /Q A, ∀x, y ∈ A, ∃t ∈ T, s.t., zt ∈ / It (xt , yt ) A ∈ t∈T CIt Q ∀t ∈ T, At ∈ CIt , where t∈T At = A ∀t ∈ T, ∀h ∈ / At , ∀p, q ∈ At , h ∈ / It (p, q).

Next, we give the generalization of Theorem 5.9. Theorem 5.10. Let (X, I) be the product of Q a family of M -fuzzifying interval spaces {(Xt , It )}t∈T , where T is finite. Then t∈T CIt = CI . Proof. For all A ∈ 2X , by Theorems 3.6 and 5.4, we have V V CI (A) = y)(z))0 / Vx,y∈A (I(x, Vz∈A V 0 = / Vx,y∈A (W t∈T It (xt , yt )(zt )) Vz∈A 0 = z ∈A / x,y∈A t∈T (It (xt , yt )(zt )) . Hence, for each a ∈ J(M ), A ∈ (CI )[a]

⇔ ⇔

CI (A) ≥ a ∀z ∈ / A, ∀x, y ∈ A, ∃t ∈ T, s.t., (It (xt , yt )(zt ))0 ≥ a ⇔ ∀z ∈ / A, ∀x, y ∈ A, ∃t ∈ T, 0 s.t., zt ∈ / (It (xt , yt ))(a ) ⇔ A ∈ CI (a0 ) .

By Theorem 5.8, for each a ∈ J(M ), Q Q A ∈ ( t∈T CIt )[a] ⇔ W t∈T CIt (A)V≥ a Q ⇔ t∈T CIt (Bt ) ≥ a t∈T Bt =A Q ⇔ ∃{Bt |t ∈VT }, s.t., t∈T Bt = A, and ∀t ∈ T, CIt (Bt ) ≥ a V 0 ⇔ ∀t ∈ T, h∈A / p,q∈Bt (It (p, q)(h)) ≥ a 0 ⇔ ∀t ∈ T, ∀h ∈ / A, ∀p, q ∈ Bt , h ∈ / It (p, q)(a ) ⇔ ∀t ∈ T, Bt ∈ CI (a0 ) t Q ⇔ A ∈ t∈T CI (a0 ) . t

Q By Theorem 5.6, for each a ∈ J(M ), CI (a0 ) = t∈T CI (a0 ) . Hence for each a ∈ J(M ), t Q Q ( t∈T CIt )[a] = (CI )[a] . Therefore t∈T CIt = CI .


161

6. Conclusion In this paper, the notion of M -fuzzifying interval spaces is introduced. This approach to the fuzzification of interval spaces preserves many basic properties of crisp interval spaces. It is proved that the category MYCSA2 can be embedded in the category MYIS as a reflective subcategory. Moreover, the formula of the M -fuzzifying hull operator of an M -fuzzifying convex structure induced by an M -fuzzifying interval space is given. Equivalent characterizations of M -fuzzifying convexity preserving functions and M -fuzzifying convexity-to-convexity functions are obtained by means of M -fuzzifying hull operators. Subspaces and product spaces of M -fuzzifying interval spaces are presented and some of their fundamental properties are obtained. These facts will be useful to help further investigations of M -fuzzifying convex structures. Acknowledgements. Authors would like to express their sincere thanks to the anonymous reviewers for their valuable comments. This work is supported by the National Natural Science Foundation of China (11371002), Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048) and Project (KYTZ201631,CRF201611) Supported by the Scientific Research Foundation of CUIT References [1] M. Berger, Convexity, American Mathematical Monthly, 97(8) (1990), 650–678. [2] P. Dwinger, Characterizations of the complete homomorphic images of a completely distributive complete lattice I, Indagationes Mathematicae (Proceedings), 85 (1982), 403–414. [3] J. Eckhoff, Helly, Radon, and Caratheodory type theorems, Handbook of convex geometrry, Vol. A, B, North-Holland (1993), 389–448. [4] J. M. Fang, Sums of L-fuzzy topological spaces, Fuzzy Sets and Systems, 157 (2005), 739–754. [5] G. Gierz, et al., Continuous lattices and domains, Encyclopedia of Mathematics and its Applications, 93, Cambridge University Press, Cambridge, 2003. [6] T. E. Gantner, R. C. Steinlage and R. H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl., 62 (1978), 547–562. [7] H. L. Huang and F. G. Shi, L-fuzzy numbers and their properties, Information Sciences, 178 (2008), 1141–1151. [8] U. H¨ ohle, Probabilistsche Metriken auf der Menge nicht negativen verteilungsfunktionen, Aequationes Math., 18 (1978), 345–356. [9] W. Kubis, Abstract convex structures in topology and set theory, PhD thesis, University of Silesia Katowice, 1999. [10] N. N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets and Systems, 27 (1988), 351–372. [11] Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 1641 (2009), 22–37. [12] C. V. Negoita and D. A. Ralescu, Applications of fuzzy sets to systems analysis, Interdisciplinary Systems Research Series, vol. 11, Birkhauser, Basel, Stuttgart and Halsted Press, New York, 1975. [13] S. Philip, Studies on fuzzy matroids and related topics, PhD thesis, Cochin University of Science and Technology, 2010. [14] S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and compactifications, in: U.¨ oHle, S.E. Rodabaugh (Eds.), Mathematics of fuzzy sets: logic, topology, and measure theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Dordrecht, (1999), 481–552.

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