Convex structures in a new kind of ordered fuzzy group 1

Abstract

By means of a fuzzy binary operation defined on partially ordered sets, a new kind of ordered fuzzy group is proposed in this paper. Some properties of this ordered fuzzy group are studied. Following that, its substructures, such as subgroup and convex subgroup, as well as its homomorphisms, along with their properties are explored. It is shown that each family of these substructures forms a convex structure, where the convex hull of a subset is exactly the (convex) subgroup generated by itself, and the homomorphisms between two ordered fuzzy groups are convexity-preserving mappings between the corresponding convex spaces. In addition, when these substructures are extended to fuzzy setting, several L-convex structures are constructed and investigated.

Keywords

Fuzzy binary operation ordered fuzzy group subgroup convex structure convexity-preserving

1 Introduction

Since fuzzy sets theory was first introduced in 1965 by Zadeh [37], it has been widely applied in many branches of mathematics. The researches of fuzzy algebra originated from the study of fuzzy subgroups defined by Rosenfeld [22]. From then on, there are increasing results about fuzzy subgroups and fuzzy subrings, one can be referred to [17]. However, the existing literatures are actually focused on fuzzy subalgebra under the framework of algebra, and there are few studies on the fuzzification of groups or other algebraic structures themselves from the perspective of binary operations. In 1999, Demirci [7] firstly introduced vague groups based on vague binary operations whose essence was the fuzzification of binary operations. A few years later, Demirci [8] proposed smooth groups based on smooth binary operations. Immediately, the researches of Demirci were followed by other studies [9 , 35]. In fact, both vague binary operations and smooth binary operations are defined on fuzzy equalities, which are not fit for a crisp set. Besides, the identity element and the inverse element may not be unique in smooth group, which results in the structures are far away from the classical case. To overcome the disadvantages of smooth groups, Yuan [36] proposed a new kind of fuzzy group, but this fuzzy group fails to better reflect the characteristics of the degree between fuzzy binary operations. Recently, in order to get rid of fuzzy equalities,Liu and Shi [15] introduced an M-fuzzy function on crisp sets, and provided a new approach to the fuzzification of groups. From the viewpoint of fuzzy logic, fuzzy function should be able to describe the degree of mapping each point in a set to each point in another set. Unfortunately, M-fuzzy function only gives the degree of mapping each point to only one point, which does not meet the requirements of fuzzy logic. On account of this consideration, we redefine fuzzy functions on crisp sets, and introduce L-binary operations on posets. By means of L-binary operations, we import a new kind of ordered fuzzy group and start a series of studies about their properties, homomorphisms, subgroups and L-subgroups.

Convexity theory [30] originated from the study about the shape of some figures, such as circles and polyhedrons in 2 or 3-dimensional Euclidean spaces. It has been accepted to be of increasing importance in the study of extremum problems in areas of applied mathematics. Convex structures exist in so many mathematical research areas, such as vector spaces, metric spaces, graphs, matroids, lattices, groups and so on (relevant literatures can be referred to [11– 14 , 38]). In our present paper, when we discuss the substructures and their properties of the ordered fuzzy group, we obtain an interesting conclusion, that is each type of the substructures can induce a convex structure or an L-convex structure. Following that, we further investigate the (L-)convex hull and (L-)convexity-preserving mappings in the corresponding convex spaces.

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 a new kind of ordered fuzzy group via L-binary operations, and then discuss its subgroups and homomorphisms along with their properties. In Section 4, we study convex subgroups of this ordered fuzzy group, and investigate the convex structures induced by subgroups and convex subgroups. In Section 5, we extend subgroups to L-subgroups and normal L-subgroups, and investigate their corresponding L-convex structures. In Section 6, we consider convex L-subgroups and the induced L-convex structures. In the final section, we summarize the results and draw some conclusions.

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∈Bb) = ⋁ _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.

For any non-empty set X, the family of all subsets of X is denoted by 2^X, and the family of all L-subsets of X is denoted by L^X. We say a family {A_i | i ∈ I} ⊆2^X (or ⊆L^X) is up-directed provided for each A₁, A₂ ∈ {A_i | i ∈ I}, there exists A₃ ∈ {A_i | i ∈ I} such that A₁ ⊆ A₃ and A₂ ⊆ A₃ (or A₁ ≤ A₃ and A₂ ≤ A₃).

For each map f : X ⟶ Y, there exists a well-known forward powerset operator f^→ : 2^X ⟶ 2^Y defined by f^→ (A) = {f (x) | x ∈ A} for each A ⊆ X, and an L-forward operator $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ defined by $f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x)$ for each A ∈ L^X, y ∈ Y. As well as that, there exists a backward powerset operator f^← : 2^Y ⟶ 2^X defined by f^← (B) = {x ∈ X | f (x) ∈ B} for each B ⊆ Y, and an L-backward operator $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ defined by $f_{L}^{\leftarrow} (B) (x) = B (f (x))$ for each B ∈ L^Y, x ∈ X.

Definition 2.1. [26] Let A, B ∈ L^X. We define the product of A and B, in symbols A × B, to be an L-subset of X × Y:

$(A \times B) (x, y) = A (x) \land B (y) (\forall (x, y) \in X \times Y) .$

Proposition 2.2. [38] Let L be a complete Heyting algebra. If {A_i | i ∈ I} ⊆ L^X is non-empty and totally ordered, then (⋁ _i∈IA_i (x) ) ∧ (⋁ _j∈IA_j (y) ) = ⋁ _i∈I (A_i (x) ∧ A_i (y) ) for all x, y ∈ X.

An L-relation E on X is an L-subset of X × X. An L-relation E on X is called an L-preorder if it satisfies

(Ref) ∀x ∈ X, E (x, x) =1,

(Tran) ∀x, y, z ∈ X, E (x, y) ∧ E (y, z) ≤ E (x, z).

An L-preorder E on X is called an L-equivalence if it satisfies

(Sym) ∀x, y ∈ X, E (x, y) = E (y, x).

An L-equivalence E on X is called an L-equality if ∀x, y ∈ X, E (x, y) =1 implies x = y.

For a given L-equality E on X, E (x, y) can be regarded as the degree of the equality of x and y. For a classical equality (denoted by =) on X, there exists an L-equality E^* : X × X → X defined by $\forall x, y \in X, E^{*} (x, y) = {\begin{matrix} 1 & x = y, \\ 0 & x \neq y . \end{matrix}$

In the sequel of this section, let us review the definitions of convexity and L-convexity.

Definition 2.3. [30] A subset $C \subseteq 2^{X}$ is called a convex structure, if it satisfies the following conditions:

(C1) $\emptyset, X \in C$ ,

(C2) $⋂_{i \in I} A_{i} \in C$ for all non-empty family ${A_{i} | i \in I} \subseteq C$ ,

(C3) $⋃_{i \in I} A_{i} \in C$ for all family ${A_{i} | i \in I} \subseteq C$ which is non-empty and totally ordered.

The pair $(X, C)$ is called a convex space. The members of $C$ are called convex sets. For each A ∈ L^X, $co (A) = ⋂ {C \in L^{X} | A \subseteq C \in C}$ is called (convex) hull of A, which is the smallest convex set including A. It is easily seen that $A \in C \Leftrightarrow co (A) = A$ .

Definition 2.4. [30] Let $(X, C_{X})$ and $(Y, C_{Y})$ be convex spaces and f : X ⟶ Y is a mapping. Then

(1) f is said to be convexity-preserving if $f^{\leftarrow} (D) \in C_{X}$ for each $D \in C_{Y}$ .

(2) f is said to be convex-to-convex if $f^{\to} (C) \in C_{Y}$ for each $C \in C_{X}$ .

Definition 2.5. [16] A subset $C \subseteq L^{X}$ is called an L-convex structure, if it satisfies the following conditions:

(LC1) $\bar{0}, \bar{1} \in C$ ;

(LC2) $⋀_{i \in I} A_{i} \in C$ for all non-empty family ${A_{i} | i \in I} \subseteq C$ ;

(LC3) $⋁_{i \in I} A_{i} \in C$ for all family ${A_{i} | i \in I} \subseteq C$ 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, $cl (A) = ⋀ {C \in L^{X} | A \leq C \in C}$ is called L-convex hull of A, which is the smallest L-convex set including A. It is easily seen that $A \in C \Leftrightarrow cl (A) = A$ .

Definition 2.6. [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 $f_{L}^{\leftarrow} (D) \in C_{X}$ for each $D \in C_{Y}$ . f is said to be L-convex-to-convex if $f_{L}^{\to} (C) \in C_{Y}$ for each $C \in C_{X}$ .

It is worth noticing that the condition the family ${A_{i} | i \in I} \subseteq C$ is totally ordered which is mentioned in (C3) and (LC3) can be replaced by the family is up-directed.

3 Ordered fuzzy groups and subgroups

The representative research work of adding order relation to algebraic structures was the discussion about an ordered group [1, 2], which was defined to be a group (G, ·) together with an order ≤ on G satisfying x ≤ y ⇒ a · x · b ≤ a · y · b for all x, y, a, b ∈ G. Later on, Borzooei [3 –5] imported fuzzy orders on groups, and proposed an L-ordered group, which was a classical group with a fuzzy order consistent with it. Inspired by that, we intend to study a new kind of a fuzzy group which is compatible with a classical order. In order to that, we start with the discussion of group fuzzification.

To fuzzify a group, it is necessary to recall the following well-known definition in classical group theory [10].

Definition 3.1. X together with a binary operation ·, denoted by (X, ·), is a semi-group iff it satisfies the associative law:

(G1) a · (b · c) = (a · b) · c, ∀ a, b, c ∈ X .

A semi-group (X, ·) is a monoid iff

(G2) There exists an element e ∈ X, called the (two-sided) identity element of (X, ·), such that e · a = a and a · e = a for each a ∈ X.

A monoid (X, ·) is a group iff

(G3) For each a ∈ X, there exists an element of X, denoted by a^-1 and called the (two-sided) inverse element of a, such that a^-1 · a = e and a · a^-1 = e.

From the definition, it is not difficult to see that a classical group is based on a binary operation, the essence of which is a function. Analogous to that, to propose a fuzzy group, it is inevitable to introduce a fuzzy function first.

A fuzzy function was firstly defined by Demirci [6] as a fuzzy relation f on X × Y w.r.t. the fuzzy equalities E_X on X and E_Y on Y, where its membership function μ_f : X × Y ⟶ I satisfies:

(F1) ∀x ∈ X, ∃ y ∈ Y such that μ_f (x, y) >0,

(F2) ∀x, y ∈ X, ∀ z, w ∈ Y, μ_f (x, z) ∧ f (y, w) ∧ E_X (x, y) ≤ E_Y (z, w).

Let $E_{X} = E_{X}^{*}$ and $E_{Y} = E_{Y}^{*}$ , then Liu and Shi [13] proposed fuzzy functions on ordinary sets as follows.

Definition 3.2. [13] For two non-empty sets X and Y, an M-fuzzy relation f ∈ M^X×Y is called an M-fuzzy function from X to Y, if f satisfies the following conditions:

(MF1) ∀x ∈ X we have ∨_y∈Yf (x, y) ≠0,

(MF2) ∀x ∈ X, ∀ y, z ∈ Y, f (x, y) ∧ f (x, z) ≠0 ⇒ y = z.

In this definition, the condition (MF2) is too rigorous. For example, if M = [0, 1], then for each x ∈ X, under the M-fuzzy function f, {y ∈ Y|f (x, y) ≠0} must be a singleton set. It indicates that f only gives the description to map x to only one point in Y to a certain extent. As we all known, f (x, y) can be regarded as the degree of mapping x to y. From the point of view of fuzzy logic, the fuzzy function should be able to describe the degree of mapping x to each y ∈ Y. For this consideration, we modify the definition of fuzzy function as following.

Definition 3.3. For two non-empty sets X and Y, an L-relation f : X × Y → L is called an L-function if it satisfies the following conditions:

(LF1) ∀x ∈ X, ∃ y ∈ Y, such that f (x, y) >0,

(LF2) ∀x ∈ X, ∀ y, z ∈ Y, f (x, y) ∧ f (x, z) = 1 ⇒ y = z.

Particularly, if ∀x ∈ X, ∃ y ∈ Y, such that f (x, y) =1, then we say f is a strong L-function.

Combining this fuzzy function and order, we get an ordered fuzzy operation on a partially order set.

Definition 3.4. Let (X, ≤) be a poset. Then a strong L-function f from X × X to X is called an ordered fuzzy (binary) operation on (X, ≤) if it satisfies:

(OFO) ∀a, b, c, m, q ∈ X and a ≤ b, then f ((a, c) , m) ∧ f ((b, c) , q) =1 or f ((c, a) , m) ∧ f ((c, b) , q) =1 implies m ≤ q.

By adding the properties of group operation on this order fuzzy operation, we can give the definition of ordered fuzzy (semi-)group.

Definition 3.5. Let (X, ≤) be a poset, and * is an ordered fuzzy (binary) operation on (X, ≤). Then (X, ≤ , *) is called an ordered fuzzy semi-group if it satisfies:

(OFG1) ∀a, b, c, d, q, m ∈ X, * ((a, b) , q) ∧ * ((q, c) , m) ∧ * ((b, c) , d) ≤ * ((a, d) , m), and * ((b, c) , d) ∧ * ((a, d) , m) ∧ * ((a, b) , q) ≤ * ((q, c) , m).

Definition 3.6. An ordered fuzzy semi-group (X, ≤ , *) is called an ordered fuzzy group if it satisfies:

(OFG2) There exists an (two-sided) identity element 1^* ∈ X, such that * ((1^*, a) , a) ∧ * ((a, 1^*) , a) =1 for each a ∈ X,

(OFG3) For each a ∈ X, there exists an (two-sided) inverse element a^-1 ∈ X, such that * ((a^-1, a) , 1^*) ∧ * ((a, a^-1) , 1^*) =1.

Example 3.7. For the real number set $ℝ$ with ordinary order ≤ and L = [0, 1], there exists an L-relation * from $ℝ \times ℝ$ to $ℝ$ defined as $* ((x, y), z) = {\begin{matrix} 1 & x + y = z, \\ 0.5 & x + y \neq z . \end{matrix} (\forall x, y, z \in ℝ)$ It is easy to check that * is an ordered fuzzy binary operation, and * satisfies (OFG1) since the addition operation satisfies the associative law. Moreover, there exists $0 \in ℝ$ such that for each $a \in ℝ$ , it has * ((0, a) , a) ∧ * ((a, 0) , a) =1 and * ((- a, a) , 0) ∧ * ((a, - a) , 0) =1. As a result, $(ℝ, \leq, *)$ is an ordered fuzzy group with the identity element 1^* = 0 and the inverse element a^-1 = - a for each $a \in ℝ$ .

Besides, we can also construct an ordered fuzzy group from a common group as the following example.

Example 3.8. Let (G, ·) be a classical group, L = [0, 1] and 0 ≤ α ≤ 1. Define ≤ : G × G → L as x ≤ y ⇔ x = y for all x, y ∈ G, and * : (G × G) × G → L as $* ((x, y), z) = {\begin{matrix} 1 & x \cdot y = z, \\ α & x \cdot y \neq z . \end{matrix} (\forall x, y, z \in G)$ Then it is easy to check that (X, ≤ , *) is an ordered fuzzy group with the identity element 1^* and the inverse element a^-1 being, respectively, the identity element and the inverse of a in the classical group (G, ·).

Remark 3.9. In fact, for each ordered (classical) group (G, ≤ , ·), the binary operation · : (G × G) → G can be regarded as the fuzzy operation as $\cdot ((x, y), z) = {\begin{matrix} 1 & x \cdot y = z, \\ 0 & x \cdot y \neq z . \end{matrix} (\forall x, y, z \in G)$ Then an ordered (classical) group (G, ≤ , ·) can be reconsidered as a special ordered fuzzy group, and it indicates that the ordered fuzzy group we proposed is an appropriate generalization of an ordered (classical) group.

For a given ordered fuzzy group, there exists a close relationship between elements in it, which can be presented through the following two propositions.

Proposition 3.10. Let (X, ≤ , *) be an ordered fuzzy group. Then for all a, b, c ∈ X, we have $* ((a, b), c) = * ((a^{- 1}, c), b) = * ((c, b^{- 1}), a) .$

Proof. Since (X, ≤ , *) is an ordered fuzzy group, then for any a, b, c ∈ X, we have * ((a^-1, a) , 1^*) = * ((a, a^-1) , 1^*) = * ((1^*, b) , b) = * ((1^*, c) , c) =1. Using (OFG1), we obtain * ((a, b) , c) = * ((a^-1, a) , 1^*) ∧ * ((1^*, b) , b) ∧ * ((a, b) , c) ≤ * ((a^-1, c) , b) and * ((a^-1, c) , b) = * ((a, a^-1) , 1^*) ∧ * ((1^*, c) , c) ∧ * ((a^-1, c) , b) ≤ * ((a, b) , c) at the same time. Hence, we have * ((a, b) , c) = * ((a^-1, c) , b).

Similarly, the proof of * ((a, b) , c) = * ((c, b^-1) , a) can be constructed.□

Proposition 3.11. Let (X, ≤ , *) be an ordered fuzzy group. Then for any a, b, c, d ∈ X, we have

(1) * ((a, b) , d) ∧ * ((a, c) , d) =1 ⇒ b = c;

(2) * ((b, a) , d) ∧ * ((c, a) , d) =1 ⇒ b = c.

Proof. (1) For any a, b, c, d ∈ X, if * ((a, b) , d) ∧ * ((a, c) , d) =1, then by Proposition 3.10, we see that * ((a^-1, d) , b) ∧ * ((a^-1, d) , c) =1. Notice that * is a strong L-function, so we have b = c.

(2) Similarly, we obtain * ((d, a^-1) , b) ∧ * ((d, a^-1) , c) = * ((b, a) , d) ∧ * ((c, a) , d) =1, and it results in b = c because of * is a strong L-function.

□

In what follows, we are going to discuss the product of ordered fuzzy groups. For that, the product of fuzzy binary operations should be considered firstly.

Let * be a fuzzy binary operation both on X₁ and X₂. Then the fuzzy binary operation ⊛ on (X₁ × X₂) defined by ⊛ ((a₁, a₂) , (b₁, b₂) , (c₁, c₂)) = * ((a₁, b₁) , c₁) ∧ * ((a₂, b₂) , c₂) is called the product of *. It is easy to check that if * is an ordered fuzzy binary operation on (X₁, ≤) and (X₂, ≤), then ⊛ is also an ordered fuzzy binary operation on (X₁ × X₂, ≤).

Proposition 3.12. Let (X₁, ≤ , *), (X₂, ≤ , *) be ordered fuzzy groups. Then (X₁ × X₂, ≤ , ⊛) is an ordered fuzzy group. We call it the product of (X₁, ≤ , *) and (X₂, ≤ , *).

Proof. (OFG1) is obvious.

(OFG2). Since (X₁, ≤ , *) and (X₂, ≤ , *) are ordered fuzzy groups, so there exist identity element $1_{X_{1}}^{*}$ in (X₁, ≤ , *) and identity element $1_{X_{2}}^{*}$ in (X₂, ≤ , *). Then $(1_{X_{1}}^{*}, 1_{X_{2}}^{*}) \in X_{1} \times X_{2}$ , and we can easily prove that $(1_{X_{1}}^{*}, 1_{X_{2}}^{*})$ is the identity element of (X₁ × X₂, ≤ , ⊛).

(OFG3). For each (a₁, a₂) ∈ X₁ × X₂, there exist $a_{1}^{- 1} \in X_{1}$ and $a_{2}^{- 1} \in X_{2}$ . It can be easily checked that $(a_{1}^{- 1}, a_{2}^{- 1})$ is the inverse of (a₁, a₂).□

In order to discuss subgroups and their related properties of this ordered fuzzy group, the closeness of subsets under an ordered fuzzy binary operation should be mentioned. For a given ordered fuzzy binary operation f on poset (X, ≤), we say that a crisp subset H ⊆ X is closed under f if $f ((a, b), c) = 1 \Rightarrow c \in H (\forall a, b \in H, \forall c \in X) .$ In this case, it is not difficult to observe that f |_H×H×H is also an ordered fuzzy binary operation on (H, ≤).

Definition 3.13. Let (X, ≤ , *) be an ordered fuzzy group. Then H ⊆ X is said to be a subgroup of X if it is closed under * and (H, ≤ , * |_H×H×H) itself is an ordered fuzzy group.

Example 3.14. For the ordered fuzzy group $(ℝ, \leq, *)$ as mentioned in Example 3.7. Let $ℤ \subseteq ℝ$ be the integer set, then it is easily seen that $ℤ$ is a subgroup of $ℝ$ .

Theorem 3.15. Let (X, ≤ , *) be an ordered fuzzy group and H ⊆ X be a non-empty set. Then H is a subgroup of Xiff * ((a, b^-1) , c) =1 ⇒ c ∈ H (∀ a, b ∈ H, ∀ c ∈ X).

Proof. (⇒). Assume that H is a subgroup of X, ∀a, b ∈ H. Then b^-1 ∈ H, and sequentially, for all c ∈ X, we have * ((a, b^-1) , c) =1 implies c ∈ H since H is closed under *.

(⇐). Because H≠ ∅, there exists u ∈ H ⊆ X, then for the identity element 1^* of (X, ≤ , *), it has * ((u, u^-1) , 1^*) =1. Then 1^* ∈ H by the known conditions. As a result, for each a ∈ H, we have a^-1 ∈ H since * ((1^*, a^-1) , a^-1) =1. Based on these analyses and regarding that (X, ≤ , *) is an ordered fuzzy group, it can be easily seen that (H, ≤ , * |_H×H×H) satisfies (OFG1)-(OFG3). Nextly, we are about to show that H is closed under *.

For any a, b ∈ H and c ∈ X such that * ((a, b) , c) =1. Then b^-1 ∈ H and * ((a, (b^-1) ^-1) , c) = * ((a, b) , c) =1. By the known conditions, we have c ∈ H. It implies that H is closed under *.

Summarize the two aspects above, then H is a subgroup of X.□

Theorem 3.16. Let (X, ≤ , *) be an ordered fuzzy group and H ⊆ X be a non-empty set. Then H is a subgroup of Xiff

(1) H is closed under *;

(2) For each a ∈ H, a^-1 ∈ H.

Proof. The necessity is obvious, so we only need to show the sufficiency. Suppose ∀a, b ∈ H, ∀c ∈ X and * ((a, b^-1) , c) =1. Then by the condition (2), we have b^-1 ∈ H. Since H is closed under *, so c ∈ H. By Theorem 3.15, it can be easily seen that H is a subgroup of X.□

In the sequel of this section, we introduce the homomorphisms between two ordered fuzzy groups, and investigate their properties.

Definition 3.17. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ : X → Y be a function in the classical sense. Then φ is called an ordered fuzzy group homomorphism iff φ not only is order-preserving, but also satisfies $* ((a, b), c) = * ((φ (a), φ (b)), φ (c)) (\forall a, b, c \in X) .$ Furthermore, an injective ordered fuzzy group homomorphism is called an ordered fuzzy group monomorphism, a surjective ordered fuzzy group homomorphism is called an ordered fuzzy group epimorphism, and a bijective ordered fuzzy group homomorphism is called an ordered fuzzy group isomorphism.

Example 3.18. Let $φ : ℝ \to ℝ$ be defined by φ (x) =2 · x, where · is the ordinary multiplication on $ℝ$ . Then φ is an ordered fuzzy group homomorphism form $(ℝ, \leq, *)$ to $(ℝ, \leq, *)$ .

Proposition 3.19. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ : X → Y be an ordered fuzzy group homomorphism. Then

(1) $φ (1_{X}^{*}) = 1_{Y}^{*}$ ,

(2) φ (a) ^-1 = φ (a^-1) for each a ∈ X.

Proof. (1) For each a ∈ X, since φ is an ordered fuzzy group homomorphism, so $* ((φ (a), φ (1_{X}^{*})), φ (a)) = * ((a, 1_{X}^{*}), a) = 1$ . Besides, $* ((φ (a), 1_{Y}^{*})), φ (a)) = 1$ naturally holds. Using Proposition 3.11, we obtain $φ (1_{X}^{*}) = 1_{Y}^{*}$ .

(2) For each a ∈ X, we have a^-1 ∈ X such that $* ((a, a^{- 1}), 1_{X}^{*}) \land * ((a^{- 1}, a), 1_{X}^{*}) = 1$ . Noting that φ is an ordered fuzzy group homomorphism, then we have

$* ((φ (a), φ (a^{- 1})), φ (1_{X}^{*}) \land * ((φ (a^{- 1}), φ (a)), φ (1_{X}^{*}) = 1$ . That is,

$* ((φ (a), φ (a^{- 1})), 1_{Y}^{*}) \land * ((φ (a^{- 1}), φ (a)), 1_{Y}^{*}) = 1$ . It implies φ (a) ^-1 = φ (a^-1).□

4 (Convex) subgroups and the convex structures

In this section, we discuss the convexity of subgroups from the perspective of order relation, and then construct and study some convex structures induced by (convex) subgroups of the ordered fuzzy group.

Generally, in a partially ordered set (X, ≤), we say H ⊆ X is convex or a convex set if for all x, y ∈ H and z ∈ X, x ≤ z ≤ y implies z ∈ H.

Definition 4.1. Let (X, ≤ , *) be an ordered fuzzy group and H ⊆ X. Then H is called a convex subgroup if it is not only a subgroup of X but also a convex set with respect to ≤.

Example 4.2. Let $G = ℤ \times ℤ$ with the pointwise order relation ≤, L = [0, 1] and $H = ℤ \times {0}$ . Now, define * : (G × G) × G ⟶ L as, for all x = (x₁, x₂) , y = (y₁, y₂) , z = (z₁, z₂) ∈ G, $* ((x, y), z) = {\begin{matrix} 1 & if x_{1} + y_{1} = z_{1}, x_{2} + y_{2} = z_{2}, \\ 0.5 & else . \end{matrix}$ Analogous to Example 3.7, it can be seen that (G, ≤ , *) is an ordered fuzzy group with the identity element 1^* = (0, 0) and the inverse element (x₁, x₂) ^-1 = (- x₁, - x₂) for each x ∈ G. What’s more, H is a convex subgroup of G.

From now on, for an ordered fuzzy group (X, ≤ , *), we use $C_{X}$ and $C_{X}^{C}$ to denote the set of all subgroups and convex subgroups of X, respectively. That is $C_{X} = {H \subseteq X | H is a subgroup of X},$ $C_{X}^{C} = {H \subseteq X | H is a convex subgroup of X} .$

Proposition 4.3. Let (X₁, ≤ , *), (X₂, ≤ , *) be ordered fuzzy groups, H₁ ⊆ X₁ and H₂ ⊆ X₂. Then

(1) $H_{1} \in C_{X_{1}}, H_{2} \in C_{X_{2}} \Rightarrow H_{1} \times H_{2} \in C_{X_{1} \times X_{2}}$ .

(2) $H_{1} \in C_{X_{1}}^{C}, H_{2} \in C_{X_{2}}^{C} \Rightarrow H_{1} \times H_{2} \in C_{X_{1} \times X_{2}}^{C}$ .

Proof. (1) For each (a₁, a₂) , (b₁, b₂) ∈ H₁ × H₂, and ∀ (c₁, c₂) ∈ X₁ × X₂ such that ⊛ (((a₁, a₂) , (b₁, b₂) ^-1) , (c₁, c₂)) =1. Then we have $* ((a_{1}, b_{1}^{- 1}), c_{1}) \land * ((a_{2}, b_{2}^{- 1}), c_{2}) = 1$ . By Theorem 3.15, we have c₁ ∈ H₁ and c₂ ∈ H₂ since H₁ and H₂ are subgroups. It implies (c₁, c₂) ∈ H₁ × H₂, and using Theorem 3.15 again, we obtain H₁ × H₂ is a subgroup of X₁ × X₂.

(2) Based on (1), it is sufficient to show H₁ × H₂ is a convex set. Assume x = (x₁, x₂) , y = (y₁, y₂) ∈ H₁ × H₂ and ∀z = (z₁, z₂) ∈ X₁ × X₂ such that x ≤ z ≤ y. Then we have x₁ ≤ z₁ ≤ y₁ and x₂ ≤ z₂ ≤ y₂. Since H₁ and H₂ are convex sets, so z₁ ∈ H₁ and z₂ ∈ H₂. It implies that z ∈ H₁ × H₂, and so H₁ × H₂ is a convex set with respect to the order. Therefore, $H_{1} \times H_{2} \in C_{X_{1} \times X_{2}}^{C}$ .□

Proposition 4.4. Let (X, ≤ , *) be an ordered fuzzy group and {H_i | i ∈ I} ⊆2^X. Then

(1) ${H_{i} | i \in I} \subseteq C_{X} \Rightarrow ⋂_{i \in I} H_{i} \in C_{X}$ .

(2) ${H_{i} | i \in I} \subseteq C_{X}^{C} \Rightarrow ⋂_{i \in I} H_{i} \in C_{X}^{C}$ .

Proof. (1) Firstly, for any a, b ∈ ⋂ _i∈IH_i and c ∈ X such that * ((a, b, c) =1. Then c ∈ H_i for each i ∈ I since H_i is closed under *. Thus, c ∈ ⋂ _i∈IH_i, and it implies that ⋂_i∈IH_i is also closed under *.

Moreover, for each a ∈ ⋂ _i∈IH_i, then a ∈ H_i for each i ∈ I. Since H_i is a subgroup of X, so a^-1 ∈ H_i for each i ∈ I. Hence, a^-1 ∈ ⋂ _i∈IH_i. Then ⋂_i∈IH_i is a subgroup of X follows from Theorem 3.16.

(2) Using (1), then it just needs to prove that ⋂_i∈IH_i is a convex set. Suppose ∀x, y ∈ ⋂ _i∈IH_i, ∀z ∈ X and x ≤ z ≤ y. Then for each i ∈ I, we have x, y ∈ H_i. Thus, z ∈ H_i since $H_{i} \in C_{X}$ . It implies that z ∈ ⋂ _i∈IH_i, and so $⋂_{i \in I} H_{i} \in C_{X}$ .□

Proposition 4.5. Let (X, ≤ , *) be an ordered fuzzy group and {H_i | i ∈ I} ⊆2^X is an up-directed family. Then

(1) ${H_{i} | i \in I} \subseteq C_{X} \Rightarrow ⋃_{i \in I} H_{i} \in C_{X}$ .

(2) ${H_{i} | i \in I} \subseteq C_{X}^{C} \Rightarrow ⋃_{i \in I} H_{i} \in C_{X}^{C}$ .

Proof. (1) For any a, b ∈ ⋃ _i∈IH_i and c ∈ X such that * ((a, b, c) =1. Then ∃i_a, i_b ∈ I such that a ∈ H_{i
_a}, b ∈ H_{i
_b}. Since {H_i | i ∈ I} ⊆2^X is up-directed, there exists i₀ ∈ I such that H_{i
_a} ⊆ H_{i
₀}, H_{i
_b} ⊆ H_{i
₀}. So a ∈ H_{i
₀}, b ∈ H_{i
₀}, and so c ∈ H_{i
₀} since H_{i
₀} is closed under *. Thus, c ∈ ⋃ _i∈IH_i, and it implies that ⋃_i∈IH_i is also closed under *.

Moreover, for each a ∈ ⋃ _i∈IH_i, then a ∈ H_i for some i ∈ I. Since H_i is a subgroup of X, so a^-1 ∈ H_i. Hence, a^-1 ∈ ⋃ _i∈IH_i. Then ⋃_i∈IH_i is a subgroup of X follows from Theorem 3.16.

(2) Based on (1), it just needs to prove that ⋃_i∈IH_i is a convex set. Suppose ∀x, y ∈ ⋃ _i∈IH_i, ∀z ∈ X and x ≤ z ≤ y. Then there exists i₀ ∈ I such that x, y ∈ H_{i
₀}. Thus, z ∈ H_{i
₀} since H_{i
₀} is a convex set. It implies that z ∈ ⋃ _i∈IH_i, and so $⋃_{i \in I} H_{i} \in C_{X}^{C}$ .□

Definition 4.6. Let (X, ≤ , *) be an ordered fuzzy group and S ⊆ X be a non-empty subset, and ${H_{i} | i \in I} \subseteq C_{X}$ is the set of all subgroups containing S. Then ⋂_i∈IH_i is called the subgroup generated by S, denoted by 〈S〉. Similarly, the intersection of all convex subgroups containing S is said to be the convex subgroup generated by S, denoted by 〈S〉^C. In other words, $〈 S 〉 = ⋂ {H_{i} \in C_{X} | S \subseteq H_{i}}$ and $〈 S 〉^{C} = ⋂ {H_{i} \in C_{X}^{C} | S \subseteq H_{i}}$ .

Proposition 4.7. Let (X, ≤ , *) be an ordered fuzzy group and A ⊆ X. Then

(1) $(X, C_{X})$ is a convex space with the (convex) hull co (A) = 〈A〉.

(2) $(X, C_{X}^{C})$ is a convex space with the (convex) hull co (A) = 〈A〉^C.

Proof. It follows from Proposition 4.4, Proposition 4.5 and Definition 4.6.□

Proposition 4.8. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ : X → Y be an ordered fuzzy group homomorphism. Then

(1) $H \in C_{X} \Rightarrow φ^{\to} (H) \in C_{Y}$ .

(2) $K \in C_{Y} \Rightarrow φ^{\leftarrow} (K) \in C_{X}$ .

(3) $K \in C_{Y}^{C} \Rightarrow φ^{\leftarrow} (K) \in C_{X}^{C}$ .

Proof. (1) Assume H is a subgroup of X, and ∀a, b ∈ φ^→ (H) , ∀ c ∈ Y such that * ((a, b^-1) , c) =1. Then it is sufficient to show that c ∈ φ^→ (H) by Theorem 3.15. Since a, b ∈ φ^→ (H), there exist u, v ∈ H such that a = φ (u) , b = φ (v). Regarding that * is a strong L-function, there exists w ∈ X such that * ((u, v^-1) , w) =1. Because H is a subgroup of X, we have w ∈ H, and so φ (w) ∈ φ^→ (H). Furthermore, since φ is an ordered fuzzy group homomorphism and using Proposition 3.19(2), it holds $\begin{matrix} 1 & = & * ((u, v^{- 1}), w) = * ((φ (u), φ (v^{- 1})), φ (w)) \\ = & * ((φ (u), φ (v)^{- 1}), φ (w)) \\ = & * ((a, b^{- 1}), φ (w)) . \end{matrix}$ Since * ((a, b^-1) , c) =1 and * is a strong L-function, we get c = φ (w). That is, c ∈ φ^→ (H), as desired.

(2) Suppose ∀a, b ∈ φ^← (K) , ∀ c ∈ X and * ((a, b^-1) , c) =1. Then we have φ (a) , φ (b) ∈ K, and so φ (b) ^-1 ∈ K since K is a subgroup. Meanwhile, since φ is an ordered fuzzy group homomorphism, we have * ((φ (a) , φ (b) ^-1) , φ (c)) = * ((φ (a) , φ (b^-1)) , φ (c)) = * ((a, b^-1) , c) =1. It indicates that φ (c) ∈ K since K is closed under *. Hence, c ∈ φ^← (K), and it implies that φ^← (K) is a subgroup.

(3) Suppose $K \in C_{Y}^{C}$ , then $φ^{\leftarrow} (K) \in C_{X}$ by (2). Furthermore, for each x, y ∈ φ^← (K) and ∀z ∈ X such that x ≤ z ≤ y. We have φ (x) , φ (y) ∈ K and φ (x) ≤ φ (z) ≤ φ (y). Since K is convex, so we have φ (z) ∈ K. It implies that z ∈ φ^← (K), and so φ^← (K) is convex. Hence, $φ^{\leftarrow} (K) \in C_{X}^{C}$ .□

Proposition 4.9. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ be an ordered fuzzy group homomorphism. Then

(1) φ is convexity-preserving from $(X, C_{X}^{C})$ to $(Y, C_{Y}^{C})$ .

(2) φ is not only convexity-preserving but also convex-to-convex from $(X, C_{X})$ to $(Y, C_{Y})$ .

Proof. Straightforward from Proposition 4.8.□

5 (Normal) L-subgroups and the L-convex structures

In this section, we introduce L-subgroups and normal L-subgroups of the ordered fuzzy group, and then get some L-convex structures by discussing their properties. Sequentially, we study L-convex hull and L-convexity-preserving mappings of the corresponding convex spaces.

Definition 5.1. Let (X, ≤ , *) be an ordered fuzzy group and A ∈ L^X. Then A is said to be an L-subgroup of X if it satisfies the following conditions for all x, y, z ∈ X,

(1) A (x) ∧ A (y) ∧ * ((x, y) , z) ≤ A (z),

(2) A (x) ≤ A (x^-1).

Example 5.2. For the ordered fuzzy group $(ℝ, \leq, *)$ as mentioned in Example 3.7. Let $ℤ \subseteq ℝ$ be the integer set and A_α ∈ L^X for any α ∈ L, where for all $x \in ℝ$ , $A_{α} (x) = {\begin{matrix} 1 & x \in ℤ, \\ α & x \notin ℤ . \end{matrix}$ Then it is easy to check that if α ≥ 0.5, then A_α is an L-subgroup of $ℝ$ .

Definition 5.3. Let (X, ≤ , *) be an ordered fuzzy group. Then an L-subgroup A ∈ L^X is said to be normal if it satisfies the following condition for each x, y, m, z ∈ X,

A (y) ∧ * ((x, y) , m) ∧ * ((m, x^-1) , z) ≤ A (z).

We use $C_{X}^{L}$ and $C_{X}^{NL}$ respectively, to denote the set of all L-subgroups and normal L-subgroups of an ordered fuzzy group (X, ≤ , *). That is $C_{X}^{L} = {A \in L^{X} | A is an L - subgroup of X},$ $C_{X}^{NL} = {A \in L^{X} | A is a normal L - subgroup of X} .$ It is obvious that $C_{X}^{NL} \subseteq C_{X}^{L}$ . The converse, however, may not hold. The following theorem specifies the condition required for the converse to make sense.

Theorem 5.4. Let (X, ≤ , *) be an ordered fuzzy group, and $A \in C_{X}^{L}$ . Then $A \in C_{X}^{NL}$ iff A (m) ∧ * ((x, y) , m) ∧ * ((y, x) , w) ≤ A (w) for each x, y, m, w ∈ X.

Proof. (⇒). Since $A \in C_{X}^{NL}$ , then for each x, y, m, w ∈ X, we have $\begin{matrix} A (m) \land * ((x, y), m) \land * ((y, x), w) \\ = & A (m) \land * ((x^{- 1}, m), y) \land * ((y, x), w) \\ \leq & A (w) . \end{matrix}$ (⇐). For each x, y, m, z ∈ X, based on the assumption, we have A (y) ∧ * ((x, y) , m) ∧ * ((m, x^-1) , z) = A (y) ∧ * ((x^-1, m) , y) ∧ * ((m, x^-1) , z) ≤ A (z). Then $A \in C_{X}^{NL}$ according to Definition 5.3.□

Proposition 5.5. Let (X, ≤ , *) be an ordered fuzzy group, and $A \in C_{X}^{L}$ . Then A (x) ≤ A (1^*) for each x ∈ X.

Proof. For each x ∈ X, since A is an L-subgroup of X, then we have $\begin{matrix} A (x) & = & A (x) \land A (x^{- 1}) \\ = & A (x) \land A (x^{- 1}) \land * ((x, x^{- 1}), 1^{*}) \\ \leq & A (1^{*}) . \end{matrix}$

□

Theorem 5.6. Let (X, ≤ , *) be an ordered fuzzy group, and A ∈ L^X. Then $A \in C_{X}^{L}$ iff A (x) ∧ A (y) ∧ * ((x, y^-1) , z) ≤ A (z) for all x, y, z ∈ X.

Proof. (⇒). Since A is an L-subgroup, then for each x, y, z ∈ X, $\begin{matrix} A (x) \land A (y) \land * ((x, y^{- 1}), z) \\ \leq A (x) \land A (y^{- 1}) \land * ((x, y^{- 1}), z) \leq A (z) . \end{matrix}$ (⇐). For each x ∈ X, we have A (x) = A (x) ∧ A (x) ∧ * ((x, x^-1) , 1^*) ≤ A (1^*). Then $\begin{matrix} A (x) & = & A (1^{*}) \land A (x) \\ = & A (1^{*}) \land A (x) \land * ((1^{*}, x^{- 1}), x^{- 1}) \\ \leq & A (x^{- 1}) . \end{matrix}$ Consequently, for each x, y, z ∈ X, we have $\begin{matrix} A (x) \land A (y) \land * ((x, y), z) \\ \leq A (x) \land A (y^{- 1}) \land * ((x, y), z) \leq A (z) . \end{matrix}$ Hence, $A \in C_{X}^{L}$ by Definition 5.1.□

Proposition 5.7. Let (X₁, ≤ , *), (X₂, ≤ , *) be ordered fuzzy groups. Then

(1) $A_{1} \in C_{X_{1}}^{L}$ and $A_{2} \in C_{X_{2}}^{L}$ implies $A_{1} \times A_{2} \in C_{X_{1} \times X_{2}}^{L}$ ;

(2) $A_{1} \in C_{X_{1}}^{NL}$ and $A_{2} \in C_{X_{2}}^{NL}$ implies $A_{1} \times A_{2} \in C_{X_{1} \times X_{2}}^{NL}$ .

Proof. (1) For each x = (x₁, x₂) , y = (y₁, y₂) , z = (z₁, z₂) ∈ X₁ × X₂, we have $\begin{matrix} (A_{1} \times A_{2}) (x) \land (A_{1} \times A_{2}) (y) \land ⊛ ((x, y^{- 1}), z) \\ = A_{1} (x_{1}) \land A_{2} (x_{2}) \land A_{1} (y_{1}) \land A_{2} (y_{2}) \\ \land * ((x_{1}, y_{1}^{- 1}), z_{1}) \land * ((x_{2}, y_{2}^{- 1}), z_{2}) \\ = (A_{1} (x_{1}) \land A_{1} (y_{1}) \land * ((x_{1}, y_{1}^{- 1}), z_{1})) \\ \land (A_{2} (x_{2}) \land A_{2} (y_{2}) \land * ((x_{2}, y_{2}^{- 1}), z_{2})) \\ \leq A_{1} (z_{1}) \land A_{2} (z_{2}) \\ = (A_{1} \times A_{2}) (z) . \end{matrix}$ By Theorem 5.6, we have $A_{1} \times A_{2} \in C_{X_{1} \times X_{2}}^{L}$ .

(2) For each x = (x₁, x₂) , y = (y₁, y₂) , m = (m₁, m₂) , w = (w₁, w₂) ∈ X, it is sufficient to show (A₁ × A₂) (m) ∧ ⊛ ((x, y) , m) ∧ ⊛ ((y, x) , w) ≤ (A₁ × A₂) (w). In fact, $\begin{matrix} (A_{1} \times A_{2}) (m) \land ⊛ ((x, y), m) \land ⊛ ((y, x), w) \\ = A_{1} (m_{1}) \land A_{2} (m_{2}) \land * ((x_{1}, y_{1}), m_{1})) \land * ((x_{2}, \\ y_{2}), m_{2})) \land * ((y_{1}, x_{1}), w_{1}) \land * ((y_{2}, x_{2}), w_{2}) \\ = (A_{1} (m_{1}) \land * ((x_{1}, y_{1}), m_{1})) \land * ((y_{1}, x_{1}), w_{1})) \\ \land (A_{2} (m_{2}) \land * ((x_{2}, y_{2}), m_{2})) \land * ((y_{2}, x_{2}), w_{2})) \\ \leq A_{1} (w_{1}) \land A_{2} (w_{2}) \\ = (A_{1} \times A_{2}) (w) . \end{matrix}$

□

Proposition 5.8. Let (X, ≤ , *) be an ordered fuzzy group. Then

(1) ${A_{i} | i \in I} \subseteq C_{X}^{L} \Rightarrow ⋀_{i \in I} A_{i} \in C_{X}^{L}$ .

(2) ${A_{i} | i \in I} \subseteq C_{X}^{NL} \Rightarrow ⋀_{i \in I} A_{i} \in C_{X}^{NL}$ .

Proof. (1). For each x, y, z ∈ X, we have $\begin{matrix} (⋀_{i \in I} A_{i}) (x) \land (⋀_{i \in I} A_{i}) (y) \land * ((x, y^{- 1}), z) \\ = & ⋀_{i \in I} (A_{i} (x) \land A_{i} (y) \land * ((x, y^{- 1}), z)) \\ \leq & ⋀_{i \in I} A_{i} (z) . \end{matrix}$ According to Theorem 5.6, $⋀_{i \in I} A_{i} \in C_{X}^{L}$ .

(2) For each x, y, m, w ∈ X, we have $\begin{matrix} (⋀_{i \in I} A_{i}) (m) \land * ((x, y), m) \land * ((y, x), w) \\ = & ⋀_{i \in I} (A_{i}) (m) \land * ((x, y), m) \land * ((y, x), w) \\ \leq & ⋀_{i \in I} A_{i} (w) \\ = & (⋀_{i \in I} A_{i}) (w) . \end{matrix}$ By Theorem 5.4, $⋀_{i \in I} A_{i} \in C_{X}^{NL}$ .□

Proposition 5.9. Let (X, ≤ , *) be an ordered fuzzy group, and {A_i ∈ L^X | i ∈ I} be totally ordered. Then

(1) ${A_{i} | i \in I} \subseteq C_{X}^{L} \Rightarrow ⋁_{i \in I} A_{i} \in C_{X}^{L}$ ;

(2) ${A_{i} | i \in I} \subseteq C_{X}^{NL} \Rightarrow ⋁_{i \in I} A_{i} \in C_{X}^{NL}$ .

Proof. (1). For each x, y, z ∈ X, by applying Proposition 2.2 and Theorem 5.6, we have $\begin{matrix} (⋁_{i \in I} A_{i}) (x) \land (⋁_{i \in I} A_{i}) (y) \land * ((x, y^{- 1}), z) \\ = & ⋁_{i \in I} (A_{i} (x) \land A_{i} (y) \land * ((x, y^{- 1}), z)) \\ \leq & ⋁_{i \in I} A_{i} (z) . \end{matrix}$ Using Theorem 5.6 again, we have $⋁_{i \in I} A_{i} \in C_{X}^{L}$ .

(2) For each x, y, m, w ∈ X, we have $\begin{matrix} (⋁_{i \in I} A_{i}) (m) \land * ((x, y), m) \land * ((y, x), w) \\ = & ⋁_{i \in I} (A_{i}) (m) \land * ((x, y), m) \land * ((y, x), w) \\ \leq & ⋁_{i \in I} A_{i} (w) \\ = & (⋁_{i \in I} A_{i}) (w) . \end{matrix}$ By Theorem 5.4, $⋁_{i \in I} A_{i} \in C_{X}^{NL}$ .□

Definition 5.10. Let (X, ≤ , *) be an ordered fuzzy group and S ∈ L^X, ${A_{i} \in C_{X}^{L} | i \in I}$ is the set of all L-subgroups containing S. Then ⋀_i∈IA_i is said to be the L-subgroup generated by S, denoted by 〈S〉^L. Similarly, the intersection of all the normal L-subgroups containing S is called the normal L-subgroup generated by S, denoted by 〈S〉^NL. In other words, $〈 S 〉^{L} = ⋀_{i \in I} {A_{i} \in C_{X}^{L} | S \leq A_{i}}$ , $〈 S 〉^{NL} = ⋀_{i \in I} {A_{i} \in C_{X}^{NL} | S \leq A_{i}}$ .

In combination with Definition 5.10, Proposition 5.8 and Proposition 5.9, we immediately come to the following conclusions.

Proposition 5.11. Let (X, ≤ , *) be an ordered fuzzy group, and A ∈ L^X. Then

(1) $(X, C_{X}^{L})$ is an L-convex space, and cl (A) = 〈A〉^L.

(2) $(X, C_{X}^{NL})$ is an L-convex space, and cl (A) = 〈A〉^NL.

Proof. It follows from Proposition 5.8 and Proposition 5.9.□

Proposition 5.12. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ : X → Y be an ordered fuzzy group homomorphism. Then

(1) $B \in C_{Y}^{L} \Rightarrow φ_{L}^{\leftarrow} (B) \in C_{X}^{L}$ .

(2) $B \in C_{Y}^{NL} \Rightarrow φ_{L}^{\leftarrow} (B) \in C_{X}^{NL}$ .

(3) If φ is epimorphism, then $A \in C_{X}^{L} \Rightarrow φ_{L}^{\to} (A) \in C_{Y}^{L}$ .

(4) If φ is epimorphism, then $A \in C_{X}^{NL} \Rightarrow φ_{L}^{\to} (A) \in C_{Y}^{NL}$ .

Proof. (1) To show that $φ_{L}^{\leftarrow} (B)$ is an L-subgroup, it is sufficient to show that for each x, y, z ∈ X, $φ_{L}^{\leftarrow} (B) (x) \land φ_{L}^{\leftarrow} (B) (y) \land * ((x, y^{- 1}), z) \leq φ_{L}^{\leftarrow} (B) (z)$ . In fact, considering φ is an ordered fuzzy group homomorphism and B is an L-subgroup, we have $\begin{matrix} φ_{L}^{\leftarrow} (B) (x) \land φ_{L}^{\leftarrow} (B) (y) \land * ((x, y^{- 1}), z) \\ = & B (φ (x)) \land B (φ (y)) \land * ((φ (x), φ (y)^{- 1}), φ (z)) \\ \leq & B (φ (z)) = φ_{L}^{\leftarrow} (B) (z) . \end{matrix}$

(2) Based on (1), we just need to show $φ_{L}^{\leftarrow} (B) (m) \land * ((x, y), m) \land * ((y, x), w) \leq φ_{L}^{\leftarrow} (B) (w)$ for each x, y, m, w ∈ X. Since φ is an ordered fuzzy group homomorphism and $B \in C_{Y}^{NL}$ , we have $\begin{matrix} φ_{L}^{\leftarrow} (B) (m) \land * ((x, y), m) \land * ((y, x), w) \\ = & B (φ (m)) \land * ((φ (x), φ (y)), φ (m)) \\ \land * ((φ (y), φ (x)), φ (w)) \\ \leq & B (φ (w)) = φ_{L}^{\leftarrow} (B) (w) . \end{matrix}$ Therefore, $φ_{L}^{\leftarrow} (B) \in C_{X}^{NL}$ .

(3) For each m, n, p ∈ Y, it is sufficient to show that $φ_{L}^{\to} (A) (m) \land φ_{L}^{\to} (A) (n) \land * ((m, n^{- 1}), p) \leq φ_{L}^{\to} (A) (p)$ . Since φ is surjective, there exists z ∈ X such that φ (z) = p. Then

$\begin{matrix} φ_{L}^{\to} (A) (m) \land φ_{L}^{\to} (A) (n) \land * ((m, n^{- 1}), p) \\ = & ⋁_{φ (x) = m} A (x) \land ⋁_{φ (y) = n} A (y) \land * ((m, n^{- 1}), φ (z)) \\ = & ⋁_{φ (x) = m} ⋁_{φ (y) = n} (A (x) \land A (y) \land * ((φ (x), φ (y^{- 1})), φ (z))) \\ = & ⋁_{φ (x) = m} ⋁_{φ (y) = n} (A (x) \land A (y) \land * ((x, y^{- 1}), z)) \\ \leq & A (z) \leq ⋁_{φ (z) = p} A (z) \\ = & φ_{L}^{\to} (A) (p) . \end{matrix}$

(4) Suppose ∀x, y, m, w ∈ Y, then based on (3), it is sufficient to show $φ_{L}^{\to} (A) (m) \land * ((x, y), m) \land * ((y, x), w) \leq φ_{L}^{\to} (A) (w)$ . Since φ is epimorphism, there exist x₀, y₀, m₀, w₀ ∈ X such that φ (x₀) = x, φ (y₀) = y, φ (m₀) = m, φ (w₀) = w. Regarding that φ is an ordered fuzzy group homomorphism and $A \in C_{X}^{NL}$ , then we have $\begin{matrix} φ_{L}^{\to} (A) (m) \land * ((x, y), m) \land * ((y, x), w) \\ = & ⋁_{φ (z) = m} A (z) \land * ((x_{0}, y_{0}), m_{0}) \land * ((y_{0}, x_{0}), w_{0}) \\ \leq & A (m_{0}) \land * ((x_{0}, y_{0}), m_{0}) \land * ((y_{0}, x_{0}), w_{0})) \\ \leq & A (w_{0}) \leq φ_{L}^{\to} (A) (w) . \end{matrix}$

□

Applying this theorem, we get the following two propositions immediately.

Proposition 5.13. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ be an ordered fuzzy group homomorphism. Then φ is an L-convexity-preserving mapping from $(X, C_{X}^{L})$ to $(Y, C_{Y}^{L})$ , as well as that from $(X, C_{X}^{NL})$ to $(Y, C_{Y}^{NL})$ .

Proof. Straightforward from Proposition 5.12.□

Proposition 5.14. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ be epimorphism. Then φ is not only an L-convexity-preserving mapping but also an L-convex-to-convex mapping from $(X, C_{X}^{L})$ to $(Y, C_{Y}^{L})$ , as well as that from $(X, C_{X}^{NL})$ to $(Y, C_{Y}^{NL})$ .

Proof. Straightforward from Proposition 5.12.□

6 Convex L-subgroups and the corresponding L-convex structure

Taking into account the order, we put forward convex L-subgroups of an ordered fuzzy group in this section. We start with the discussion of some of their properties and go into the study of the corresponding L-convex structure.

Definition 6.1. Let (X, ≤ , *) be an ordered fuzzy group, and A ∈ L^X be an L-subgroup of X. Then A is said to be a convex L-subgroup if for each x, y, z ∈ X, x ≤ z ≤ y ⇒ A (x) ∧ A (y) ≤ A (z). The set of all convex L-subgroups is denoted by $C_{X}^{CL}$ .

Example 6.2. Let $G = ℤ \times ℤ$ with the pointwise order relation ≤, L = [0, 1] and A ∈ L^G defined by $A (x_{1}, x_{2}) = {\begin{matrix} 1 & if x_{2} = 0, \\ 0.5 & else . \end{matrix} (\forall x = (x_{1}, x_{2}) \in G)$ Now, define * : (G × G) × G ⟶ L as, for all x = (x₁, x₂) , y = (y₁, y₂) , z = (z₁, z₂) ∈ G, $* ((x, y), z) = {\begin{matrix} 1 & if x_{1} + y_{1} = z_{1}, x_{2} + y_{2} = z_{2}, \\ 0.5 & else . \end{matrix}$ Then (G, ≤ , *) is an ordered fuzzy group and A is a convex L-subgroup of G.

Proposition 6.3. Let (X₁, ≤ , *), (X₂, ≤ , *) be ordered fuzzy groups, $A_{1} \in C_{X_{1}}^{CL}$ and $A_{2} \in C_{X_{2}}^{CL}$ . Then $A_{1} \times A_{2} \in C_{X_{1} \times X_{2}}^{CL}$ .

Proof. First of all, $A_{1} \times A_{2} \in C_{X_{1} \times X_{2}}^{L}$ is immediately accessible from Proposition 5.7(1). Next, we will prove that (A₁ × A₂) (x) ∧ (A₁ × A₂) (y) ≤ (A₁ × A₂) (z) for each x = (x₁, x₂) , y = (y₁, y₂) , z = (z₁, z₂) ∈ X₁ × X₂ such that x ≤ z ≤ y. Noting that $A_{1} \in C_{X_{1}}^{CL}, A_{2} \in C_{X_{2}}^{CL}$ and x ≤ z ≤ y, we have $\begin{matrix} (A_{1} \times A_{2}) (x) \land (A_{1} \times A_{2}) (y) \\ = & A_{1} (x_{1}) \land A_{2} (x_{2}) \land A_{1} (y_{1}) \land A_{2} (y_{2}) \\ = & (A_{1} (x_{1}) \land A_{1} (y_{1})) \land (A_{2} (x_{2}) \land A_{2} (y_{2})) \\ \leq & A_{1} (z_{1}) \land A_{2} (z_{2}) \\ = & (A_{1} \times A_{2}) (z) . \end{matrix}$ Thus, $A_{1} \times A_{2} \in C_{X_{1} \times X_{2}}^{CL}$ .□

Proposition 6.4. Let (X, ≤ , *) be an ordered fuzzy group, ${A_{i} | i \in I} \subseteq C_{X}^{CL}$ . Then $⋀_{i \in I} A_{i} \in C_{X}^{CL}$ .

Proof. By Proposition 5.8(1), we have $⋀_{i \in I} A_{i} \in C_{X}^{L}$ . So it is sufficient to show that x ≤ z ≤ y ⇒ (⋀ _i∈IA_i) (x) ∧ (⋀ _i∈IA_i) (y) ≤ (⋀ _i∈IA_i) (z) for each x, y, z ∈ X. Since for each i ∈ I, $A_{i} \in C_{X_{1}}^{CL}$ . Thus, $\begin{matrix} (⋀_{i \in I} A_{i}) (x) \land (⋀_{i \in I} A_{i}) (y) \\ = & ⋀_{i \in I} (A_{i} (x) \land A_{i} (y)) \\ \leq & ⋀_{i \in I} A_{i} (z) = (⋀_{i \in I} A_{i}) (z) . \end{matrix}$

□

Proposition 6.5. Let (X, ≤ , *) be an ordered fuzzy group, and ${A_{i} | i \in I} \subseteq C_{X}^{CL}$ be an up-directed family. Then $⋁_{i \in I} A_{i} \in C_{X}^{CL}$ .

Proof. Assume ${A_{i} | i \in I} \subseteq C_{X}^{CL}$ , then $⋁_{i \in I} A_{i} \in C_{X}^{L}$ according to Proposition 5.9 (1). For each x, y, z ∈ X such that x ≤ z ≤ y, then it is sufficient to show that (⋁ _i∈IA_i) (x) ∧ (⋁ _i∈IA_i) (y) ≤ (⋁ _i∈IA_i) (z). In fact, $\begin{matrix} (⋁_{i \in I} A_{i}) (x) \land (⋁_{i \in I} A_{i}) (y) \\ = & ⋁_{i \in I} (A_{i}) (x) \land A_{i} (y)) \\ \leq & ⋁_{i \in I} A_{i} (z) = (⋁_{i \in I} A_{i}) (z) . \end{matrix}$

□

Definition 6.6. Let (X, ≤ , *) be an ordered fuzzy group and S ∈ L^X, ${A_{i} \in C_{X}^{CL} | i \in I}$ is the set of all convex L-subgroups containing S. Then ⋀_i∈IA_i is said to be the convex L-subgroup generated by S, denoted by 〈S〉^CL. In other words, $〈 S 〉^{CL} = ⋀_{i \in I} {A_{i} \in C_{X}^{CL} | S \leq A_{i}}$ .

In combination with Definition 6.6, Proposition 6.4 and Proposition 6.5, we immediately come to the following proposition.

Proposition 6.7. Let (X, ≤ , *) be an ordered fuzzy group. Then $(X, C_{X}^{CL})$ is an L-convex space with cl (A) = 〈A〉^CL for each A ∈ L^X.

Proof. It follows from Proposition 6.4 and Proposition 6.5.□

Proposition 6.8. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, φ : X → Y be an ordered fuzzy group homomorphism and $B \in C_{Y}^{CL}$ . Then $φ_{L}^{\leftarrow} (B) \in C_{X}^{CL}$ .

Proof. Suppose $B \in C_{Y}^{CL}$ , then $φ_{L}^{\leftarrow} (B) \in C_{X}^{L}$ by Proposition 5.12 (1). So it is sufficient to show $φ_{L}^{\leftarrow} (B) (x) \land φ_{L}^{\leftarrow} (B) (y) \leq φ_{L}^{\leftarrow} (B) (z)$ for all x, y, z ∈ X such that x ≤ z ≤ y. Since φ is order-preserving, then we have φ (x)) ≤ φ (z) ≤ φ (y). Therefore, $\begin{matrix} φ_{L}^{\leftarrow} (B) (x) \land φ_{L}^{\leftarrow} (B) (y) \\ = & B (φ (x)) \land B (φ (y)) \\ \leq & B (φ (z)) = φ_{L}^{\leftarrow} (B) (z) . \end{matrix}$

□

Following proposition 6.8, we obtain the next proposition right away.

Proposition 6.9. Let (X, ≤ , *) and (Y, ≤ , *) be ordered fuzzy groups, and φ be an ordered fuzzy group homomorphism. Then φ is an L-convexity-preserving mapping from $(X, C_{X}^{CL})$ to $(Y, C_{Y}^{CL})$ .

Proof. Straightforward from Proposition 6.8.□

7 Conclusions

In this paper, we propose a new kind of ordered fuzzy group via a fuzzy binary operation defined on posets, then we study their related algebraic characteristic and ordered fuzzy group homomorphisms. After that, we introduce (convex) subgroup and L-subgroups of this ordered fuzzy group. By analyzing the properties, the corresponding convex spaces induced by these substructures are constructed and characterized. All the conclusions in the present paper are based on the order fuzzy binary operation, which combines the properties of fuzzy binary operation and order. It is worth noting that the order under our consideration is general partial order. If we consider fuzzy partial order, then how will be it going? In other words, we can further consider how to define L-ordered fuzzy group and study its performance in the future.

Footnotes

Acknowledgment

This work is supported by National Natural Science Foundation of China (11901358) and Shandong Province Social Science Planning Research Project (14DGLJ06).

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