Convexity on fuzzy partially ordered sets

1 Introduction

The initial concept of convex structures is mainly defined and studied in ℝ n in the pioneering works of Newton, Minkowski and others [3, 6], and it has been accepted to be of increasing importance in the study of extremum problems in areas of applied mathematics. Actually, convex structures exists in so many mathematical research areas, such as vector spaces, metric spaces, graphs, matroids, median algebras, lattices and so on.

Based on axiomatizing the properties of convex sets in ℝ n , in literature [26], the notion of convex structures (also called abstract convexity) on a non-empty set X is defined to be a subfamily of 2 ^X which contains both the empty set ∅ and X itself and which is closed under arbitrary intersections and nested unions. Soon later, the notion of convex structures has been extended to fuzzy setting by means of the fuzzy set theory which was introduced by Zadeh [39]. In 1994, Rosa [19, 20] proposed the concept of a fuzzy convex structure as a subfamily of [0, 1] ^X . In 2009, Y. Maruyama developed the definition of a fuzzy convex structure by generalizing the lattice [0, 1] to a completely distributive lattice L in [16], which is called an L-convex structure. For this kind of fuzzy convex structure, there are some other studies [10 , 21] in the last two years. In 2014, Shi and Xiu [23] introduced a new approach to the fuzzification of convex structures, which is called an M-fuzzifying convex structure. There is also some other research work on M-fuzzifying convexity, such as [24 , 28–32]. Recently, Shi and Xiu [25] present a more general approach to the the fuzzification of convex structures, which is called an (L, M)-fuzzy convex structure. It is a generalization of an L-convex structure and an M-fuzzifying convex structure. Very soon, Li [3] investigates (L, M)-fuzzy convex structure from the point of view of category.

Order convexity is first studied on partially ordered sets [26], it can also be imported to semilattices and lattices. Based on that, Zhong and Shi lately introduced a kind of L-convex structure on partially ordered sets [43], it is proved that such an ordered L-convex structure and its L-convexity-preserving mapping has the similar properties as an ordered convex structure. Besides, it is also imported to semilattices and lattices as well as the way of that order convexity imported to semilattices and lattices. Almost at the same time, Li and Shi [12] presented an L-fuzzy convex structure on a lattice by studying the L-convex fuzzy sublattice degrees.

With the development of fuzzy mathematics and fuzzy logic, fuzzy order has attracted more and more attentions. There are so many literatures related to it [8 , 40–42]. In literatures [4], in order to construct a quotient L-ordered group, convex L-subgroups of L-ordered groups are introduced and some characterizations are given. After that, literatures [5] discussed how to construct a convex L-subgroup via an L-subset in an L-ordered group. The present paper investigates the properties of convex sets in fuzzy order structure without a consideration of group structure, then proposes an L-ordered L-convex structure and studies its corresponding (convex) hull, convexity-preserving mappings and the properties of the product of L-ordered L-convex sets. Furthermore, we consider the degree to which an L-subset of an L-poset is a convex set and obtain an L-ordered (L, L)-fuzzy convex structure, and also analyze their convexity-preserving mappings and convex-to-convex mappings as well as the convexity degree of the product of finite L-subsets.

The paper is organized as follows. In Section 2, we recall some necessary definitions and results which are needed later on. In Section 3, we introduce L-ordered L-convex structure on L-posets, and investigate its L-convex hulls, the properties of the product of L-ordered L-convex sets and L-convexity-preserving mappings. In Section 4, we introduce L-ordered (L, L)-fuzzy convexity on L-posets, then study its characterizations and discuss its product and (L, L)-fuzzy convexity-preserving mappings. In the final section, we summarize the results and draw a conclusion.

2 Preliminaries

Throughout this paper, unless otherwise stated, L always denotes a complete Heyting algebra or a frame. In other words, L is a complete lattice satisfying the infinite distributive law of finite meets over arbitrary joins i.e., a ∧ (⋁ _b∈B b) = ⋁ _b∈B (a ∧ b) for any a ∈ L and B ⊆ L. The smallest element and the largest element in L are denoted by 0 and 1, respectively. An element a ∈ L is called a co-prime element if a ≤ b ∨ c implies a ≤ b or a ≤ c, and the set of non-zero co-prime elements in L is denoted by J (L). An element a ∈ L is called a prime element if b ∧ c ≤ a implies b ≤ a or c ≤ a, and the set of non-unit prime elements in L is denoted by P (L). For all a, b ∈ L, we say that a is wedge below b, in symbols a ≺ b, if for every subset D ⊆ L with b ≤ ⋁ D, there exists d ∈ D such that a ≤ d. Similarly, we define b ≺ ^op a if and only if for every subset D ⊆ L with ⋀D ≤ b, there exists d ∈ D such that d ≤ a.

Let β (b) = {a ∈ L | a ≺ b}, β ^∗ (b) = β (b) ∩ J (L); α (b) = {a ∈ L | b ≺ ^op a}, α ^∗ (b) = α (b) ∩ P (L).

In particular, L is a completely distributive [7, 9] if and only if b = ⋁ β (b) for each b ∈ L. In this case, b = ⋁ β (b) = ⋁ β ^∗ (b) = ⋀ α (b) = ⋀ α ^∗ (b), and β (⋁ _i∈I b _i ) = ⋃ _i∈I β (b _i ), α (⋀ _i∈I b _i ) = ⋃ _i∈I α (b _i ) (refer to [27]).

Lemma 2.1. Let L be a completely distributive lattice. Then the following statements are equivalent:

(1) x ≤ y;

(2) for each a ∈ J (L), a ≤ x ⇒ a ≤ y;

(3) for each a ∈ P (L), xnleqa ⇒ ynleqa;

(4) for each a ∈ P (L), a ∉ α (x) ⇒ a ∉ α (y);

(5) for each a ∈ J (L), a ∈ β (x) ⇒ a ∈ β (y).

Proof. The proof is similar to that of Lemma 2.1 in [12].□

For any non-empty set X, let 2 ^X be the family of all subsets of X, and L ^X the set of all L-subsets of X. Then L ^X is also a complete lattice by defining ≤ on L ^X pointwisely. We always do not discriminate an element a ∈ L with the constant function a ¯ : X ⟶ L such that a ¯ ( x ) = a for all x ∈ X.

For each A ∈ L ^X and a ∈ L, we can define

A _[a] = {x ∈ X | A (x) ≥ a},

A ^(a) = {x ∈ X | A (x) nleqa}.

If L is a completely distributive lattice, we can define

A ^[a] = {x ∈ X | a ∉ α (A (x))},

A _(a) = {x ∈ X | a ∈ β (A (x))}.

Definition 2.2. [22] Let A ∈ L ^X and B ∈ L ^X . We define the product of A and B, in symbols A × B, to be an L-subset of X × Y: ( A × B ) ( x , y ) = A ( x ) ∧ B ( y ) ( ∀ ( x , y ) ∈ X × Y ) .

Proposition 2.3. [43] Let L be a complete Heyting algebra. If {A _i | i ∈ I} ⊆ L ^X is non-empty and totally ordered, then (⋁ _i∈I A _i (x)) ∧ (⋁ _j∈I A _j (y)) = ⋁ _i∈I (A _i (x) ∧ A _i (y)) for all x, y ∈ X.

For each map f : X ⟶ Y, there is a well-known L-forward powerset operator f L → : L X ⟶ L Y as f L → ( A ) ( y ) = ⋁ f ( x ) = y A ( x ) for all y ∈ Y, A ∈ L ^X , and an L-backward powerset operator f L ← : L Y ⟶ L X as f L ← ( B ) = B ∘ f for each B ∈ L ^Y .

Next, we recall the definitions of L-convexity and (L, M)-fuzzy convexity, respectively.

Definition 2.4. ([16]) A subset C ⊆ L X is called an L-convex structure, if it satisfies the following conditions: (LC1) 0 ¯ , 1 ¯ ∈ C ; (LC2) ⋀ i ∈ I A i ∈ C for all non-empty family { A i | i ∈ I } ⊆ C ; (LC3) ⋁ i ∈ I A i ∈ C for all family { A i | i ∈ I } ⊆ C which is non-empty and totally ordered.

The pair ( X , C ) is called an L-convex space. The members of C are called L-convex sets. For each A ∈ L ^X , co ( A ) = ⋀ { C ∈ L X | A ≤ C ∈ C } is called L-convex hull of A, which is the smallest L-convex set including A. It is easily seen that A ∈ C ⇔ co ( A ) = A .

Definition 2.5. ([16]) Let ( X , C X ) and ( Y , C Y ) be L-convex spaces. A mapping f : X ⟶ Y is said to be L-convexity-preserving if f L ← ( D ) ∈ C X for each D ∈ C Y . f is said to be L-convex-to-convex if f L → ( C ) ∈ C Y for each C ∈ C X .

Definition 2.6. ([25]) A mapping C : L X ⟶ M is called an (L, M)-fuzzy convex structure on X if it satisfies: (LMC1) C ( 0 ¯ ) = C ( 1 ¯ ) = 1 ; (LMC2) C ( ⋀ i ∈ I A i ) ≥ ⋀ i ∈ I C ( A i ) for all non-empty family { A i | i ∈ I } ⊆ C ; (LMC3) C ( ⋁ i ∈ I A i ) ≥ ⋀ i ∈ I C ( A i ) for all family { A i | i ∈ I } ⊆ C which is non-empty and totally ordered.

The pair ( X , C ) is called an (L, M)-fuzzy convex space.

Definition 2.7.([25]) Let ( X , C X ) and ( Y , C Y ) be (L, M)-fuzzy convex spaces. A mapping f : X ⟶ Y is said to be (L, M)-fuzzy-convexity-preserving if C X ( f L ← ( D ) ) ≥ C Y ( D ) for all D ∈ L ^Y . f is said to be (L, M)-fuzzy-convex-to-convex if C X ( D ) ≤ C Y ( f L → ( D ) ) for all D ∈ L ^X .

In particular, if M = 2, then an (L, 2)-fuzzy convexity is an L-convexity in Definition 2.4. If L = 2, then an (2, M)-fuzzy convexity is called an M-fuzzifying convexity in [23]. If M = L = 2, then an (2, 2)-fuzzy convexity is exactly a convexity in [26].

In the sequel of this section, let us recall some concepts and results about fuzzy partial order which are needed later on.

Definition 2.8. ([8 , 40]) A fuzzy preorder e (also called an L-preorder) on X is an L-relation satisfying:

∀x ∈ X, e (x, x) =1;

Remark 2.9. The original definition of a fuzzy partial order proposed by Fan and Zhang in [8, 40] is based on complete Heyting algebras, and it is generalized onto complete residuated lattices in [37]. In order to study fuzzy relational systems, Bělohlávek also defines and studies an L-order on a set [1, 2], and in [33, 34], Yao verifies that a fuzzy partial order in the sense of Fan and Zhang is equivalent to an L-order in the sense of Bělohlávek.

Let (X, e) be an L-poset. Then it is easy to check that (X, e _[a]) and (X, e ^(a)) are preordered sets for each a ∈ L. In particular, if L is a completely distributive lattice. Then (X, e ^[a]) is also an preordered set for each a ∈ L. In addition, if L satisfies β (b ∧ c) = β (b) ∩ β (c) for all b, c ∈ L. Then (X, e _(a)) is also a preordered set for each a ∈ L.

Proposition 2.10. ([11, 34]) Let {(X _i , e _i ) | i ∈ I} be a non-empty family of L-posets. Put X = ∏ i ∈ I X i and define e : X × X ⟶ L by ∀x = (x _i ) _i∈I , y = (y _i ) _i∈I ∈ X: e ( x , y ) = ⋀ i ∈ I e i ( x i , y i ) . Then (X, e) is an L-poset, called the product L-poset of {(X _i , e _i ) | i ∈ I}.

Definition 2.11. ([11, 40]) Let (X, e _X ), (Y, e _Y ) be L-posets. A mapping f : X ⟶ Y is said to be L-order-preserving (resp., L-order-reversing) if e _X (x, y) ≤ e _Y (f (x), f (y)) (resp., e _X (x, y) ≤ e _Y (f (y), f (x))) for all x, y ∈ X.

Example 2.12. By the definition of product of L-posets, it is easy to see that each projection p _i : Π _i∈I X _i ⟶ X _i is L-order-preserving.

3 L-ordered L-convex structure

The concept of ordered convex structure [26] is first proposed on a partially ordered set (poset for short) (X, ≤) via ordered convex sets, where A ⊆ X is an order convex set provided z ∈ A whenever x ≤ z ≤ y and x, y ∈ A. In fact, the definition is also adapt to a preordered set. When an order convex set was generalized to a fuzzy subset in a poset, then an ordered L-convex structure is obtained [43]. Both the ordered convex structures and ordered L-convex structures are relative to the classical order. There arises a question naturally: How to define convex structures with a fuzzy partially order? Inspired by the fact ordered convex structures is constructed by ordered convex sets, we will propose an L-ordered L-convex structure by means of L-convex sets in L-posets which introduced in [4, 5]. Definition 3.1. [4, 5] Let (X, e) be an L-poset and A ∈ L ^X . Then A is called an L-ordered L-convex set if e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. From now on, we use C e to denote the set of all L-ordered L-convex sets. Theorem 3.2. C e is an L-convex structure, where we call it L-ordered L-convex structure. Then ( X , C e ) is called an L-ordered L-convex space. Proof. (LC1) is obvious. To prove (LC2). Suppose { A i | i ∈ I } ⊆ C e is non-empty. Then for all x, y, z ∈ X, we have e ( x , z ) ∧ e ( z , y ) ∧ ( ⋀ i ∈ I A i ) ( x ) ∧ ( ⋀ i ∈ I A i ) ( y ) = e ( x , z ) ∧ e ( z , y ) ∧ ⋀ i ∈ I ( A i ( x ) ∧ A i ( y ) ) = ⋀ i ∈ I ( e ( x , z ) ∧ e ( z , y ) ∧ A i ( x ) ∧ A i ( y ) ) ≤ ⋀ i ∈ I A i ( z ) = ( ⋀ i ∈ I A i ) ( z ) . It implies that ⋀ i ∈ I A i ∈ C e .□ To prove (LC3). Let { A i | i ∈ I } ⊆ C e be non-empty and totally ordered. Then for all x, y, z ∈ X, we have e ( x , z ) ∧ e ( z , y ) ∧ ( ⋁ i ∈ I A i ) ( x ) ∧ ( ⋁ i ∈ I A i ) ( y ) = e ( x , z ) ∧ e ( z , y ) ∧ ⋁ i ∈ I ( A i ( x ) ∧ A i ( y ) ) = ⋁ i ∈ I ( e ( x , z ) ∧ e ( z , y ) ∧ A i ( x ) ∧ A i ( y ) ) ≤ ⋁ i ∈ I A i ( z ) = ( ⋁ i ∈ I A i ) ( z ) . It implies that ⋁ i ∈ I A i ∈ C e . Theorem 3.3. Let (X, e) be an L-poset and A ∈ L ^X . Then for all x ∈ X, co C e ( A ) ( x ) =

⋁ x 1 , x 2 ∈ X e ( x 1 , x ) ∧ e ( x , x 2 ) ∧ A ( x 1 ) ∧ A ( x 2 ) .

Proof. Denote C (x) = ⋁ _{x 1,x 2∈X} e (x ₁, x) ∧ e (x, x ₂) ∧ A (x ₁) ∧ A (x ₂), then it suffices to prove that C is the smallest L-ordered L-convex set including A. First. A ≤ C is obvious. Second. For all x, y, z ∈ X,

e ( x , z ) ∧ e ( z , y ) ∧ C ( x ) ∧ C ( y ) = e ( x , z ) ∧ e ( z , y ) ∧ ⋁ x 1 , x 2 ∈ X ( e ( x 1 , x ) ∧ e ( x , x 2 ) ∧ A ( x 1 ) ∧ A ( x 2 ) ) ∧ ⋁ y 1 , y 2 ∈ X ( e ( y 1 , y ) ∧ e ( y , y 2 ) ∧ A ( y 1 ) ∧ A ( y 2 ) ) ≤ ⋁ x 1 , y 2 ∈ X ( e ( x , z ) ∧ e ( z , y ) ∧ e ( x 1 , x ) ∧ e ( y , y 2 ) ∧ A ( x 1 ) ∧ A ( y 2 ) ) ≤ ⋁ x 1 , y 2 ∈ X ( e ( x 1 , z ) ∧ e ( z , y 2 ) ∧ A ( x 1 ) ∧ A ( y 2 ) ) = C ( z ) .

Hence C is an L-ordered L-convex set by Definition 3.1. Finally, suppose D ∈ C e and A ≤ D, it needs to show D ≥ C. For every x ∈ X, since D ∈ C e , so D (x) ≥ e (y, x) ∧ e (x, z) ∧ D (y) ∧ D (z) for all y, z ∈ X. Hence D (x) ≥ e (y, x) ∧ e (x, z) ∧ A (y) ∧ A (z), and it implies D (x) ≥ C (x). That is D ≥ C, as required.□ Theorem 3.4. Let (X, e) be an L-poset and A ∈ L ^X . Then for all x ∈ X,

co C e ( A ) ( x ) = ⋁ k = 0 ∞ P k ( A ) ( x ) ,

where P ₀ (A) = A and for k ≥ 0, P _k+1 (A) (x) = ⋁ _{x 1,x 2∈X} e (x ₁, x) ∧ e (x, x ₂) ∧ P _k (A) (x ₁) ∧ P _k (A) (x ₂). Proof. First of all, it is easy to see that P _k+1 (A) ≥ P _k (A) and P k ( A ) ∈ C e for each k ≥ 1. In other words, { P k ( A ) | k = 1 , 2 ⋯ } ⊆ C e is an ascending chain. Let C ( x ) = ⋁ k = 0 ∞ P k ( A ) ( x ) , then C ≥ A and C ∈ C e obviously. Moreover, C is the smallest L-ordered convex set including A. In fact, for every D ∈ C e and A ≤ D, we have D ≥ P ₀ (A). Suppose D ≥ P _k (A), then for all x, x ₁, x ₂ ∈ X, D ( x ) ≥ e ( x 1 , x ) ∧ e ( x , x 2 ) ∧ D ( x 1 ) ∧ D ( x 2 ) ≥ e ( x 1 , x ) ∧ e ( x , x 2 ) ∧ P k ( A ) ( x 1 ) ∧ P k ( A ) ( x 2 ) .

That implies D (x) ≥ P _k+1 (A) (x), and so D ( x ) ≥ C ( x ) = ⋁ k = 0 ∞ P k ( A ) ( x ) . Hence, C is the smallest L-ordered L-convex set including A, that is to say, co C e ( A ) ( x ) = C ( x ) = ⋁ k = 0 ∞ P k ( A ) ( x ) .□

Next, we will give some equivalent descriptions for L-ordered L-convex sets.

Proposition 3.5. Let (X, e) be an L-poset. If A ∈ L ^X is an L-ordered L-convex set, then (1) ∀a ∈ L, A _[a] is an ordered convex set in (X, e _[a]), (2) ∀a ∈ P (L), A ^(a) is an ordered convex set in (X, e ^(a)).

Proof. To prove (1). For each a ∈ L, suppose x, y ∈ A _[a] and (x, z), (z, y) ∈ e _[a]. Then A (x) ≥ a, A (y) ≥ a and e (x, z) ≥ a, e (z, y) ≥ a. Since A is an L-ordered L-convex set, that is, e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z). Therefore A (z) ≥ a, in other words, z ∈ A _[a]. Hence A _[a] is an ordered convex set in (X, e _[a]) by definition.

To prove (2). For each a ∈ P (L), suppose x, y ∈ A ^(a) and (x, z), (z, y) ∈ e ^(a). Then A (x) nleqa, A (y) nleqa and e (x, z) nleqa, e (z, y) nleqa. It results e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) nleqa because of a ∈ P (L). Note that A is an L-ordered L-convex set, that is, e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z). Then A (z) nleqa, in other words, z ∈ A ^(a). Thus A ^(a) is an ordered convex set in (X, e ^(a)).□

Theorem 3.6. Let (X, e) be an L-poset. Then A ∈ L ^X is an L-ordered L-convex set if and only if A _[a] is an ordered convex set in (X, e _[a]) for all a ∈ L.

Proof. Necessity. Straightforward from Proposition 3.5. Sufficiency. To prove A is L-ordered L-convex, we need to prove e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. It suffices to prove e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≥ a implies A (z) ≥ a for each a ∈ L. In fact, e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≥ a implies x, y ∈ A _[a] and (x, z), (z, y) ∈ e _[a]. So z ∈ A _[a] since A _[a] is an ordered convex set in (X, e _[a]), that is, A (z) ≥ a, as needed.

Theorem 3.7. Let (X, e) be an L-poset with L be a completely distributive lattice, and A ∈ L ^X . Then the following conditions are equivalent: (1) A is an L-ordered L-convex set;(2) A _[a] is an ordered convex set in (X, e _[a]) for all a ∈ L, (3) A _[a] is an ordered convex set in (X, e _[a]) for all a ∈ P (L), (4) A ^(a) is an ordered convex set in (X, e ^(a)) for all a ∈ P (L), (5) A ^[a] is an ordered convex set in (X, e ^[a]) for all a ∈ L, (6) A ^[a] is an ordered convex set in (X, e ^[a]) for all a ∈ P (L).

Proof. By Proposition 3.5, we have (1) ⇒ (2), (4), and (2) ⇒ (3), (5) ⇒ (6) is obvious. So we just need to prove (3) ⇒ (1), (4) ⇒ (1), (1) ⇒ (5) and (6) ⇒ (1).

(3) ⇒ (1) is similar to the proof of the sufficiency in Theorem 3.6. (4) ⇒ (1).

To prove that A is L-ordered L-convex, it suffices to prove e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. According to Lemma 2.1, it suffices to prove for each a ∈ P (L), e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) nleqa implies A (z) nleqa. In fact, e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) nleqa implies x, y ∈ A ^(a) and (x, z), (z, y) ∈ e ^(a). So z ∈ A ^(a) since A ^(a) is an convex in (G, e ^(a)), that is, A (z) nleqa,

as needed. (1) ⇒ (5). For each a ∈ L, suppose x, y ∈ A ^[a] and (x, z), (z, y) ∈ e ^[a]. Then a ∉ α (e (x, z)), a ∉ α (e (z, y)) and a ∉ α (A (x)), a ∉ α (A (y)). Thus a ∉ α (e (x, z)) ∪ α (e (z, y)) ∪ α (A (x)) ∪ α (A (y)) = α (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)). Since A is an L-ordered L-convex set, that is, e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z), then we have a ∉ α (A (z)) by Lemma 2.1. It implies z ∈ A ^[a], and so A ^[a] is an ordered convex set in (X, e ^[a]).

(6) ⇒ (1). To prove that A is L-ordered L-convex, it suffices to prove e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. According to Lemma 2.1, it suffices to prove for each a ∈ P (L), a ∉ α (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)) implies a ∉ α (A (z)). In fact, a ∉ α (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)) = α (e (x, z)) ∩ α (e (z, y)) ∩ α (A (x)) ∩ α (A (y)) implies a ∉ α (e (x, z)), a ∉ α (e (z, y)) and a ∉ α (A (x)), a ∉ α (A (y)). In other words, x, y ∈ A ^[a] and (x, z), (z, y) ∈ e ^[a]. So z ∈ A ^[a] since A ^[a] is an ordered convex set in (X, e ^[a]). That is, a ∉ α (A (z)), as desired.□

Theorem 3.8. Let (X, e) be an L-poset with L be a completely distributive lattice, and A ∈ L ^X . If β (b ∧ c) = β (b) ∩ β (c) for all b, c ∈ L, then the following conditions are equivalent: (1) A is an L-ordered L-convex set; (2) ∀a ∈ L, A _(a) is an ordered convex set in (X, e _(a)); (3) ∀a ∈ J (L), A _(a) is an ordered convex set in (X, e _(a)).

Proof. (1) ⇒ (2). For each a ∈ L, suppose x, y ∈ A _(a) and (x, z), (z, y) ∈ e _(a), it suffices to prove z ∈ A _(a). In fact, x, y ∈ A _(a) and (x, z), (z, y) ∈ e _(a) implies a ∈ β (A (x)), a ∈ β (A (y)) and a ∈ β (e (x, z)), a ∈ β (e (z, y)). Since β (x ∧ y) = β (x) ∩ β (y) for all x, y ∈ L, so we have a ∈ β (A (x)) ∩ β (A (y)) ∩ β (e (x, z)) ∩ β (e (z, y)) = β (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)). Because A is an L-ordered L-convex set, that is, e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z), then we have a ∈ β (A (z)) by Lemma 2.1. It implies z ∈ A _(a),

as required. (2) ⇒ (3) is obvious.

(3) ⇒ (1). To prove A is L-ordered L-convex, it suffices to prove e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. According to Lemma 2.1, it suffices to prove a ∈ β (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)) implies a ∈ β (A (z)) for each a ∈ J (L). In fact, a ∈ β (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)) implies a ∈ β (e (x, z)), a ∈ β (e (z, y)), a ∈ β (A (x)), a ∈ β (A (y)). In other words, x, y ∈ A _(a) and (x, z), (z, y) ∈ e _(a). So z ∈ A _(a) since A _(a) is an ordered convex set in (X, e _(a)). That is, a ∈ β (A (z)), as needed.□

In what follows, we are about to discuss the product of L-ordered L-convex sets.

Proposition 3.9. Let (X, e _X ) and (Y, e _Y ) be L-posets. If A ∈ L ^X and B ∈ L ^Y are L-ordered L-convex sets, then A × B is an L-ordered L-convex set in the product L-poset (X × Y, e).

Proof. ∀x = (x ₁, x ₂), y = (y ₁, y ₂), z = (z ₁, z ₂) ∈ X × Y, we have

Therefore, A × B is an L-ordered L-convex set in (X × Y, e).□

The converse of Proposition 3.9 does not hold, please see the following example.

Example 3.10. Let L = {0, a, b, 1} with ≤ defined as 0 ≤ a, b ≤ 1, a ∨ b = 1 and a ∧ b = 1. (X, e _X ) and (Y, e _Y ) be L-posets, where X = {x ₁, x ₂, x ₃} and Y = {y ₁, y ₂, y ₃}, e _X : X × X ⟶ L and e _Y : Y × Y ⟶ L are as shown in Table 1 and Table 2:

Let A ∈ L ^X defined by A (x ₁) = A (x ₃) = a, A (x ₂) =0 and B ∈ L ^Y defined by B (y ₁) = B (y ₃) = b, B (y ₂) =0. Then for each (x, y) ∈ X × Y, we have (A × B) (x, y) =0, it results that A × B is L-ordered L-convex in (X × Y, e). But e _X (x ₁, x ₂) ∧ e _X (x ₂, x ₃) ∧ A (x ₁) ∧ A (x ₃) = anleqA (x ₂), it implies that A is not L-ordered L-convex in (X, e _X ).

The product L-posets of (X, e _X ) and (Y, e _Y ) can easily generalized to the finite case, so we have the following theorem.

Theorem 3.11. Let { ( X i , e i ) } i = 1 n be a non-empty family of L-posets, and for each 1 ≤ i ≤ n, A _i ∈ L ^{X _i} be an L-ordered L-convex set. Then Π i = 1 n A i is an L-ordered L-convex set in the product L-poset ( ∏ i = 1 n X i , e ) .

In the last of this section, we are about to discuss L-convexity-preserving mappings and L-convex-to-convex mappings in L-ordered L-convex space.

Theorem 3.12. Let (X, e _X ) and (Y, e _Y ) be L-posets, and ( X , C e X ) and ( Y , C e Y ) be L-ordered L-convex spaces. If the mapping f : X ⟶ Y is L-order-preserving, then f is L-convexity-preserving.

Proof. Suppose f : X ⟶ Y is L-order-preserving and D ∈ C e Y . Then it suffices to show f L ← ( D ) ∈ C e X . In fact, for all x, y, z ∈ X, we have e X ( x , z ) ∧ e X ( z , y ) ∧ f L ← ( D ) ( x ) ∧ f L ← ( D ) ( y ) ≤ e Y ( f ( x ) , f ( z ) ) ∧ e Y ( f ( z ) , f ( y ) ) ∧ D ( f ( x ) ) ∧ D ( f ( y ) ) ≤ D ( f ( z ) ) = f L ← ( D ) ( z ) .

It implies that f L ← ( D ) ∈ C e X , as desired.□

Employing this result, it accesses immediately the following corollary.

Corollary 3.13. If (X, e) is the product L-poset of {(X _i , e _i ) | i ∈ I}. Then each projection p _i : (Π _i∈I X _i , e) ⟶ (X _i , e _i ) is L-convexity-preserving.

Proof. It follows from Theorem 3.12.□

Remark 3.14. By definition 3.1, we can see that the essence of L-order L-convexity will not change by reversing L-order. So an L-order-reversing mapping, as well as an L-order-preserving mapping, is also L-convexity-preserving. But not vice versa.

Example 3.15. Let (X, e _X ) and (Y, e _Y ) be L-posets, where X = {x ₁, x ₂, x ₃} and Y = {y _i | i = 1, 2, 3, 4}, L = [0, 1], e _X : X × X ⟶ L and e _Y : Y × Y ⟶ L are as shown in Table 3 and Table 4 respectively:

A mapping f : X ⟶ Y defined by f (x ₁) = y ₃, f (x ₂) = f (x ₃) = y ₂, then f is L-convexity-preserving. In fact, ∀D ∈ L ^Y , since e X ( x 1 , x 2 ) ∧ e X ( x 2 , x 3 ) ∧ f L ← ( D ) ( x 1 ) ∧ f L ← ( D ) ( x 3 ) ≤ f L ← ( D ) (x ₂). Therefore, f L ← ( D ) ∈ C e X , and it implies that f is L-convexity-preserving. But e _X (x ₁, x ₂) nleqe _Y (f (x ₁), f (x ₂)), it results that f is not L-order-preserving.

It is worth noting that although an L-order-preserving mapping must be L-convexity-preserving, but it may not be an L-convex-to-convex mapping. Please see the following example.

Example 3.16.Let (X, e _X ) and (Y, e _Y ) be L-posets defined as Example 3.10, a mapping f : Y ⟶ X defined by f (y ₁) = f (y ₂) = f (y ₃) = x ₁ and f (y ₄) = x ₃. It is easily check that f is L-order-preserving, but it is not L-convex-to-convex. Because when we choose C ∈ L ^Y with C (y ₁) = C (y ₂) = C (y ₄) =0.5 and C (y ₃) =0.6, then C ∈ C e Y . But e X ( x 1 , x 2 ) ∧ e X ( x 2 , x 3 ) ∧ f L → ( C ) ( x 1 ) ∧ f L → ( C ) ( x 3 ) = 0.5 f L → ( C ) ( x 2 ) = 0 , which implies that f L → ( C ) is not an L-ordered L-convex set. Thus f is not L-convex-to-convex.

From the above section, we can see that for an L-poset (X, e), an L-subset A is either an L-ordered L-convex set when the inequality e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) holds for all x, y, z ∈ X, or not an L-ordered L-convex set when the inequality does not hold. There is no other case. But from the view of fuzzy logic, the inequality can be established to some extent, that is to say, A may be an L-ordered L-convex set to some extent. Based on this consideration, we are about to propose a new kind of convex structure on L-posets to describe the degree to which an L-subset is an L-ordered L-convex set. That is L-ordered (L, L)-fuzzy convex structure we will introduce in this section.

First of all, on a complete Heyting algebras L, there exists a well-known operator → : L × L ⟶ L as the right adjoint for the operator ∧ by a → b = ⋁ { x ∈ L | a ∧ x ≤ b }

Definition 4.1. Let (X, e) be an L-poset. Define C e ∗ : L X ⟶ L by C e ∗ ( A ) = ⋀ x , y , z ∈ X ( e ( x , z ) ∧ e ( z , y ) ∧ A ( x ) ∧ A ( y ) → A ( z ) ) .

Theorem 4.2. C e ∗ is an (L, L)-fuzzy convex structure, which is called an L-ordered (L, L)-fuzzy convex structure. Then ( X , C e ∗ ) is called an L-ordered (L, L)-fuzzy convex space, and C e ∗ ( A ) can be regarded as the degree to which A is an L-ordered L-convex set.

Proof. (LMC1). C e ∗ ( 0 ¯ ) = C e ∗ ( 1 ¯ ) = 1 is obvious.

(LMC2). For all {A _i | i ∈ I} ⊆ L ^X , we have C e ∗ ( ⋀ i ∈ I A i ) = ⋀ x , y , z ∈ X ( e ( x , z ) ∧ e ( z , y ) ∧ ( ⋀ i ∈ I A i ( x ) ) ( ⋀ i ∈ I A i ( y ) ) → ( ⋀ i ∈ I A i ( z ) ) ) ≥ ⋀ i ∈ I ⋀ x , y , z ∈ X ( e ( x , z ) ∧ e ( z , y ) ∧ A i ( x ) ∧ A i ( y ) → A i ( z ) ) = ⋀ i ∈ I C e ∗ ( A i ) . (LMC3). Suppose {A _i | i ∈ I} ⊆ L ^X is non-empty and totally ordered, then C e ∗ ( ⋁ i ∈ I A i ) = ⋀ x , y , z ∈ X ( e ( x , z ) ∧ e ( z , y ) ∧ ( ⋁ i ∈ I A i ( x ) ) ∧ ( ⋁ i ∈ I A i ( y ) ) → ( ⋁ i ∈ I A i ( z ) ) ) = ⋀ x , y , z ∈ X ( e ( x , z ) ∧ e ( z , y ) ∧ ⋁ i ∈ I ( A i ( x ) ∧ A i ( y ) ) → ( ⋁ i ∈ I A i ( z ) ) ) ≥ ⋀ i ∈ I ⋀ x , y , z ∈ X ( e ( x , z ) ∧ e ( z , y ) ∧ A i ( x ) ∧ A i ( y ) → A i ( z ) ) = ⋀ i ∈ I C e ∗ ( A i ) . □

Remark 4.3. From [25], we all know the relationship between (L, M)-fuzzy convex structure and L-convex structure. According to the existing results, we can conclude that ( C e ∗ ) [ a ] is an L-convex structure for each a ∈ L. It is worth noting that the L-ordered L-convex structure we defined in section 3 is exactly extracted from L-ordered (L, L)-fuzzy convex structure C e ∗ . In fact, C e = ( C e ∗ ) [ 1 ] .

Now, let us give some equivalent descriptions for L-ordered (L, L)-fuzzy convex structure.

Proposition 4.4. Let (X, e) be an L-poset, and A ∈ L ^X . Then C e ∗ ( A ) = ⋁ { a ∈ L | a ∧ e ( x , z ) ∧ e ( z , y ) ∧ A ( x ) ∧ A ( y ) ≤ A ( z ) , ∀ x , y , z ∈ X } .

Proof. The proof is trivial and omitted.□

Theorem 4.5. Let (X, e) be an L-poset, and A ∈ L ^X . Then C e ∗ ( A ) = ⋁ { a ∈ L | ∀ b ≤ a , A [ b ] is convex in ( X , e [ b ] ) } .

Proof. Suppose a ∈ L with a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. For each b ≤ a, if x, y ∈ A _[b] and (x, z), (z, y) ∈ e _[b]. Then e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≥ b. It results A (z) ≥ (b ∧ a) = b. So z ∈ A _[b], and it implies A _[b] is convex in (X, e _[b]). According to Proposition 4.4, it is easily seen that C e ∗ ( A ) ≤ ⋁ { a ∈ L | ∀ b ≤ a , A [ b ] is convex in ( X , e [ b ] ) } .

Conversely, for every a ∈ L with A _[b] is convex in (X, e _[b]) for all b ≤ a, we need to prove that a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. Suppose a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≥ b, then we have b ≤ a, x, y ∈ A _[b] and (x, z), (z, y) ∈ e _[b]. Therefore, z ∈ A _[b] since A _[b] is convex. That is A (z) ≥ b, so a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z).

Theorem 4.6. If L is a completely distributive lattice, and (X, e) is an L-poset. Then for all A ∈ L ^X , we have C e ∗ ( A ) = ⋁ { a ∈ L | ∀ b ∈ P ( L ) , a b , A ( b ) is convex in ( X , e ( b ) ) } .□

Proof. Suppose a ∈ L with a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. For each b ∈ P (L) with anleqb, if x, y ∈ A ^(b) and (x, z), (z, y) ∈ e ^(b), then e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) nleqb. It results A (z) nleqb. So z ∈ A ^(b), and it implies A ^(b) is convex in (X, e ^(b)). Thus C e ∗ ( A ) ≤ ⋁ { a ∈ L | ∀ b ∈ P ( L ) , a b , A ( b ) is convex in ( X , e ( b ) ) } .

Conversely, for every a ∈ L with A ^(b) is convex in (X, e ^(b)) for all b ∈ P (L) such that anleqb, we need to prove that a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. Suppose b ∈ P (L) and a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) nleqb, then we have anleqb, x, y ∈ A ^(b) and (x, z), (z, y) ∈ e ^(b). Therefore, z ∈ A ^(b) since A ^(b) is convex. That is A (z) nleqb, so a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) by Lemma 2.1.

Theorem 4.7. If L is a completely distributive lattice, and (X, e) is an L-poset. Then for all A ∈ L ^X , we have C e ∗ ( A ) = ⋁ { a ∈ L | ∀ b ∉ α ( a ) , A [ b ] is convex in ( X , e [ b ] ) } .

Proof. Suppose a ∈ L with a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. For each b ∉ α (a), if x, y ∈ A ^[b] and (x, z), (z, y) ∈ e ^[b]. Then b ∉ α (A (x)), b ∉ α (A (y)), b ∉ α (e (x, z)), b ∉ α (e (z, y)). It implies b ∉ α (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)). By Lemma 2.1, we have b ∉ α (A (z)). That is, z ∈ A ^[b]. So A ^[b] is convex in (X, e ^[b]). According to Proposition 4.4, we have C e ∗ ( A ) ≤ ⋁ { a ∈ L | ∀ b ≤ a , A [ b ] is convex in ( X , e [ b ] ) } .

Conversely, for every a ∈ L with A ^[b] is convex in (X, e ^[b]) for all b ∉ α (a), we need to prove that a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. Suppose b ∉ α (a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)), then we have b ∉ α (a), b ∉ α (A (x)), b ∉ α (A (y)), b ∉ α (e (x, z)), b ∉ α (e (z, y)). In other words, x, y ∈ A ^[b] and (x, z), (z, y) ∈ e ^[b]. Therefore, z ∈ A ^[b] since A ^[b] is convex. That is b ∉ α (A (z)), so a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) by Lemma 2.1, as required.

Theorem 4.8. If L is a completely distributive lattice with β (a ∧ b) = β (a) ∩ β (b) for all a, b ∈ L, and (X, e) is an L-poset. Then for all A ∈ L ^X , we have C e ∗ ( A ) = ⋁ { a ∈ L | ∀ b ∈ β ( a ) , A ( b ) is convex in ( X , e ( b ) ) } .

Proof. Suppose a ∈ L with a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. For each b ∈ β (a), if x, y ∈ A _(b) and (x, z), (z, y) ∈ e _(b). Then b ∈ β (A (x)), b ∈ β (A (y)), b ∈ β (e (x, z)), b ∈ β (e (z, y)). It implies b ∈ β (e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)). By Lemma 2.1, we have b ∈ β (A (z)). That is, z ∈ A _(b). So A _(b) is convex in (X, e _(b)). According to Proposition 4.3, we have C e ∗ ( A ) ≤ ⋁ { a ∈ L | ∀ b ≤ a , A ( b ) is convex in ( X , e ( b ) ) } .

Conversely, for every a ∈ L with A _(b) is convex in (X, e _(b)) for all b ∈ β (a), we need to prove that a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) for all x, y, z ∈ X. Suppose b ∈ β (a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y)), then we have b ∈ β (a), b ∈ β (A (x)), b ∈ β (A (y)), b ∈ β (e (x, z)), b ∈ β (e (z, y)). It results x, y ∈ A _(b) and (x, z), (z, y) ∈ e _(b). Therefore, z ∈ A _(b) since A _(b) is convex. That is b ∈ β (A (z)), so a ∧ e (x, z) ∧ e (z, y) ∧ A (x) ∧ A (y) ≤ A (z) by Lemma 2.1, as required.

Theorem 4.9. Let (X, e _X ), (Y, e _Y ) be L-posets, and (X × Y, e) be their product. Then for all A ∈ L ^X and B ∈ L ^Y ,

Proof. C e ∗ ( A × B ) = ⋀ x , y , z ∈ X × Y ( e ( x , z ) ∧ e ( z , y ) ∧ ( A × B ) ( x ) ∧ ( A × B ) ( y ) → ( A × B ) ( z ) ) = ⋀ x , y , z ∈ X × Y ( e X ( x 1 , z 1 ) ∧ e Y ( x 2 , z 2 ) ∧ e X ( z 1 , y 1 ) ∧ e Y ( z 2 , y 2 ) ∧ A ( x 1 ) ∧ B ( x 2 ) ∧ A ( y 1 ) ∧ B ( y 2 ) → A ( z 1 ) ∧ B ( z 2 ) ) ≥ ⋀ x 1 , y 1 , z 1 ∈ X ( e X ( x 1 , z 1 ) ∧ e X ( z 1 , y 1 ) ∧ A ( x 1 ) ∧ A ( y 1 ) → A ( z 1 ) ) ⋀ x 2 , y 2 , z 2 ∈ Y ( e Y ( x 2 , z 2 ) ∧ e Y ( z 2 , y 2 ) ∧ B ( x 2 ) ∧ B ( y 2 ) → B ( z 2 ) ) = C e X ∗ ( A ) ∧ C e Y ∗ ( B ) .

Theorem 4.10. Let (X, e _X ) and (Y, e _Y ) be L-posets, ( X , C e X ∗ ) and ( Y , C e Y ∗ ) be L-ordered (L, L)-fuzzy convex spaces. If the mapping f : X ⟶ Y is L-order-preserving, then f is (L, L)-fuzzy-convexity-preserving.

Proof. For every B ∈ L ^Y , we have C e X ∗ ( f L ← ( B ) ) = ⋀ x 1 , x , x 2 ∈ X ( e X ( x 1 , x ) ∧ e X ( x , x 2 ) ∧ f L ← ( B ) ( x 1 ) ∧ f L ← ( B ) ( x 2 ) → f L ← ( B ) ( x ) ) ≥ ⋀ x 1 , x , x 2 ∈ X ( e Y ( f ( x 1 ) , f ( x ) ) ∧ e Y ( f ( x ) , f ( x 2 ) ) ∧ B ( f ( x 1 ) ) ∧ B ( f ( x 2 ) ) → B ( f ( x ) ) ) ≥ ⋀ y 1 , y , y 2 ∈ Y ( e Y ( y 1 , y ) ∧ e Y ( y , y 2 ) ∧ B ( y 1 ) ∧ B ( y 2 ) → B ( y ) ) = C e Y ∗ ( B ) . By Definition 2.7, f is (L, L)-fuzzy-convexity-preserving.□

5 Conclusions

In this paper, we first extend ordered convex structure on posets to L-ordered L-convex structure on L-posets and give its hull formula. We also discuss the equivalent characterizations for an L-ordered L-convex set and analyze L-convexity-preserving mappings and L-convex-to-convex mappings. Furthermore, we introduce an L-ordered (L, L)-fuzzy convex structure which is induced by the L-convex sets degrees, and give some properties about it and study the corresponding (L, L)-fuzzy-convexity-preserving mappings.