On a new kind of ordered fuzzy group

Abstract

This paper aims to further study the new kind of ordered fuzzy group named ordered L-group, which is put forward in literature [20]. Some algebraic properties of ordered L-groups, such as the relationship between elements, the equivalent characterizations and the products of these groups are discussed. Following that, the properties of substructures including characterization theorems, the convexity, the products of (normal) subgroups maintain the original substructure, along with the properties of ordered L-group homomorphisms are explored. The discussion of ordered fuzzy groups in this paper is from the perspective of fuzzy binary operation, which is different from the commonly method that just discuss the fuzzification of substructures in the research of fuzzy algebra. It can better reflect the essence of fuzzy groups logically just like that of classical groups.

Keywords

convex structure subgroup

1 Introduction

Fuzzy sets theory has been widely applied in plenty of branches of mathematics since it was first imported by Zadeh [44]. The application of fuzzy set theory to Algebra stems from the researches on fuzzy subgroups by Rosenfeld [28]. Later on, Biswas [6] invented and studied intuitionistic fuzzy subgroup. Since then, more and more studies and results about fuzzy algebras which are collected in [23] have emerged. In 2004, Zhan and Tan [45] introduced intuitionistic M-fuzzy groups. In recent years, with the proposal of Pythagorean fuzzy set [42] and its application [39], Pythagorean fuzzy subgroup have attracted the attention of many researchers [3 –5]. However, the existing literatures mostly aim at the fuzzification of substructures of algebras, and there is little work has been done on the fuzzification of algebraic structures themselves from the perspective of binary operations in a manner just similar to the classical algebra. In 1999, Demirci [12] introduced vague groups based on the vague binary operation, which is the first attempt to the fuzzification of binary operations. A few years later, Demirci [13] proposed smooth groups by means of the smooth binary operation which is another kind of fuzzy binary operation. Since then, more and more other studies about smooth groups, vague groups and other fuzzy algebras via vague or smooth binary operations have emerged [14 , 41]. In fact, both vague binary operations and smooth binary operations are dependent on fuzzy equalities, which are not fit for a crisp set. Besides, both the identity element and the inverse element may not be unique in smooth groups, which lead to the differences between the structures of smooth groups and classical groups. To overcome the disadvantages of smooth groups, Yuan [43] proposed a new kind of fuzzy group, but it fails to reflect the characteristics of the degree of fuzzy binary operations well. Recently, in order to get rid of the dependence on fuzzy equalities, Liu and Shi [21] raised an M-fuzzy function on crisp sets, and then defined a fuzzy binary operation and provided a new method to fuzzify groups. Unfortunately, M-fuzzy function only assigns to each point in X a unique point of Y to some extent, which does not meet the requirements of fuzzy logic. Generally, from the viewpoint of fuzzy logic, the fuzzy function is supposed to describe the degree of mapping any point in a set X to each point in another set Y. Taking that into account, a new kind of fuzzy function was reconsidered, and consequently L-binary operation and a new kind of ordered fuzzy group based on it was presented in [20].

In the previous work of literature [20], we mainly investigated the convex structures formed by subgroups and L-subgroups of this ordered L-group. But there are still many algebraic properties, such as the equivalent characterizations, the normal subgroups and the cosets as well as the properties of the products and homomorphisms, are not involved. So in this paper, we intend to further study the ordered L-group in a similar way to the classical group. First of all, the properties of the elements and the relationship between them in an ordered L-group are investigated, then some equivalent descriptions for ordered L-groups are given and meanwhile the notion of products is defined. Right after that, the substructures of an ordered L-group are studied, through a series of analyses and demonstration, some characterization theorems for subgroups or normal subgroups are obtained, the convexity of (normal) subgroups is clarified. It is also shown that the products of (normal) subgroups remains a (normal) subgroup. Last, the properties of ordered L-group homomorphisms are discussed.

The paper is organized as follows. In Section 2, we recollect some necessary definitions and conclusions needed later on. In Section 3, we first discuss the properties of the elements of an ordered L-group and the relationship between them, and then give equivalent descriptions and define the products of ordered L-groups. In Section 4, we focus on the substructures of ordered L-groups, and investigate the algebraic properties of subgroups, normal subgroups and the cosets. In Section 5, we study the properties of ordered L-groups homomorphisms. In the final section, we make a summary and draw some conclusions.

2 Preliminaries

Throughout this paper, unless otherwise stated, L always denotes a complete Heyting algebra or a frame. That is to say, L is a complete lattice with the smallest element 0 and the largest element 1, 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.

For any non-empty set X, the notation 2^X denotes the family of all subsets of X. A family {A_i | i ∈ I} ⊆2^X is said to be up-directed if for any A₁, A₂ ∈ {A_i | i ∈ I}, there exists A₃ ∈ {A_i | i ∈ I} such that A₁ ⊆ A₃ and A₂ ⊆ A₃.

For a given map f : X ⟶ Y, usually there exists a forward powerset operator f^→ : 2^X ⟶ 2^Y and a backward powerset operator f^← : 2^Y ⟶ 2^X which are defined by f^→ (A) = {f (x) | x ∈ A} (∀ A ⊆ X) and f^← (B) = {x ∈ X | f (x) ∈ B} (∀ B ⊆ Y).

An L-relation E on X is an L-subset of X × X, and it 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).

In particular, an L-preorder E on X is called an L-equivalence if it satisfies

(Sym) ∀x, y ∈ X, E (x, y) = E (y, x).

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

In fact, for any non-empty set X with a classical equality =, there exists a natural L-equality E^* : X × X → X defined as $\begin{matrix} \forall x, y \in X, E^{*} (x, y) = {\begin{matrix} 1 & x = y \\ 0 & x \neq y \end{matrix} \end{matrix}$

Now, we are about to introduce new approaches to the fuzzification of groups. To fuzzify a group, it is necessary to review some well-known concepts in classical group theory.

Definition 2.1. [15] A semigroup (X, ·) is a non-empty set X associated with a binary operation · satisfying the associative property:

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

A semigroup (X, ·) is called a monoid if it also satisfies the following condition.

(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, ·) satisfying the condition (G3) is called a group.

(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.

By the definition above, it is easily seen that a (classical) group is based on a binary operation whose essence is a function. Similarly, to get a fuzzy group, we should start with the fuzzification of functions.

A fuzzy function is firstly imported by Demirci [11] as a fuzzy relation f on X × Y w.r.t. the fuzzy equalities E_X, E_Y on X and Y, respectively, 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).

When setting $E_{X} = E_{X}^{*}$ and $E_{Y} = E_{Y}^{*}$ , Liu and Shi [18] proposed fuzzy functions on crisp sets immediately.

Definition 2.2. [18] 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. Suppose M = [0, 1], then ∀x ∈ X, {y ∈ Y|f (x, y) ≠0} must be a singleton set. In other words, f only assigns to any x ∈ X a unique point y ∈ Y to some extent. But from the viewpoint of fuzzy logic, the fuzzy function is supposed to describe the degree of mapping any point of X to each point of Y. Unfortunately, M-fuzzy function fails completely. In this respect, we redefined a new kind of fuzzy functions called L-function in [20].

Definition 2.3. [20] For two non-empty sets X and Y, an L-relation f : X × Y → L is called an L-function from X to Y 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 for any x ∈ X, there exists y ∈ Y such that f (x, y) =1, then f is said to be a strong L-function. When f is a strong L-function from X × X to X, it is usually called an L-operation on X.

It is well known that an ordered group [1, 7] is 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, which indicates that the binary operation and order are in harmony. Following this idea, Borzooei [8 –10] studied L-ordered group, which is a group together with L-order being harmonious with the binary operation. As always, coordinating the L-operation and order, we defined ordered L-operations and then imported ordered L-groups consequently in [20].

Definition 2.4. [20] Let (X, ≤) be a poset and f an L-operation on X. Then f is called an ordered L-operation on (X, ≤) if it satisfies:

(OLO) ∀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.

Definition 2.5. [20] Let * be an ordered L-operation on a poset (X, ≤). Then (X, ≤ , *) is called a L-semigroup if it satisfies:

(OLG1) * ((a, b), q) ∧ * ((q, c), m) ∧ * ((b, c), d) ≤ * ((a, d), m), * ((b, c), d) ∧ * ((a, d), m) ∧ * ((a, b), q) ≤ * ((q, c), m) (∀a, b, c, d, q, m ∈ X),

Definition 2.6. [20] An ordered L-semigroup (X, ≤ , *) is called an ordered L-group if it satisfies:

(OLG2) there exists an (two-sided) identity element 1^* ∈ X, such that * ((1^*, a), a) ∧ * ((a, 1^*), a) =1 for each a ∈ X,

(OLG3) 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 2.7. [20] Let $ℝ$ be the set of all real numbers with the ordinary order ≤ and addition operation. Define $* : (ℝ \times ℝ) \times ℝ \to L$ as $\begin{matrix} * ((x, y), z) = {\begin{matrix} 1 & x + y = z \\ 0.5 & x + y \neq z \end{matrix} \end{matrix}$

Then it can be easily seen that (X, ≤ , *) is an ordered fuzzy group with the identity element 1^* = 0 and the inverse element a^-1 = - a for each $a \in ℝ$ .

In the sequel of this section, we will introduce some knowledge of convexity theory, which is originally inspired by some elementary geometric problems such as shapes of circles or polytopes in low dimensional Euclidean spaces, and was systematically studied in 1990 when Minkowski used geometric arguments to prove theorems about numbers [2]. Later in 1993, Van De Vel collected the theory of convexity systematically in [35]. In fact, convexity exists in many mathematical research areas, such as vector spaces, metric spaces, graphs, matroids, lattices, groups and so on, and it has been accepted to be of increasing importance in recent years in the study of extremum problems in areas of applied mathematics. With the continuous development of the theory, there are more and more studies on convexity theory [16–20 , 46].

Definition 2.8. [35] 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 2.9. [35] 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 $C \in C_{X}$ .

It is worth noticing that the condition (C3) in the definition of convex structure is equivalent to the following condition.

$⋃_{i \in I} A_{i} \in C$ for all non-empty family ${A_{i} | i \in I} \subseteq C$ which is up-directed.

3 The algebraic characteristics of ordered L-groups

In [20], we introduced ordered L-groups by means of L-functions on posets, then we mainly discussed the convex structures and convex spaces constructed by their substructures. Now we are about to study the algebraic properties of ordered L-groups.

Proposition 3.1. Let (X, ≤ , *) be an ordered L-group. Then the identity element 1^* and the inverse element a^-1 of each a ∈ X is unique.

Proof. Assume that 1^* and $1_{0}^{*}$ are the identity elements. Then according to the definition of identity element, we have $* ((1^{*}, 1_{0}^{*}), 1^{*}) = * ((1^{*}, 1_{0}^{*}), 1_{0}^{*}) = 1$ . Thus $1^{*} = 1_{0}^{*}$ since * is a strong L-function.

For each a ∈ X, suppose a^-1 and $a_{0}^{- 1}$ are the inverse of a, then $a^{- 1} = a_{0}^{- 1}$ can be obtained in a manner similar to that of $1^{*} = 1_{0}^{*}$ . □ It is worth stressing that as a consequence of the uniqueness of the identity and the inverse element, we have (a^-1) ^-1 = a for each a in an ordered L-group (X, ≤ , *).

Proposition 3.2. Let (X, ≤ , *) be an ordered L-group and ∀a, b ∈ X. Then a ≤ b ⇔ b^-1 ≤ a^-1.

Proof. (⇒). For any a, b ∈ X such that a ≤ b, there exists m ∈ X such that * ((a^-1, b), m) =1. Since * ((a^-1, a), 1^*) =1 and a ≤ b, thus we have 1^* ≤ m according to (OFO). Moreover, by (OFG1) we have 1 = * ((b, b^-1), 1^*) ∧ * ((a^-1, 1^*), a^-1) ∧ * ((a^-1, b), m) ≤ * ((m, b^-1), a^-1). Noting that * ((1^*, b^-1), b^-1) =1 and using (OFO) again, we have b^-1 ≤ a^-1, as required.

For the other direction (⇐), just follows from the proved result and the fact that (a^-1) ^-1 = a for ∀a ∈ X. □ As pointed out in Remark 3.9 [20], each ordered group (G, ≤ , ·) can be regarded as a special ordered L-group. Conversely, for a poset (X, ≤) and an L-operation f : (X × X) × X → L, there exists a binary operation f₁ : X × X → X defined by xf₁y = z ⇔ f ((x, y), z) =1 for all x, y ∈ X. Considering this binary operation on (X, ≤), we have the following proposition.

Proposition 3.3. If (X, ≤ , *) is an ordered L-group, then (X, ≤ , * ₁) is an ordered group, but not vice versa.

Proof. First, for any a, b, c ∈ X, suppose a * ₁b = u, u * ₁c = m, b * ₁c = q and a * ₁q = w, then we have * ((a, b), u) = * ((u, c), m) = * ((b, c), q) = * ((a, q), w) =1. Using (OFG1), we get 1 = * ((a, b), u) ∧ * ((u, c), m) ∧ * ((b, c), q) ≤ * ((a, q), m). Thus * ((a, q), w) ∧ * ((a, q), m) =1, and so m = q. In other words, (a * ₁b) * ₁c = a * ₁ (b * ₁c).

Second, since (X, ≤ , *) is an ordered L-group, then it exists identity element 1^* such that * ((1^*, a), a) = * ((a, 1^*), a) =1 for each a ∈ X. That is, 1^* * ₁a = a * ₁1^* = 1 for each a ∈ X. Therefore 1^* is also the identity element with respect to (X, ≤ , * ₁).

Third, ∀a ∈ X, we have a^-1 ∈ X, and * ((a^-1, a), 1^*) = * ((a, a^-1), 1^*) =1. It indicates that a^-1 * ₁a = a * ₁a^-1 = 1^*. Hence a^-1 is also the inverse of a with respect to (X, ≤ , * ₁).

Finally, suppose x, y ∈ X, x ≤ y, and for any a, b ∈ X, assume a * ₁x = u, u * ₁b = m, a * ₁y = v, v * ₁b = w. Then * ((a, x), u) ∧ * ((a, y), v) =1, and it implies u ≤ v by (OLG2). Considering the hypothesis u * ₁b = m, v * ₁b = w and using (OLG2) again, then we obtain m ≤ w.

Based on the above analysis, it comes to the conclusion that (X, ≤ , * ₁) is an ordered group. □ To demonstrate the other direction, we give the following counterexample.

Example 3.4. Let $ℝ$ be the set of all real numbers with the ordinary order ≤ and addition operation +. Define $* : (ℝ \times ℝ) \times ℝ \to L$ as $\begin{matrix} * ((x, y), z) = {\begin{matrix} 1 & x + y = z; \\ 0.5 & x + y \neq z, x = y; \\ 0.3 & x + y \neq z, x \neq y . \end{matrix} \end{matrix}$ Then it is easy to check that (X, ≤ , * ₁) is an ordered group. But 0.5 = * ((1, 2), 3) ∧ * ((3, 3), 7) ∧ * ((2, 3), 5) ≰ * ((1, 5), 7) =0.3, it indicates (OLG1) doesn’t hold. Thus (X, ≤ , *) is not an ordered L-group.

Lemma 3.5. [20] Let (X, ≤ , *) be an ordered L-group. Then for all a, b, c ∈ X, we have $* ((a, b), c) = * ((a^{- 1}, c), b) = * ((c, b^{- 1}), a)$

Applying the lemma above repeatedly, we get the following proposition which reveals the close relationship between elements of an ordered L-group.

Proposition 3.6. Let (X, ≤ , *) be an ordered L-group, and ∀a, b, c ∈ X. Then * ((a, b), c) = * ((a^-1, c), b) = * ((c, b^-1), a) = * ((b, c^-1), a^-1) = * ((c^-1, a), b^-1) = * ((b^-1, a^-1), c^-1).

Proposition 3.7. [20] Let (X, ≤ , *) be an ordered L-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.

In an ordered L-semigroup (X, ≤ , *), if there exists $1_{L}^{*} \in X (1_{R}^{*} \in X)$ such that $* ((1_{L}^{*}, a), a) = 1 (* ((a, 1_{R}^{*}), a) = 1)$ for each a ∈ X, then $1_{L}^{*} (1_{R}^{*})$ is called a left (right) identity element of (X, ≤ , *). Furthermore, for an ordered L-semigroup (X, ≤ , *) with the left (right) identity element $1_{L}^{*} (1_{R}^{*})$ , for any a ∈ X, if there exists an element $a_{L}^{- 1} \in X (a_{R}^{- 1} \in X)$ such that $* ((a_{L}^{- 1}, a), 1_{L}^{*}) = 1 (* ((a, a_{R}^{- 1}), 1_{R}^{*}) = 1)$ , then $a_{L}^{- 1} (a_{R}^{- 1})$ is called a left (right) inverse of a.

Proposition 3.8. Let (X, ≤ , *) be an ordered L-group. Then

(1) $1_{L}^{*} = 1_{R}^{*} = 1^{*}$ ;

(2) $a_{L}^{- 1} = a_{R}^{- 1} = a^{- 1}$ for any a ∈ X.

Proof. (1) Since (X, ≤ , *) is an ordered L-group, for each a ∈ X, $* ((1_{L}^{*}, a), a) \land * ((1^{*}, a), a) = 1$ . Then $1_{L}^{*} = 1^{*}$ straightforward by Proposition 3.7, and so does $1_{R}^{*} = 1^{*}$ .

(2) For any a ∈ X, we possess $* ((a_{L}^{- 1}, a), a) \land * ((a^{- 1}, a), a) = 1$ and $* ((a, a_{R}^{- 1}), a) \land * ((a, a^{- 1}), a) = 1$ . By Proposition 3.7, it implies $a_{L}^{- 1} = a_{R}^{- 1} = a^{- 1}$ . □

Proposition 3.9. Let (X, ≤ , *) be an ordered L-group, and ∀a, b, c, d ∈ X.

(1) * ((a, b), 1^*) =1 ⇒ b = a^-1, a = b^-1;

(2) * ((a, b), 1^*) =1 ⇒ * ((b, a), 1^*) =1;

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

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

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

(6) * ((a, a), a) =1 ⇒ a = 1^*;

(7) If * ((x, x), 1^*) = x for each x ∈ X, then * ((a, b), c) = ((b, a), c).

Proof. (1) * ((a, b), 1^*) =1 ⇒ * ((a, b), 1^*) ∧ * ((a, a^-1), 1^*) =1 ⇒ b = a^-1 by Proposition 3.7. Similarly, * ((a, b), 1^*) ∧ * ((b^-1), b, 1^*) =1 ⇒ a = b^-1.

(2) * ((a, b), 1^*) =1 ⇒ b = a^-1 ⇒ * ((b, a), 1^*) = * ((a^-1, a), 1^*) =1.

(3) Since * ((a, b), c) ∧ * ((b^-1, a^-1), d) =1, so we have $\begin{matrix} 1 & = * ((b, b^{- 1}), 1^{*}) \land * ((a, 1^{*}), a) \land * ((a, b), c) \\ \leq * ((c, b^{- 1}), a) . \end{matrix}$ It results in

$\begin{matrix} 1 & = * ((c, b^{- 1}), a) \land * ((a, a^{- 1}), 1^{*}) \land * ((b^{- 1}, a^{- 1}), d) \\ \leq * ((c, d), 1^{*}), \end{matrix}$

Then d = c^-1, c = d^-1 according to (1).

(4) Applying Proposition 3.6 and Proposition 3.7, it holds naturally that ((a, b), c) ∧ * ((d, c), b) =1 ⇒ * ((a^-1, c), b) ∧ * ((d, c), b) =1 ⇒ d = a^-1, and ((a, b), c) ∧ * ((d, c), b) =1 ⇒ * ((a, b), c) ∧ * ((d^-1, b), c) =1 ⇒ a = d^-1 .

(5) Analogous to (4), we have $\begin{matrix} ((a, b), c) & \land * ((c, d), a) = 1 \\ \Rightarrow & * ((c, b^{- 1}), a) \land * ((c, d), a) = 1 \\ \Rightarrow & d = b^{- 1}, \\ ((a, b), c) & \land * ((c, d), a) = 1 \\ \Rightarrow & * ((a, b), c) \land * ((a, d^{- 1}), c) = 1 \\ \Rightarrow & b = d^{- 1} . \end{matrix}$

(6) If * ((a, a), a) =1, so ((a, a), a) ∧ ((1^*, a), a) =1. Thus a = 1^*.

(7) Following from (1), we can declare that x = x^-1 for each x ∈ X. Then * ((a, b), c) = * ((b^-1, a^-1), c^-1) = * ((b, a), c) by Proposition 3.6.

□

Proposition 3.10. Let (X, ≤ , *) be an ordered L-semigroup.

(1) If (X, ≤ , *) has a left identity element $1_{L}^{*}$ and, for each a ∈ X, there exists a left inverse $a_{L}^{- 1}$ . Then $* ((c, c), c) \leq * ((1_{L}^{*}, c), 1_{L}^{*})$ for any c ∈ X.

(2) If (X, ≤ , *) has a right identity element $1_{R}^{*}$ and, for each a ∈ X, there exists a right inverse $a_{R}^{- 1}$ . Then $* ((c, c), c) \leq * ((c, 1_{R}^{*}), 1_{R}^{*})$ for any c ∈ X.

Proof. (1) Since (X, ≤ , *) has a left identity element $1_{L}^{*}$ and a left inverse for each element in X, then for any c ∈ X, we have $* ((c_{L}^{- 1}, c), 1_{L}^{*}) = 1$ . Therefore, $* ((c, c), c) = * ((c, c), c) \land * ((c_{L}^{- 1}, c), 1_{L}^{*}) \land * ((c_{L}^{- 1}, c), 1_{L}^{*}) \leq * ((1_{L}^{*}, c), 1_{L}^{*})$ by (OLG1).

(2) The proof is similar to that of (1). □

Theorem 3.11. Let (X, ≤ , *) be an ordered L-semigroup. Then

(1) (X, ≤ , *) is an ordered L-group ⇔ it has a left identity element $1_{L}^{*}$ and, for each a ∈ X, there exists a left inverse $a_{L}^{- 1}$ .

(2) (X, ≤ , *) is an ordered L-group ⇔ it has a right identity element $1_{R}^{*}$ and, for each a ∈ X, there exists a right inverse $a_{R}^{- 1}$ .

Proof. (1) The implication (⇒) is obvious.

To prove the converse implication (⇐). For each a ∈ X, there exists a left inverse $a_{L}^{- 1}$ such that $* ((a_{L}^{- 1}, a), 1_{L}^{*}) = 1$ . Since * is a strong L-function, there exists u, w ∈ X such that $* ((a, a_{L}^{- 1}), u) = * ((a, 1_{L}^{*}), w) = 1$ . Therefore,

$\begin{matrix} 1 & = & * ((a_{L}^{- 1}, a), 1_{L}^{*}) \land * ((a, 1_{L}^{*}), w) \land * ((a, a_{L}^{- 1}), u) \\ \leq & * ((u, a), w) \\ 1 & = & * ((1_{L}^{*}, a_{L}^{- 1}), a_{L}^{- 1}) \land * ((a, a_{L}^{- 1}), u) \land * ((a, 1_{L}^{*}), w) \\ \leq & * ((w, a_{L}^{- 1}), u), \end{matrix}$

and so $\begin{matrix} 1 & = & * ((u, a), w) \land * ((w, a_{L}^{- 1}), u) \land * ((a, a_{L}^{- 1}), u) \\ \leq & * ((u, u), u) . \end{matrix}$

According to Proposition 3.10 (1), we have

$* ((1_{L}^{*}, u), 1_{L}^{*}) = 1$ , and then $\begin{matrix} 1 & = & * ((1_{L}^{*}, 1_{L}^{*}), 1_{L}^{*}) \land * ((1_{L}^{*}, u), u) \land * ((1_{L}^{*}, u), 1_{L}^{*}) \\ \leq & * ((1_{L}^{*}, 1_{L}^{*}), u) . \end{matrix}$ As a result, $* ((1_{L}^{*}, 1_{L}^{*}), 1_{L}^{*}) \land * ((1_{L}^{*}, 1_{L}^{*}), u) = 1$ , and it implies that $u = 1_{L}^{*}$ since * is a strong L-function, and so we have $* ((a, a_{L}^{- 1}), 1_{L}^{*}) = 1$ . Now, let us show that $1_{L}^{*}$ is also a right identity element, in other words, for each a ∈ X, $* ((a, 1_{L}^{*}), a) = 1$ . It just results from the fact that

$\begin{matrix} 1 & = & * ((a, a_{L}^{- 1}), 1_{L}^{*})) \land * ((1_{L}^{*}, a), a) \land * ((a_{L}^{- 1}, a), 1_{L}^{*}) \\ \leq & * ((a, 1_{L}^{*}), a) . \end{matrix}$

So we can conclude that $1_{L}^{*}$ is a two-sided identity element of (X, ≤ , *).

Considering the result $* ((a, a_{L}^{- 1}), 1_{L}^{*}) = 1$ proved before, it is straightforward that $a_{L}^{- 1}$ is a two-sided inverse element of a. Hence (X, ≤ , *) is an ordered fuzzy group.

(2) It can be proved similarly. □

Theorem 3.12. Let (X, ≤ , *) be an ordered L-semigroup. Then (X, ≤ , *) is an ordered L-group ⇔ ∀a, b ∈ X, there exists x, y ∈ X such that * ((a, x), b) ∧ * ((y, a), b) =1.

Proof. (⇒). Suppose (X, ≤ , *) is an ordered L-group, and ∀a, b ∈ X. Then we have a^-1 ∈ X, and there exists x, y ∈ X such that * ((a^-1, b), x) = * ((b, a^-1), y) =1. According to Proposition 3.6, $\begin{matrix} * ((a, x), b) \land * ((y, a), b) \\ = & * ((a^{- 1}, b), x) \land * ((b, a^{- 1}), y) = 1 . \end{matrix}$ (⇐). By the assumption, for a fixed m ∈ X and ∀a ∈ X, there exists $x, 1_{m}^{*} \in X$ such that $* ((m, x), a) = * ((1_{m}^{*}, m), m) = 1$ . Then by (OLG1), we have $\begin{matrix} 1 & = & * ((1_{m}^{*}, m), m) \land * ((m, x), a) \land * ((m, x), a) \\ \leq & * ((1_{m}^{*}, a), a) . \end{matrix}$ It implies that $1_{m}^{*}$ is the left identity element of (X, ≤ , *).

On the other hand, by the hypothesis, for each b ∈ X, there exists b^* ∈ X such that $* ((b^{*}, b), 1_{m}^{*}) = 1$ , and it indicates that b has the left inverse b^*. Then (X, ≤ , *) is an ordered L-group follows from Theorem 3.11. □ In the next of the Section, we are going to discuss the product of ordered L-groups.

Let *_X be an ordered L-operation on (X, ≤), and *_Y be an ordered L-operation on (Y, ≤). Then there exists an L-operation ⊛ on (X × Y) defined by $\begin{matrix} ⊛ (((x_{1}, y_{1}), (x_{2}, y_{2})), (x_{3}, y_{3})) \\ = & *_{X} ((x_{1}, x_{2}), x_{3}) \land *_{Y} ((y_{1}, y_{2}), y_{3}) . \end{matrix}$

It is easy to check that ⊛ is an ordered L-operation on (X × Y, ≤).

Proposition 3.13. Let (X, ≤ , * _X), (Y, ≤ , * _Y) be ordered L-groups. Then (X × Y, ≤ , ⊛) is an ordered L-group, which is called the product of (X, ≤ , * _X) and (Y, ≤ , * _Y).

Proof. (OLG1) is obvious.

(OLG2) Because (X, ≤ , * _X) and (Y, ≤ , * _Y) are ordered L-groups, we suppose that $1_{X}^{*}$ is the identity element of (X, ≤ , * _X) and $1_{Y}^{*}$ is the identity element of (Y, ≤ , * _Y). Then $(1_{X}^{*}, 1_{Y}^{*}) \in X \times Y$ , and it can be easily seen that $(1_{X}^{*}, 1_{Y}^{*})$ is the identity element of (X × Y, ≤ , ⊛).

(OLG3) For each (x, y) ∈ X × Y, there exists x^-1 ∈ X and y^-1 ∈ Y. It can be easily checked that (x^-1, y^-1) is the inverse of (x, y). □

4 The subgroups and normal subgroups of ordered L-groups

In this section, we are going to study the subgroups and normal subgroups of an ordered L-group. In addition to the convexity, the properties of the products and the algebraic properties of the cosets as well as the equivalent characterization theorems are explored.

Let f be an ordered L-operation on a poset (X, ≤), then a crisp subset H ⊆ X is said to be closed under f if $f ((a, b), c) = 1 \Rightarrow c \in H (\forall a, b \in H, \forall c \in X) .$ Under such circumstances, it is not difficult to conclude that f |_H×H×H is also an ordered L-operation on (H, ≤).

Definition 4.1. [20] Let (X, ≤ , *) be an ordered L-group and H ⊆ X. If H is closed under * and meanwhile (H, ≤ , * |_H×H×H) is an ordered L-group, then H is called a subgroup of X if it.

we use $C_{X}$ to denote the set of all subgroups of an ordered L-group (X, ≤ , *).

Theorem 4.2. [20] Let (X, ≤ , *) be an ordered L-group and H ⊆ X be non-empty. Then H is a subgroup if and only if * ((a, b^-1), c) =1 ⇒ c ∈ H (∀ a, b ∈ H, ∀ c ∈ X).

Theorem 4.3. [20] Let (X, ≤ , *) be an ordered L-group and H ⊆ X be non-empty. Then H is a subgroup if and only if the following conditions hold.

(1) H is closed under *;

(2) For each a ∈ H, a^-1 ∈ H.

Proposition 4.4. Let (X, ≤ , *) be an ordered L-group, and H ⊆ X is a subgroup of X. Then

(1) the identity element of (H, ≤ , * |_H×H×H) is the same as that of (X, ≤ , *);

(2) for each a ∈ H ⊆ X, a has the same inverse in (H, ≤ , * |_H×H×H) and (X, ≤ , *).

Proof. (1) Suppose $1_{H}^{*}$ and $1_{X}^{*}$ are respectively, the identity element of (H, ≤ , * |_H×H×H) and (X, ≤ , *). Then for any a ∈ H ⊆ X, we have $* ((1_{H}^{*}, a), a) \land * ((1_{X}^{*}, a), a) = 1$ . It is straightforward that $1_{H}^{*} = 1_{X}^{*}$ since * is a strong L-function. That is, the identity element of (H, ≤ , * |_H×H×H) is the same as that of (X, ≤ , *), and we usually denote it by 1^* conveniently.

(2) For any a ∈ H ⊆ X, let’s assume that it has the inverse $a_{H}^{- 1}$ and $a_{X}^{- 1}$ in (H, ≤ , * |_H×H×H) and (X, ≤ , *) respectively. Then we obtain $* ((a_{H}^{- 1}, a), 1^{*}) \land * ((a_{X}^{- 1}, a), 1^{*}) = 1$ , and it implies that $a_{H}^{- 1} = a_{X}^{- 1}$ . For convenience, we use a^-1 to denote the same inverse of a. □

Proposition 4.5. Let (X, ≤ , *) be an ordered L-group and {H_i | i ∈ I} be a non-empty family of non-empty subgroups of X. Then ∩_i∈IH_i is a subgroup of X.

Proof. For any a, b ∈ ∩ _i∈IH_i and ∀c ∈ X such that * ((a, b^-1), c) =1. Then for each i ∈ I, a, b ∈ H_i. noticing Theorem 4.2 and the fact that H_i is a subgroup, then we have c ∈ H_i. Thus c ∈ ∩ _i∈IH_i, and then ∩_i∈IH_i is a subgroup resulting from Theorem 4.2. □

Proposition 4.6. Let (X, ≤ , *) be an ordered L-group and {H_i | i ∈ I} be a non-empty and up-directed family of non-empty subgroups of X. Then ∪_i∈IH_i is a subgroup of X.

Proof. For any a, b ∈ ∪ _i∈IH_i and ∀c ∈ X such that * ((a, b^-1), c) =1. Then there exist i₁, i₂ ∈ I such that a ∈ H_{i
₁} and b ∈ H_{i
₂}. Because {H_i | i ∈ I} is up-directed, then ∃i₀ ∈ I such that H_{i
₁} ∪ H_{i
₂} ⊆ H_{i
₀}, which indicates a, b ∈ H_{i
₀}. Applying Theorem 4.2, we have c ∈ H_{i
₀}. Thus c ∈ ∪ _i∈IH_i, and it results in ∪_i∈IH_i being a subgroup. □

Definition 4.7. Let (X, ≤ , *) be an ordered L-group and S ⊆ X be a non-empty subset. Let {H_i | i ∈ I} be the family of all subgroups of X containing S. Then ∩_i∈IH_i is called the subgroup generated by S, and it is denoted by 〈S〉. In other words, $〈 S 〉 = \cap {H_{i} | S \subseteq H_{i}, H_{i} \in C_{X}}$ .

Proposition 4.8. Let (X, ≤ , *) be an ordered L-group. Then $(X, C_{X})$ is a convex space, where the (convex) hull co (A) = 〈A〉 for each A ⊆ X.

Proof. It follows from Proposition 4.5, Proposition 4.6, Definition 2.8 and Definition 4.7. □

Proposition 4.9. Let (X, ≤ , * _X), (Y, ≤ , * _Y) be ordered L-groups, H₁ and H₂ be subgroups of X and Y respectively. Then H₁ × H₂ is a subgroup of (X × Y, ≤ , ⊛).

Proof. ∀ (a₁, a₂), (b₁, b₂) ∈ H₁ × H₂, and ∀ (c₁, c₂) ∈ X × Y such that $⊛ (((a_{1}, a_{2}), (b_{1}, b_{2})^{- 1}), (c_{1}, c_{2})) = 1,$ then $*_{X} ((a_{1}, b_{1}^{- 1}), c_{1}) \land *_{Y} ((a_{2}, b_{2}^{- 1}), c_{2}) = 1$ . By Theorem 4.2, we have c₁ ∈ H₁ and c₂ ∈ H₂ since H₁ and H₂ are subgroups. It implies (c₁, c₂) ∈ H₁ × H₂, and using Theorem 4.2 again, we obtain H₁ × H₂ is a subgroup of X × Y. □ For a given ordered L-group (X, ≤ , *), we can define an operation $\bar{*}$ on the powerset of X as $A \bar{*} B = {c | \exists a \in A, \exists b \in B, * ((a, b), c) = 1} (\forall A, B \subseteq X) .$ Particularly, $g \bar{*} A$ is short for ${g} \bar{*} A$ , and is called a left coset of A in X. $A \bar{*} g$ is short for $A \bar{*} {g}$ , and is called a right coset of A in X.

Proposition 4.10. Let (X, ≤ , *) be an ordered L-group and H a subgroup of X. Then $H \bar{*} H = H$ .

Proof. On one hand, for each h ∈ H, there exists 1^* ∈ H and * ((1^*, h), h) =1. It indicates $h \in H \bar{*} H$ . So $H \subseteq H \bar{*} H$ .

On the other hand, $\forall h \in H \bar{*} H = {c | \exists a, b \in H, * ((a, b), c) = 1}$ , then there exists a, b ∈ H such that * ((a, b), h) =1. Since H is closed under *, thus h ∈ H. It implies $H \bar{*} H \subseteq H$ . The proof is completed. □

Proposition 4.11. Let (X, ≤ , *) be an ordered L-group and H is a subgroup of X. Then for each a ∈ X, we have

(1) $a \in a \bar{*} H$ ;

(2) $b \in a \bar{*} H \Rightarrow \exists b_{0} \in H$ such that * ((a^-1, b), b₀) =1. (3) $a \bar{*} H \subseteq H \Leftrightarrow a \in H$ ;

(4) $a \bar{*} H = H \Leftrightarrow a \in H$ ;

(5) $a \bar{*} H$ is a subgroup if and only if a ∈ H.

Proof. (1) For each a ∈ X, there exists 1^* ∈ H and * ((a, 1^*), a) =1. So $a \in a \bar{*} H$ .

(2) If $b \in a \bar{*} H$ , then there exists b₀ ∈ H such that * ((a, b₀), b) =1. By Proposition 3.6, we have * ((a^-1, b), b₀) = * ((a, b₀), b) =1.

(3) (⇒). It holds straightforwardly from (1) and the condition $a \bar{*} H \subseteq H$ .

(⇐). If a ∈ H, then for each $c \in a \bar{*} H$ , we have b ∈ H such that * ((a, b), c) =1. Hence c ∈ H since H is closed under *. It results in $a \bar{*} H \subseteq H$ .

(4) Based on (3), it is just need to show that $a \in H \Rightarrow H \subseteq a \bar{*} H$ . In other words, it is sufficient to prove that $h \in a \bar{*} H$ for each h ∈ H along with a ∈ H. Equivalently, it is sufficient to show there exists b ∈ H such that * ((a, b), h) =1, which holds naturally according to Theorem 3.12 since H is a subgroup of X. The proof is completed.

(5) The implication (⇐) is obvious.

For the converse implication (⇒). Since $a \bar{*} H$ is a subgroup, then $1^{*} \in a \bar{*} H$ . By the definition of $a \bar{*} H$ , we observe that there exists b ∈ H such that * ((a, b), 1^*) =1. Then by Proposition 3.9(1), we obtain a = b^-1. Note that b ∈ H and H is a subgroup, then it can be seen that a ∈ H. □

Proposition 4.12. Let (X, ≤ , *) be an ordered L-group and H is a subgroup of X. Then the following statements are equivalent for each a, b ∈ X:

(1) $b \in a \bar{*} H$ ;

(2) * ((a^-1, b), c) =1 ⇒ c ∈ H for any c ∈ X;

(3) $a \bar{*} H = b \bar{*} H$ ;

(4) $a \in b \bar{*} H$ .

Proof. Let’s firstly prove the equivalence of (1) and (2), and then show (1)⇒ (3)⇒ (4)⇒ (1).

(1) ⇒ (2). If $b \in a \bar{*} H$ and * ((a^-1, b), c) =1 (∀ c ∈ X). Then there exists b₀ ∈ H such that * ((a, b₀), b) =1. Using Proposition 3.6, we have * ((a^-1, b), b₀) ∧ * ((a^-1, b), c) = * ((a, b₀), b) ∧ * ((a^-1, b), c) =1. It implies that c = b₀ since * is a strong L-function. Hence c ∈ H.

Conversely, for each a, b ∈ X, there exists c ∈ X such that * ((a^-1, b), c) =1. By (2), we know that c ∈ H. Besides, we have * ((a, c), b) = * ((a^-1, b), c) =1. It indicates $b \in a \bar{*} H$ . In summary, we have (1)⇔ (2).

(1) ⇒ (3). If $b \in a \bar{*} H$ , then there exists b₀ ∈ H such that * ((a, b₀), b) =1. Using Proposition 3.6, we have $* ((b, b_{0}^{- 1}), a) = 1$ . For each $c \in a \bar{*} H$ , there exists h₁ ∈ H such that * ((a, h₁), c) =1. Sequentially, there exists w₁ ∈ X such that $* ((b_{0}^{- 1}, h_{1}), w_{1}) = 1$ . Regarding that H is a subgroup, then we obtain w₁ ∈ H. Furthermore, we have $1 = * ((b, b_{0}^{- 1}), a) \land * ((a, h_{1}), c) \land * ((b_{0}^{- 1}, h_{1}), w_{1}) \leq * ((b, w_{1}), c)$ . It implies $c \in b \bar{*} H$ , and Thus $a \bar{*} H \subseteq b \bar{*} H$ .

For each $c \in b \bar{*} H$ , there exists h₂ ∈ H such that * ((b, h₂), c) =1, and there exists w₂ ∈ X such that * ((b₀, h₂), w₂) =1. Regarding that H is a subgroup, then we obtain w₂ ∈ H. Furthermore, we have 1 = * ((a, b₀), b) ∧ * ((b, h₂), c) ∧ * ((b₀, h₂), w₂) ≤ * ((a, w₂), c). It implies $c \in a \bar{*} H$ , and Thus $b \bar{*} H \subseteq a \bar{*} H$ .

(3) ⇒ (4). It follows from Proposition 4.11 (1).

(4) ⇒ (1). If $a \in b \bar{*} H$ , there exists h ∈ H such that * ((b, h), a) =1. Since H is a subgroup, we observe h^-1 ∈ H, and 1 = * ((h, h^-1), 1^*) ∧ * ((b, 1^*), b) ∧ * ((b, h), a) ≤ * ((a, h^-1), b). It implies that $b \in a \bar{*} H$ .

□ Taking advantages of the cosets of ordered L-groups, now we can introduce normal subgroup.

Definition 4.13. Let (X, ≤ , *) be an ordered L-group and H is a subgroup of X. Then H is called a normal subgroup if $a \bar{*} H = H \bar{*} a$ for each a ∈ X.

we denote the set of all normal subgroups of an ordered L-group (X, ≤ , *) by $C_{X}^{N}$ .

Theorem 4.14. Let (X, ≤ , *) be an ordered L-group and H is a subgroup of X. Then H is a normal subgroup iff * ((x, y), h) ∧ * ((y, x), z) =1 ⇒ z ∈ H for any h ∈ H and ∀x, y, z ∈ X.

Proof. (⇒). Suppose H is a normal subgroup and for any h ∈ H, ∀x, y, z ∈ X such that * ((x, y), h) ∧ * ((y, x), z) =1. Then we have * ((x^-1, h), y) =1, and it indicates $y \in x^{- 1} \bar{*} H = H \bar{*} x^{- 1}$ . So there exists h₀ ∈ H such that * ((h₀, x^-1), y) =1. Therefore, * ((y, x), h₀) ∧ * ((y, x), z) = * ((h₀, x^-1), y) ∧ * ((y, x), z) =1. It implies that z = h₀ ∈ H.

(⇐). By the definition of normal subgroups, it sufficient to show that $a \bar{*} H = H \bar{*} a$ for each a ∈ X. Assume $y \in a \bar{*} H$ , then ∃h ∈ H such that * ((a, h), y) =1. Besides, there exists m ∈ X such that * ((y, a^-1), m) =1 since * is a strong L-function. So * ((a^-1, y), h) ∧ * ((y, a^-1), m) = * ((a, h), y) ∧ * ((y, a^-1), m) =1. It results in m ∈ H. Noting that * ((m, a), y) = * ((y, a^-1), m) =1, we observe $y \in H \bar{*} a$ . Conversely, assume $y \in H \bar{*} a$ , then we can similarly prove $y \in a \bar{*} H$ . From the above analysis, it can be seen $a \bar{*} H = H \bar{*} a$ , and so H is a normal subgroup of X. □

Proposition 4.15. Let (X, ≤ , *) be an ordered L-group and {H_i | i ∈ I} be a non-empty family of non-empty normal subgroups of X. Then ∩_i∈IH_i is a normal subgroup of X.

Proof. First of all, ∩_i∈IH_i is a subgroup of X by Proposition 4.5. For any h ∈ ∩ _i∈IH_i and ∀x, y, z ∈ X, if * ((x, y), h) ∧ * ((y, x), z) =1, then according to Theorem 4.13, z ∈ H_i for each i ∈ I since H_i is a normal subgroup. Thus z ∈ ∩ _i∈IH_i, and it implies ∩_i∈IH_i is a normal subgroup. □

Proposition 4.16. Let (X, ≤ , *) be an ordered L-group and {H_i | i ∈ I} be a non-empty and up-directed family of non-empty normal subgroups of X. Then ∪_i∈IH_i is a normal subgroup of X.

Proof. Obviously, ∪_i∈IH_i is a subgroup of X by Proposition 4.6. For any h ∈ ∪ _i∈IH_i and ∀x, y, z ∈ X, suppose * ((x, y), h) ∧ * ((y, x), z) =1. Then ∃i₀ ∈ I such that h ∈ H_{i
₀}, so z ∈ H_{i
₀} just because H_{i
₀} is normal. As a result, z ∈ ∪ _i∈IH_i and so ∪_i∈IH_i is a normal subgroup. □

Definition 4.17. Let (X, ≤ , *) be an ordered L-group and S ⊆ X be a non-empty subset. Let {H_i | i ∈ I} be the family of all normal subgroups of X containing S. Then ∩_i∈IH_i is called the normal subgroup generated by S, which is denoted by 〈S〉^N. In other words, $〈 S 〉^{N} = \cap {H_{i} | S \subseteq H_{i}, H_{i} \in C_{X}^{N}}$ .

Proposition 4.18. Let (X, ≤ , *) be an ordered L-group. Then $(X, C_{X}^{N})$ is a convex space, where the (convex) hull co (A) = 〈A〉^N for each A ⊆ X.

Proof. It follows from Proposition 4.15, Proposition 4.16, Definition 2.8 and Definition 4.17. □ Proposition 4.19. Let (X₁, ≤ , * _{X
₁}), (X₂, ≤ , * _{X
₂}) be ordered L-groups, H₁ and H₂ be normal subgroups respectively of X₁ and X₂. Then H₁ × H₂ is a normal subgroup of (X₁ × X₂, ≤ , ⊛).

Proof. First of all, H₁ × H₂ is a subgroup of (X₁ × X₂, ≤ , ⊛) according to Proposition 4.9. For any (h₁, h₂) ∈ H₁ × H₂ and ∀ (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ X₁ × X₂ such that ⊛ (((x₁, x₂), (y₁, y₂)), (h₁, h₂)) ∧ ⊛ (((y₁, y₂), (x₁, x₂)), (z₁, z₂)) =1. Then we have *_{X
₁} ((x₁, y₁), h₁) ∧ * _{X
₁} ((y₁, x₁), z₁) ∧ * _{X
₂} ((x₂, y₂), h₂) ∧ * _{X
₂} ((y₂, x₂), z₂) =1. By Theorem 4.14, we have z₁ ∈ H₁ and z₂ ∈ H₂ since H₁ and H₂ are normal subgroups. It implies (z₁, z₂) ∈ H₁ × H₂, and so H₁ × H₂ is a normal subgroup of X₁ × X₂.

□

5 Ordered L-group homomorphisms

In this section, we will investigate the homomorphisms between two ordered L-groups. It will be shown that the homomorphisms preserve some properties related to ordered L-groups and subgroups.

Definition 5.1. Let (X, ≤ , * _X) and (Y, ≤ , * _Y) be ordered L-groups, and φ : X → Y be a function in the classical sense. Then φ is called an ordered L-group homomorphism iff φ is not only order-preserving, but also satisfies $*_{X} ((a, b), c) = *_{Y} ((φ (a), φ (b)), φ (c)) (\forall a, b, c \in X) .$ Furthermore, an injective ordered L-group homomorphism is called an ordered L-group monomorphism, a surjective ordered L-group homomorphism is called an ordered L-group epimorphism, and a bijective ordered fuzzy L-homomorphism is called an ordered L-group isomorphism.

Proposition 5.2. Let (X, ≤ , * _X) and (Y, ≤ , * _Y) be ordered L-groups, and φ : X → Y be an ordered L-group homomorphism. Then

(1) $φ (1_{X}^{*}) = 1_{Y}^{*}$ ,

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

Proof. (1) For any a ∈ X, $*_{Y} ((φ (a), 1_{Y}^{*})), φ (a)) = 1$ naturally holds. Since φ is an ordered L-groups homomorphism, $*_{Y} ((φ (a), φ (1_{X}^{*})), φ (a)) = *_{X} ((a, 1_{X}^{*}), a) = 1$ . Using Proposition 3.7, we obtain $φ (1_{X}^{*}) = 1_{Y}^{*}$ .

(2) For each a ∈ X, we have a^-1 ∈ X such that $*_{X} ((a, a^{- 1}), 1_{X}^{*}) \land * ((a^{- 1}, a), 1_{X}^{*}) = 1$ . Noting that φ is an ordered L-groups homomorphism, then we have $\begin{matrix} *_{Y} ((φ (a), φ (a^{- 1})), φ (1_{X}^{*})) & \land \\ *_{Y} ((φ (a^{- 1}), φ (a)), φ (1_{X}^{*})) & = & 1 . \end{matrix}$ That is,

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

Definition 5.3. Let (X, ≤ , * _X) and (Y, ≤ , * _Y) be ordered L-groups, and φ : X → Y be an ordered L-group homomorphism. Then the set $\ker φ = {a \in X | φ (a) = 1_{Y}^{*}}$ is called a kernel of φ.

Proposition 5.4. Let (X, ≤ , * _X) and (Y, ≤ , * _Y) be ordered L-groups, and φ : X → Y be an ordered L-group homomorphism. Then φ is injective $\Leftrightarrow \ker φ = {1_{X}^{*}}$ .

Proof. (⇒). Noticing that $φ (1_{X}^{*}) = 1_{Y}^{*}$ and φ is injective, then $\ker φ = {1_{X}^{*}}$ .

(⇐). Suppose x₁, x₂ ∈ X and x₁ ≠ x₂, then it is sufficient to prove that φ (x₁) ≠ φ (x₂). In fact, if φ (x₁) = φ (x₂), then φ (x₁) ^-1 = φ (x₂) ^-1, and $*_{Y} ((φ (x_{1}), φ (x_{2})^{- 1}), 1_{Y}^{*}) = 1$ consequently. As a result, we have $*_{X} ((x_{1}, x_{2}^{- 1}), 1_{X}^{*}) = *_{Y} ((φ (x_{1}), φ (x_{2})^{- 1}), 1_{Y}^{*}) = 1$ since φ is an ordered L-group homomorphism, which results in the fact that $x_{1} = (x_{2}^{- 1})^{- 1} = x_{2}$ . Then we can conclude that φ (x₁) ≠ φ (x₂) for any φ (x₁) = φ (x₂).

□

Proposition 5.5. Let (X, ≤ , * _X) and (Y, ≤ , * _Y) be ordered L-groups, and φ : X → Y be an ordered L-group homomorphism. Then

(1) If H is a subgroup of X, then φ^→ (H) is a subgroup of Y.

(2) If H is a normal subgroup of X and φ is surjective, then φ^→ (H) is a subgroup of Y.

(3) If K is a subgroup of Y, then φ^← (K) is a subgroup of X.

(4) If K is a normal subgroup of Y, then φ^← (K) is a normal subgroup of X.

Proof. (1) Assume H is a subgroup of X, and ∀a, b ∈ φ (H), ∀ c ∈ Y such that *_Y ((a, b^-1), c) =1, then it is sufficient to show that c ∈ φ (H) by Theorem 4.2. Since a, b ∈ φ (H), there exists u, v ∈ H such that a = φ (u), b = φ (v). Regarding that *_X is a strong L-function, there exists w ∈ X such that *_X ((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 L-group homomorphism and using Proposition 5.2 (2), it holds $\begin{matrix} 1 & = & *_{X} ((u, v^{- 1}), w) = *_{Y} ((φ (u), φ (v^{- 1})), φ (w)) \\ = & *_{Y} ((φ (u), φ (v)^{- 1}), φ (w)) \\ = & *_{Y} ((a, b^{- 1}), φ (w)) . \end{matrix}$ Using Proposition 3.7 and since *_Y ((a, b^-1), c) =1, we get c = φ (w). That is, c ∈ φ (H), as desired.

(2) As shown by (1), φ^→ (H) is a subgroup of Y. For any y₀ ∈ φ^→ (H) and ∀y₁, y₂, y₃ ∈ Y such that *_Y ((y₁, y₂), y₀) ∧ * _Y ((y₂, y₁), y₃) =1, there exist x₀ ∈ H and x₁, x₂, x₃ ∈ X such that y_i = φ (x_i) (i = 0, 1, 2, 3) since φ is surjective. Then *_X ((x₁, x₂), x₀) ∧ * _X ((x₂, x₁), x₃) =1. Thus x₃ ∈ H since H is normal. It implies y₃ ∈ φ^→ (H) and so φ^→ (H) is normal by Theorem 4.14.

(3) Assume a, b ∈ φ^← (K), c ∈ X such that *_X ((a, b^-1), c) =1, then it is sufficient to prove c ∈ φ^← (K). In other words, it is sufficient to prove φ (c) ∈ K. By the hypothesis, we have φ (a), φ (b) ∈ K. Then φ (b) ^-1 ∈ K since K is a subgroup. Meanwhile, since φ is an ordered L-groups homomorphism, we have *_Y ((φ (a), φ (b) ^-1), φ (c)) = * _Y ((φ (a), φ (b^-1)), φ (c)) = * _X ((a, b^-1), c) =1. It hints that φ (c) ∈ K since K is closed under *, and so it completes the proof.

(4) Resulting from (3) and Theorem 4.14, it is sufficient to show for any x₀ ∈ φ^← (K) and ∀x₁, x₂, x₃ ∈ X such that *_X ((x₁, x₂), x₀) ∧ * _X ((x₂, x₁), x₃) =1 implies x₃ ∈ φ^← (K). Then there exist y₀ ∈ K and y₁, y₂, y₃ ∈ Y such that φ (x_i) (i = 0, 1, 2, 3) = y_i, so *_Y ((y₁, y₂), y₀) ∧ * _Y ((y₂, y₁), y₃) =1. Because K is a normal subgroup of Y, then we have y₃ ∈ K by Theorem 4.14, which indicates x₃ ∈ φ^← (K), as required.

□

Proposition 5.6. Let (X, ≤ , * _X) and (Y, ≤ , * _Y) be ordered L-groups, and φ be an ordered L-group homomorphism. Then

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

(2) $φ : (X, C_{X}^{N}) \to (Y, C_{Y}^{N})$ is convexity-preserving. In addition, if φ is surjective, then it is also convex-to-convex.

Proof. Straightforward from Proposition 5.5. □

Proposition 5.7. Let (X, ≤ , * _X) and (Y, ≤ , * _Y) be ordered L-groups, and φ : X → Y be an ordered L-group homomorphism. Then kerφ is a normal subgroup of X.

Proof. At first, we are about to prove that kerφ is a subgroup of X. Suppose ∀a, b ∈ kerφ, ∀ c ∈ X and *_X ((a, b^-1), c) =1, then it is sufficient to show c ∈ kerφ according to Theorem 4.3. In fact, since a, b ∈ kerφ, we have $φ (a) = φ (b) = 1_{Y}^{*}$ . It implies that $φ (a)^{- 1} = φ (b)^{- 1} = 1_{Y}^{*}$ , and so $*_{Y} ((φ (a), φ (b)^{- 1}), 1_{Y}^{*}) = 1$ . Meanwhile, since φ is an ordered L-groups homomorphism, we possess

1 = * _X ((a, b^-1), c) = * _Y ((φ (a), φ (b^-1)), φ (c)) = * _Y ((φ (a), φ (b) ^-1), φ (c)). Thus we obtain $φ (c) = 1_{Y}^{*}$ by Proposition 3.7, and it implies c ∈ kerφ.

Now, we are about to prove that a * kerφ = kerφ * a for each a ∈ X. For each c ∈ a * kerφ, we have

$\begin{matrix} c & \in & a * \ker φ \Rightarrow \exists b \in X, φ (b) = 1_{Y}^{*}, *_{X} ((a, b), c) = 1 \\ \Rightarrow & *_{Y} ((φ (a), 1_{Y}^{*}), φ (c)) = *_{Y} ((φ (a), φ (b)), φ (c)) \\ = & *_{X} ((a, b), c) = 1 . \\ \Rightarrow & φ (a) = φ (c) \\ \Rightarrow & *_{Y} ((φ (b), φ (a)), φ (c)) = *_{Y} ((1_{Y}^{*}, φ (a)), φ (a)) = 1 \\ \Rightarrow & *_{X} ((b, a), c) = *_{Y} ((φ (b), φ (a)), φ (c)) = 1 \\ \Rightarrow & c \in \ker φ * a . \end{matrix}$

Similarly, for each c ∈ kerφ * a, we can verify that c ∈ a * kerφ, and so a * kerφ = kerφ * a.

In summary, kerφ is a normal subgroup of X, and it completes the proof.□

6 Conclusions

In this paper, we have deeply investigated ordered L-groups following the way of which generally used in classical case. Firstly, the properties of elements in an ordered L-group and the relationship between them along with some equivalent descriptions for ordered L-groups have been studied. Next, the subgroups and normal subgroups of an ordered L-group are discussed in detail, some characterization theorems of (normal) subgroups have been given, the convexity of substructures has been demonstrated, and the result that the product of (normal) subgroups is also a (normal) subgroup have been obtained. Finally, the algebraic properties of ordered L-group homomorphisms that they preserve identity elements, inverse elements and substructures have been proven. All the results about fuzzy group are obtained by means of fuzzy binary operation, which can better reflect the essence of fuzzy groups logically. However, our discussion does not involve quotient groups, which greatly limits our practical application. We will try to study about quotient groups and the specific applications of the ordered L-groups in our future work.

Footnotes

Acknowledgment

The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments. This work is supported by the National Natural Science Foundation of China (11901358), the Doctoral Scientific Research Foundation of Shandong Jianzhu University (XNBS1344), and Shandong Province Social Science Planning Research Project(14DGLJ06).

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