A new approach of describing L -fuzzy convexity

Abstract

In this paper, the notions of L-fuzzy convex subgroup and L-fuzzy convex subgroup degree are introduced and their characterizations are given. A new method for describing L-fuzzy convexity is obtained. Also, L-fuzzy convexity preserving mappings and L-fuzzy convex-to-convex mappings are analyzed. Besides, we provide a counter example to illustrate one of the conclusions.

1 Introduction

Convexity is originally inspired by some elementary geometric figures such as shapes of circles or polyhedrons in Euclidean spaces. Convexity has an old history and multiple applications. In 1993, M. van de Vel collected the theory of convexity systematically in the famous book [35]. With the development of fuzzy mathematics, many mathematical structures have been generalized to the fuzzy case [7 , 17]. Also, convexity has also been combined with fuzzy set theory. In 1994, Rosa [23, 24] firstly generalized convex spaces to fuzzy situation. In 2009, Y. Maruyama generalized the lattice [0,1] to a completely distributive lattice in [9]. In 2014, F.-G. Shi and Z.-Y. Xiu gave a new approach to fuzzification of convexity and proposed the concept of M-fuzzifying convexity [30]. Besides, there are some studies about M-fuzzifying convex spaces were showed in [2, 31, 37–41]. Subsequently, many properties of convexity theory are generalized to L-convexity [3 , 29]. Afterwards, abstract convexity was extended to (L, M)-fuzzy convexity in [32].

In 1971, A. Rosenfeld introduced the concept of fuzzy groups [25]. In 1979, Anthony and Sherwood redefined fuzzy groups [1]. Since then many results in group theory have been generalized to fuzzy setting [21, 28]. Some authors discussed the relations between the fuzzy subgroups and their level subgroups [4, 33]. In 2011, F.-G. Shi and X. Xin generalized the notion of degree to which a fuzzy subset is a fuzzy subgroup to L-fuzzy setting [34].

In this paper, we focus on the degree to which an L-fuzzy subgroup is an L-fuzzy convex subgroup. Then an L-fuzzy convexity on a subgroup is naturally constructed. Also, their L-fuzzy convexity preserving mappings and L-fuzzy convex-to-convex mappings are discussed.

This paper is organized as follows. In Section 2, we give some preliminaries on lattices, fuzzy sets and L-fuzzy convexities. In Section 3, the definition of L-fuzzy convex subgroup is introduced and their characterizations are given. In Section 4, the degree to which an L-fuzzy subgroup is an L-fuzzy convex subgroup is defined. Also, their characterizations and properties are presented. In Section 5, an L-fuzzy convexity is induced by L-fuzzy convex degrees on subgroup. Furthermore, we analyze the corresponding L-fuzzy convexity-preserving mappings and L-fuzzy convex-to-convex mappings.

2 Preliminaries

Throughout this paper, unless otherwise stated, L denotes a completely distributive lattice. The smallest element and the largest element in L are denoted by ⊥ and ⊤, respectively.

For a non-empty set X, each mapping A : X ⟶ L is called an L-subset on X, and we denote the collection of all L-subsets on X by L^X . L^X is also a complete lattice by defining “≤” pointwisely. Furthermore, the smallest element and the largest element in L^X are denoted by χ_∅ and χ_X, respectively.

An element a in L is called co-prime if a ⩽ b ∨ c implies a ⩽ c or a ⩽ b [5]. The set of non-zero co-prime elements in L is denoted by J (L). An element a in L is called prime if b ∧ c ⩽ a implies c ⩽ a or b ⩽ a. The set of non-unit prime elements in L is denoted by P (L). From [5] we know that each element of L is the supremum of co-prime elements and the infimum of prime elements.

For each a, b ∈ L, we say that a is wedge below b in L, in symbols a ≺ b, if for every subset D ⊆ L, b ⩽ ⋁ D implies a ⩽ d for some d ∈ D. A complete lattice L is a completely distributive if and only if b = ⋁ {a ∈ L ∣ a ≺ b} for each b ∈ L. The set {a ∈ L ∣ a ≺ b}, denoted by β (b), is called the greatest minimal family of b in the sense of [36].

For all a, b ∈ L, b ≺ ^opa if and only if for every subset D ⊆ L, ⋀D ⩽ b implies d ⩽ a for some d ∈ D. The set {b ∈ M ∣ a ≺ ^opb}, denoted by α (a), is called the greatest maximal family of a in the sense of [36].

Theorem 2.1. ([36]). Let L be a completely distributive lattice. Then for all {a_i ∣ i ∈ Ω} ⊆ L, we have

α (⋀ _i∈Ωa_i) = ⋃ _i∈Ωα (a_i), i.e., α is a ⋀ -⋃ mapping.

β (⋁ _i∈Ωa_i) = ⋃ _i∈Ωβ (a_i), i.e., β is a union-preserving mapping.

Definition 2.2. ([33]). Let A ∈ L^X and let a ∈ L. We define

A_[a] = {x ∈ X ∣ a ⩽ A (x)} ,

A^(a) = {x ∈ X ∣ A (x) ≰ a} ,

A_(a) = {x ∈ X ∣ a ∈ β (A (x))},

A^[a] = {x ∈ X ∣ a ∉ α (A (x))}.

In a completely distributive lattice L, there exists an implication operator →: L × L ⟶ L as the right adjoint for the meet operation ∧ by $a \to b = ⋁ {c \in L ∣ a \land c ⩽ b} .$

We list some properties of the implication operation in the following lemma.

Lemma 2.3. ([7]). Let (L, ∨ , ∧) be a completely distribution lattice and let → be the implication operator corresponding to ∧. Then for all a, b, c ∈ L, the following statements hold:

⊤→ a = a ;

(a→ c) ∧ (c → b) ⩽ a → b ;

(a → b) ∧ (c → d) ⩽ a ∧ c → b ∧ d .

In what follows, the concept of L-fuzzy convexity is introduced.

Definition 2.4. ([32]). A mapping 𝒞 : L^X ⟶ L is called an (L, L)-fuzzy convexity (L-fuzzy convexity for short) on X if it satisfies the following conditions:

𝒞 (χ_∅) = 𝒞 (χ_X) = ⊤ _L;

if {A_i ∣ i ∈ Ω} ⊆ L^X is nonempty, then ⋀_i∈Ω𝒞 (A_i) ⩽ 𝒞 (⋀ _i∈ΩA_i);

if {A_i ∣ i ∈ Ω} ⊆ L^X is nonempty and totally ordered by inclusion, then ⋀_i∈Ω𝒞 (A_i) ⩽ 𝒞 (⋁ _i∈ΩA_i).

The pair (X, 𝒞) is called an (L, L)-fuzzy convex space (L-fuzzy convex space for short). An (L, 2)-fuzzy convexity is an L-convexity and the pair (X, 𝒞) is an L-convex space in [9 , 19]. Given a mapping f : X ⟶ Y, define $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ and $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ by $f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x), f_{L}^{\leftarrow} (B) = B \circ f$ for all A ∈ L^X, x ∈ X, B ∈ L^Y and y ∈ Y [22].

Definition 2.5 ([32]). Let (X, 𝒞) and (Y, scripfontD) be L-fuzzy convex spaces. A mapping f : X ⟶ Y is called

L-fuzzy convexity preserving mapping provided $D (B) ⩽ C (f_{L}^{\leftarrow} (B))$ for all B ∈ L^Y.

L-fuzzy convex-to-convex mapping provided $C (A) ⩽ D (f_{L}^{\to} (A))$ for all A ∈ L^X.

Definition 2.6. ([9 , 24]). Let (X, 𝒞_X) and (Y, 𝒞_Y) be L-convex spaces. A mapping f : (X, 𝒞_X) ⟶ (Y, 𝒞_Y) is called

convexity-preserving (CP for short) provided B ∈ 𝒞_Y implies $f_{L}^{\leftarrow} (B) \in C_{X}$ ;

convex-to-convex (CC for short) provided A ∈ 𝒞_X implies $f_{L}^{\to} (A) \in C_{Y}$ .

Definition 2.7. ([25]). Let G be a group and μ ∈ L^G . μ is called an L-fuzzy subgroup if for all x, y ∈ G, μ (xy^-1) ≥ μ (x) ∧ μ (y) .

Theorem 2.8. ([33]). Let G be a group and μ ∈ L^G. Then the following conditions are equivalent.

μ is an L-fuzzy subgroup;

∀a ∈ L, μ_[a] is a subgroup on G;

∀a ∈ J (L) , μ_[a] is a subgroup on G;

∀a ∈ L, μ^[a] is a subgroup on G;

∀a ∈ P (L) , μ^[a] is a subgroup on G;

∀a ∈ P (L) , μ^(a) is a subgroup on G.

Definition 2.9. ([34]). Let G be a group and μ ∈ L^G. The subgroup degree m (μ) of μ is defined as m (μ) = ⋀ _x,y∈G ((μ (x) ∧ μ (y)) → μ (xy^-1))

Lemma 2.10. ([34]). Let G be a group and μ ∈ L^G. Then m (μ) ≥ a if and only if for any x, y ∈ G, μ (x) ∧ μ (y) ∧ a ≤ μ (xy^-1).

Theorem 2.11. ([34]). Let G be a group and μ ∈ L^G. ∀x, y ∈ G, Then

m (μ) = ⋁ {a ∈ L : (μ (x) ∧ μ (y) ∧ a ≤ μ (xy^-1)};

$m (μ) = ⋁ {a \in L | \begin{matrix} \forall b ⩽ a, \\ μ_{[b]} is a subgroup on G \end{matrix}};$

$m (μ) = ⋁ {a \in L | \begin{matrix} \forall b \notin α (a), \\ μ^{[b]} is a subgroup on G \end{matrix}};$

$m (μ) = ⋁ {a \in L | \begin{matrix} \forall b \in P (L), a b, \\ μ^{(b)} is a subgroup on G \end{matrix}};$

Theorem 2.12. ([34]). Let G be a group and μ ∈ L^G. If β (a ∧ b) = β (a) ∩ β (b) for any a, b ∈ L, then $m (μ) = ⋁ {a \in L | \begin{matrix} \forall b \in β (a), \forall x, y \in G \\ μ_{(b)} is a subgroup on G \end{matrix}} .$

Theorem 2.13. ([34]). Let f : G ⟶ H be a group homomorphism and μ ∈ L^G, λ ∈ L^H.

$m (f_{L}^{\to} (μ)) \geq m (μ)$ , and if f is injective, then $m (f_{L}^{\to} (μ)) = m (μ)$ ;

$m (f_{L}^{\leftarrow} (λ)) \geq m (λ)$ , and if f is surjective, then $m (f_{L}^{\leftarrow} (λ)) = m (λ)$ .

Theorem 2.14. ([34]). Let G and H be groups, letμ ∈ L^G and λ ∈ L^H be L-fuzzy subgroups on G and on H, respectively. Then m (μ) ∧ m (λ) ≤ m (μ × λ).

3 L-fuzzy convex subgroup

In this section, we introduce the notion of L-fuzzy convex subgroups and give their characterizations in terms of four kinds of cut sets of L-fuzzy convex subgroups.

Definition 3.1. Let G be an ordered group and μ be an L-fuzzy subgroup. μ is called an L-fuzzy convex subgroup if for all x, y, z ∈ G, x ≤ z ≤ y implies μ (x) ∧ μ (y) ≤ μ (z).

Theorem 3.2. Let G be an ordered group and μ ∈ L^G. Then the following conditions are equivalent.

μ is an L-fuzzy convex subgroup;

∀a ∈ L, μ_[a] is a convex subgroup on G;

∀a ∈ J (L) , μ_[a] is a convex subgroup on G;

∀a ∈ L, μ^[a] is a convex subgroup on G;

∀a ∈ P (L) , μ^[a] is a convex subgroup on G;

∀a ∈ P (L) , μ^(a) is a convex subgroup on G.

Proof. The proofs are trivial and omitted here. □

Theorem 3.3. Let G be an ordered group and μ ∈ L^G. If β (a ∧ b) = β (a) ∩ β (b) for all a, b ∈ L, then the following conditions are equivalent.

μ is an L-fuzzy convex subgroup;

∀a ∈ L, μ_(a) is a convex subgroup on G;

∀a ∈ J (L) , μ_(a) is a convex subgroup on G.

Proof. (1)⇒(2). Assume that μ is an L-fuzzy convex subgroup. Take x, y ∈ μ_(a) with x ≤ z ≤ y, then a ∈ β (μ (x)) and a ∈ β (μ (y)). By β (μ (x) ∧ μ (y)) = β (μ (x)) ∩ β (μ (y)), we have a ∈ β (μ (x) ∧ μ (y)). Since μ is L-fuzzy convex subgroup, we have μ (xy^-1) ≥ μ (x) ∧ μ (y) and μ (z) ≥ μ (x) ∧ μ (y). Thus, a ∈ β (μ (xy^-1)) and a ∈ β (μ (z)), i.e., xy^-1 ∈ μ_(a), z ∈ μ_(a). Therefore μ_(a) is a convex subgroup on G.

(2)⇒(3) is obvious.

(3)⇒(1). Assume that μ_(a) is a convex subgroup on G for each a ∈ J (L). For any x, y, z ∈ G with x ≤ z ≤ y, let a ∈ J (L) with a ∈ β (μ (x) ∧ μ (y)). By β (μ (x) ∧ μ (y)) = β (μ (x)) ∩ β (μ (y)), we have a ∈ β (μ (x)) and a ∈ β (μ (y)), i.e., x, y ∈ μ_(a). Since μ_(a) is a convex subgroup, we have xy^-1 ∈ μ_(a) and z ∈ μ_(a), i.e., a ∈ β (μ (xy^-1)) and a ∈ β (μ (z)). This implies β (μ (x) ∧ μ (y)) ⊆ β (μ (xy^-1)) and β (μ (x) ∧ μ (y)) ⊆ β (μ (z)). So μ (x) ∧ μ (y) = ⋁ β (μ (x) ∧ μ (y)) ≤ ⋁ β (μ (xy^-1)) = μ (xy^-1) and μ (x) ∧ μ (y) = ⋁ β (μ (x) ∧ μ (y)) ≤ ⋁ β (μ (z)) ≤ μ (z). Therefore μ is an L-fuzzy convex subgroup. □

Definition 3.4. Let G and H be two ordered groups, let f : G → H. Then f is order-preserving provided x₁ ≤ x₂ implies f (x₁) ≤ f (x₂) for all x₁, x₂ ∈ G.

Theorem 3.5. Let G and H be two ordered groups. Let f : G → H be a group homomorphism and order preserving. Then the following statements hold for all μ ∈ L^G, λ ∈ L^H

if λ is an L-fuzzy convex subgroup, then $f_{L}^{\leftarrow} (λ)$ is an L-fuzzy convex subgroup;

if μ is an L-fuzzy convex subgroup and f is surjective, then $f_{L}^{\to} (μ)$ is an L-fuzzy convex subgroup.

Proof. (1) ∀x, y ∈ G, $\begin{matrix} f_{L}^{\leftarrow} (λ) (x) \land f_{L}^{\leftarrow} (λ) (y) \\ = & λ (f (x)) \land λ (f (y)) ⩽ λ (f (x) f (y)^{- 1}) \\ = & λ (f ({xy}^{- 1})) = f_{L}^{\leftarrow} (λ) ({xy}^{- 1}) . \end{matrix}$

This implies that $f_{L}^{\leftarrow} (λ)$ is an L-fuzzy subgroup. To prove $f_{L}^{\leftarrow} (λ)$ is convex, take x₁, x₂, x₃ ∈ G with x₁ ≤ x₂ ≤ x₃, then f (x₁) ≤ f (x₂) ≤ f (x₃) and $\begin{matrix} f_{L}^{\leftarrow} (λ) (x_{1}) \land f_{L}^{\leftarrow} (λ) (x_{3}) \\ = & λ (f (x_{1})) \land λ (f (x_{3})) \leq λ (f (x_{2})) = f_{L}^{\leftarrow} (λ) (x_{2}) . \end{matrix}$ (2) $f_{L}^{\to} (μ)$ is an L-fuzzy subgroup can be proved by the following fact. Take y₁, y₂ ∈ H, we have $\begin{matrix} f_{L}^{\to} (μ) (y_{1}) \land f_{L}^{\to} (μ) (y_{2}) \\ = & ⋁_{f (x_{1}) = y_{1}} μ (x_{1}) \land ⋁_{f (x_{2}) = y_{2}} μ (x_{2}) \\ = & ⋁ {μ (x_{1}) \land μ (x_{2}) | f (x_{1}) = y_{1}, f (x_{2}) = y_{2}} \\ ⩽ & ⋁ {μ (x_{1} {x_{2}}^{- 1}) | f (x_{1}) = y_{1}, f (x_{2}) = y_{2}} \\ ⩽ & ⋁ {μ (x_{1} {x_{2}}^{- 1}) | f (x_{1} {x_{2}}^{- 1}) = y_{1} y_{2}^{- 1}} \\ ⩽ & ⋁ {μ (z) | f (z) = y_{1} y_{2}^{- 1}} \\ = & f_{L}^{\to} (μ) (y_{1} y_{2}^{- 1}) . \end{matrix}$

To prove $f_{L}^{\to} (μ)$ is convex, take y₁, y, y₂ ∈ G such that y₁ ≤ y ≤ y₂. If y₁ = y or y₂ = y, then the inequality $f_{L}^{\to} (μ) (y_{1}) \land f_{L}^{\to} (μ) (y_{2}) ⩽ f_{L}^{\to} (μ) (y)$ is obvious. Otherwise, $\begin{matrix} f_{L}^{\to} (μ) (y_{1}) \land f_{L}^{\to} (μ) (y_{2}) \\ = & ⋁_{f (x_{1}) = y_{1}} μ (x_{1}) \land ⋁_{f (x_{2}) = y_{2}} μ (x_{2}) \\ = & ⋁ {μ (x_{1}) \land μ (x_{2}) | f (x_{1}) = y_{1}, f (x_{2}) = y_{2}} \end{matrix}$

Note that G and H are ordered groups and f is surjective and order-preserving. For all y ∈ H with f (x₁) < y < f (x₂), there exists x^∗ ∈ G such that f (x^∗) = y and x₁ < x^∗ < x₂, thus

$\begin{matrix} f_{L}^{\to} (μ) (y_{1}) \land f_{L}^{\to} (μ) (y_{2}) \\ ⩽ & ⋁ {μ (x^{*}) | f (x_{1}) = y_{1} \leq y = f (x^{*}) \leq f (x_{2}) = y_{2}} \\ = & f_{L}^{\to} (μ) (y) \end{matrix}$

Therefore, $f_{L}^{\to} (μ)$ is an L-fuzzy convex subgroup. □

Definition 3.6. Let G and H be ordered groups, let μ ∈ L^G and λ ∈ L^H. Define μ × λ as follows: $\begin{matrix} (μ \times λ) (x, y) = μ (x) \land λ (y) \end{matrix}$ for all pairs (x, y) ∈ G × H.

Theorem 3.7. Let G and H be ordered groups, let μ ∈ L^G and λ ∈ L^H be L-fuzzy convex subgroups. Then μ × λ is an L-fuzzy convex subgroup on G × H.

Proof. Since μ ∈ L^G and λ ∈ L^H are L-fuzzy subgroups, it follows that μ (x_G) ∧ μ (y_G) ⩽ μ (x_Gy_G^-1) and λ (x_H) ∧ λ (y_H) ⩽ λ (x_Hy_H^-1) for any x_G, y_G ∈ G, x_H, y_H ∈ H. First we prove that μ × λ is an L-fuzzy subgroup. $\begin{matrix} (μ \times λ) (x_{G}, x_{H}) \land (μ \times λ) (y_{G}, y_{H}) \\ = & μ (x_{G}) \land λ (x_{H}) \land μ (y_{G}) \land λ (y_{H}) \\ \leq & μ (x_{G} {y_{G}}^{- 1}) \land λ (x_{H} {y_{H}}^{- 1}) \\ = & (μ \times λ) (x_{G} {y_{G}}^{- 1}, x_{H} {y_{H}}^{- 1}) \end{matrix}$ Next, we prove that μ × λ is convex. For all x_G, y_G, z_G ∈ G, x_H, y_H, z_H ∈ H with x_G ≤ z_G ≤ y_G, x_H ≤ z_H ≤ y_H, we have $\begin{matrix} (μ \times λ) (x_{G}, x_{H}) \land (μ \times λ) (y_{G}, y_{H}) \\ = & μ (x_{G}) \land λ (x_{H}) \land μ (y_{G}) \land λ (y_{H}) \\ \leq & μ (z_{G}) \land λ (z_{H}) \\ = & (μ \times λ) (z_{G}, z_{H}) \end{matrix}$ Therefore μ × λ is an L-fuzzy convex subgroup on G × H. □

In classical case, if μ × λ is convex on G × H if and only if μ and λ are convex on G and on H, respectively. However, the converse of Theorem 3.7 is not true (see Example 3.8).

Example 3.8. Let G and H be integer additive groups and L = 2^{a,b}. Define μ ∈ L^G and λ ∈ L^H by $\begin{matrix} μ (x) = {\begin{matrix} {a}, & x = - 1, 1; \\ ⊥, & x \neq - 1, 1 . \end{matrix} \end{matrix}$ $\begin{matrix} λ (y) = {\begin{matrix} {b}, & y = - 1, 1; \\ ⊥, & y \neq - 1, 1 . \end{matrix} \end{matrix}$

Take any (x_G, x_H), (y_G, y_H) ∈ G × H. Obviously $\begin{matrix} μ (x_{G}) \land μ (y_{G}) \land λ (x_{H}) \land λ (y_{H}) = ⊥ . \end{matrix}$ Thus $\begin{matrix} (μ \times λ) (x_{G}, x_{H}) \land (μ \times λ) (y_{G}, y_{H}) \\ = & μ (x_{G}) \land λ (x_{H}) \land μ (y_{G}) \land λ (y_{H}) \\ \leq & (μ \times λ) (x_{G} {y_{G}}^{- 1}, x_{H} {y_{H}}^{- 1}) . \end{matrix}$

This implies that μ × λ is an L-fuzzy subgroup. Take any (x_G, x_H), (y_G, y_H) , (z_G, z_H) ∈ G × H with x_G ≤ z_G ≤ y_G, x_H ≤ z_H ≤ y_H. Then $\begin{matrix} (μ \times λ) (x_{G}, x_{H}) \land (μ \times λ) (y_{G}, y_{H}) \\ = & μ (x_{G}) \land λ (x_{H}) \land μ (y_{G}) \land λ (y_{H}) \\ \leq & (μ \times λ) (z_{G}, z_{H}) . \end{matrix}$

This shows that μ × λ is convex on G × H.

Therefore, μ × λ is an L-fuzzy convex subgroup. But it is easy to check that $\begin{matrix} μ (- 1) \land μ (1) = a μ (1 {(- 1)}^{- 1}) = μ (2), \\ λ (- 1) \land λ (1) = b λ (1 (- 1)^{- 1}) = λ (2) . \end{matrix}$ and $\begin{matrix} μ (- 1) \land μ (1) = a μ (0), \\ λ (- 1) \land λ (1) = b λ (0) . \end{matrix}$

This shows that μ and λ are not L-fuzzy convex subgroups on G and on H, respectively.

In the following, we will generalize Theorem 3.7 to the finite case.

Theorem 3.9. Let {G_i} _1⩽i⩽n be a family of ordered groups. If μ_i ∈ L^{G
_i} is an L-fuzzy convex subgroup for all 1 ⩽ i ⩽ n, then μ₁ × μ₂ × ⋯ × μ_n is an L-fuzzy convex subgroup on G₁ × G₂ × ⋯ × G_n.

Theorem 3.10. Let G and H be ordered groups. Then for any μ ∈ L^G and λ ∈ L^H, the following conditions are equivalent:

μ × λ is an L-fuzzy convex subgroup on G× H ;

for each a ∈ J (L), (μ × λ) _[a] = μ_[a] × λ_[a] is a convex subgroup on G× H ;

for each a ∈ α (⊥), (μ × λ) ^[a] = μ^[a] × λ^[a] is a convex subgroup on G× H ;

for each a ∈ P (L), (μ × λ) ^(a) is a convex subgroup on G× H ;

for each a ∈ P (L), μ^(a) × λ^(a) is a convex subgroup on G× H ;

if β (a ∧ b) = β (a) ∩ β (b) for all a, b ∈ L, then for each a ∈ β (⊤), (μ × λ) _(a) is a convex subgroup on G× H ;

if β (a ∧ b) = β (a) ∩ β (b) for all a, b ∈ L, then for each a ∈ β (⊤), μ_(a) × λ_(a) is a convex subgroup on G× H ;

Proof. (1) ⇔ (2), (1) ⇔ (3), (1) ⇔ (4) and (1) ⇔ (6) are easily obtained from Theorem 3.2 and Theorem 3.3.

(1) ⇒ (5). The following implications show that μ^(a) × λ^(a) is a subgroup on G × H. Take any (x_G, x_H), (y_G, y_H) ∈ μ^(a) × λ^(a), i.e., x_G, y_G ∈ μ^(a) and x_H, y_H ∈ λ^(a). Then μ (x_G) ≰ a, μ (y_G) ≰ a, λ (x_H) ≰ a, λ (y_H) ≰ a. We have μ (x_G) ∧ λ (y_G) ∧ μ (x_H) ∧ λ (y_H) ≰ a, since a ∈ P (L). Note that μ × λ is an L-fuzzy subgroup, it follows that μ (x_Gy_G^-1) ∧ λ (x_Hy_H^-1) ≰ a. Then μ (x_Gy_G^-1) ≰ a and λ (x_Hy_H^-1) ≰ a, i.e., x_Gy_G^-1 ∈ μ^(a) and x_Hy_H^-1 ∈ λ^(a). Thus implies (x_Gy_G^-1, x_Hy_H^-1) ∈ μ^(a) × λ^(a). The following implications show that μ^(a) × λ^(a) is convex on G × H. Take any (x₁, x₂), (y₁, y₂) ∈ μ^(a) × λ^(a) with (x₁, x₂) ≤ (z₁, z₂) ≤ (y₁, y₂), i.e., x₁, y₁ ∈ μ^(a), x₂, y₂ ∈ λ^(a) and with x₁ ≤ z₁ ≤ y₁, x₂ ≤ z₂ ≤ y₂. Then μ (x₁) ≰ a, μ (y₁) ≰ a, λ (x₂) ≰ a, λ (y₂) ≰ a. We have μ (x₁) ∧ λ (y₁) ∧ μ (x₂) ∧ λ (y₂) ≰ a, since a ∈ P (L). Note that μ × λ is an L-fuzzy convex subgroup, it follows that (μ × λ) (z₁, z₂) ≰ a, i.e., (z₁, z₂) ∈ μ^(a) × λ^(a).

(5) ⇒ (1). In order to prove μ × λ is an L-fuzzy convex subgroup on G × H, we need to check the inequality μ (x_G) ∧ λ (x_H) ∧ μ (y_G) ∧ λ (y_H) ⩽ μ (x_Gy_G^-1) ∧ λ (x_Hy_H^-1) for all (x_G, x_H), (y_G, y_H) ∈ G × H and μ (x₁) ∧ λ (x₂) ∧ μ (y₁) ∧ λ (y₂) ⩽ μ (z₁) ∧ λ (z₂) for all (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ G × H with (x₁, x₂) ≤ (z₁, z₂) ≤ (y₁, y₂). First we prove that μ × λ is an L-fuzzy subgroup. Suppose that a ∈ P (L) with μ (x_G) ∧ λ (x_H) ∧ μ (y_G) ∧ λ (y_H) ≰ a. Then μ (x_G) ≰ a, μ (y_G) ≰ a, λ (x_H) ≰ a, λ (y_H) ≰ a, i.e., x_G, y_G ∈ μ^(a) and x_H, y_H ∈ λ^(a). This implies (x_G, x_H), (y_G, y_H) ∈ μ^(a) × λ^(a). Since μ^(a) × λ^(a) is a subgroup, it follows that (x_Gy_G^-1, x_Hy_H^-1) ∈ μ^(a) × λ^(a). Hence x_Gy_G^-1 ∈ μ^(a) and x_Hy_H^-1 ∈ λ^(a), i.e., μ (x_Gy_G^-1) ≰ a and λ (x_Hy_H^-1) ≰ a. Since a ∈ P (L), it follows that μ (x_Gy_G^-1) ∧ λ (x_Hy_H^-1) ≰ a. By the arbitrariness of a, we obtain μ (x_G) ∧ λ (x_H) ∧ μ (y_G) ∧ λ (y_H) ⩽ μ (x_Gy_G^-1) ∧ λ (x_Hy_H^-1). Next, we prove that μ × λ is convex. Suppose that a ∈ P (L) with μ (x₁) ∧ λ (x₂) ∧ μ (y₁) ∧ λ (y₂) ≰ a. Then μ (x₁) ≰ a, μ (y₁) ≰ a, λ (x₂) ≰ a, λ (y₂) ≰ a, i.e., x₁, y₁ ∈ μ^(a) and x₂, y₂ ∈ λ^(a). This implies (x₁, x₂), (y₁, y₂) ∈ μ^(a) × λ^(a). Since μ^(a) × λ^(a) is a convex subgroup, it follows that (z₁, z₂) ∈ μ^(a) × λ^(a) with (x₁, x₂) ≤ (z₁, z₂) ≤ (y₁, y₂). Hence z₁ ∈ μ^(a) and z₂ ∈ λ^(a), i.e., μ (z₁) ≰ a and λ (z₂) ≰ a. Since a ∈ P (L), it follows that μ (z₁) ∧ λ (z₂) ≰ a. By the arbitrariness of a, we obtain μ (x₁) ∧ λ (x₂) ∧ μ (y₁) ∧ λ (y₂) ⩽ μ (z₁) ∧ λ (z₂).

(1) ⇒ (7). The following implications show that μ_(a) × λ_(a) is a subgroup. Take any (x_G, x_H), (y_G, y_H) ∈ μ_(a) × λ_(a), i.e., x_G, y_G ∈ μ_(a) and x_H, y_H ∈ λ_(a). Then a ∈ β (μ (x_G)), a ∈ β (μ (y_G)), a ∈ β (λ (x_H)), a ∈ β (λ (y_H)). We have a ∈ β (μ (x_G)) ∩ β (μ (y_G)) ∩ β (λ (x_H)) ∩ β (λ (y_H)). By β (μ (x_G) ∧ μ (y_G) ∧ λ (x_H) ∧ λ (y_H)) = β (μ (x_G)) ∩ β (μ (y_G)) ∩ β (λ (x_H)) ∩ β (λ (y_H)), we obtain a ∈ β (μ (x_G) ∧ μ (y_G) ∧ λ (x_H) ∧ λ (y_H)). Since μ × λ is an L-fuzzy subgroup, it follows that a ∈ β (μ (x_Gy_G^-1) ∧ λ (x_Hy_H^-1)). Note that β is an order-preserving mapping, we obtain a ∈ β (μ (x_Gy_G^-1)) and a ∈ β (λ (x_Hy_H^-1)), i.e., x_Gy_G^-1 ∈ μ_(a) and x_Hy_H^-1 ∈ λ_(a). This shows that (x_Gy_G^-1, x_Hy_H^-1) ∈ μ_(a) × λ_(a). The following implications show that μ_(a) × λ_(a) is convex. Take any (x₁, x₂), (y₁, y₂) ∈ μ_(a) × λ_(a) with (x₁, x₂) ≤ (z₁, z₂) ≤ (y₁, y₂), i.e., x₁, y₁ ∈ μ_(a), x₂, y₂ ∈ λ_(a), and with x₁ ≤ z₁ ≤ y₁, x₂ ≤ z₂ ≤ y₂. Then a ∈ β (μ (x₁)), a ∈ β (μ (y₁)), a ∈ β (λ (x₂)), a ∈ β (λ (y₂)). We have a ∈ β (μ (x₁)) ∩ β (μ (y₁)) ∩ β (λ (x₂)) ∩ β (λ (y₂)). By β (μ (x₁) ∧ μ (y₁) ∧ λ (x₂) ∧ λ (y₂)) = β (μ (x₁)) ∩ β (μ (y₁)) ∩ β (λ (x₂)) ∩ β (λ (y₂)), we obtain a ∈ β (μ (x₁) ∧ μ (y₁) ∧ λ (x₂) ∧ λ (y₂)). Since μ × λ is an L-fuzzy convex subgroup, it follows that a ∈ β (μ (z₁) ∧ λ (z₂)). Note that β is an order-preserving mapping, we obtain a ∈ β (μ (z₁)) and a ∈ β (λ (z₂)), i.e., z₁ ∈ μ_(a) and z₂ ∈ λ_(a). This shows that (z₁, z₂) ∈ μ_(a) × λ_(a).

(7) ⇒ (1). First we prove that μ × λ is an L-fuzzy subgroup. Suppose that a ∈ β (⊤) with a ∈ β (μ (x_G) ∧ μ (y_G) ∧ λ (x_H) ∧ λ (y_H)). By β (μ (x_G) ∧ μ (y_G) ∧ λ (x_H) ∧ λ (y_H)) = β (μ (x_G)) ∩ β (μ (y_G)) ∩ β (λ (x_H)) ∩ β (λ (y_H)), we have a ∈ β (μ (x_G)) ∩ β (μ (y_G)) ∩ β (λ (x_H)) ∩ β (λ (y_H)). Then a ∈ β (μ (x_G)), a ∈ β (μ (y_G)), a ∈ β (λ (x_H)) and a ∈ β (λ (y_H)), i.e., x_G, y_G ∈ μ_(a) and x_H, y_H ∈ λ_(a). This implies (x_G, x_H), (y_G, y_H) ∈ μ_(a) × λ_(a). Since μ_(a) × λ_(a) is a subgroup, it follows that (x_Gy_G^-1, x_Hy_H^-1) ∈ μ_(a) × λ_(a). Hence a ∈ β (μ (x_Gy_G^-1)) and a ∈ β (λ (x_Hy_H^-1)). By β (a ∧ b) = β (a) ∩ β (b), we have a ∈ β (μ (x_Gy_G^-1) ∧ λ (x_Hy_H^-1)). By the arbitrariness of a, we obtain μ (x_G) ∧ λ (x_H) ∧ μ (y_G) ∧ λ (y_H) ⩽ μ (x_Gy_G^-1) ∧ λ (x_Hy_H^-1). Next, we prove that μ × λ is convex. Suppose that a ∈ β (μ (x₁) ∧ μ (y₁) ∧ λ (x₂) ∧ λ (y₂)). It is analogous to the aforesaid that we can get that a ∈ β (μ (z₁) ∧ λ (z₂)) for all (x₁, x₂), (y₁, y₂), (z₁, z₂) ∈ G × H with (x₁, x₂) ≤ (z₁, z₂) ≤ (y₁, y₂). By the arbitrariness of a, we obtain μ (x₁) ∧ λ (x₂) ∧ μ (y₁) ∧ λ (y₂) ⩽ μ (z₁) ∧ λ (z₂). □

4 L-fuzzy convex subgroup degree

In this section, the degree to which an L-fuzzy subgroup is an L-fuzzy convex subgroup is introduced and its characterizations are presented.

Definition 4.1. Let G be an ordered group. For each μ ∈ L^G, the degree to which μ is an L-fuzzy convex subgroup on G (or called L-convex subgroup degree of μ) is defined as follows: CD (μ) = m (μ) ∧ ⋀ {μ (x) ∧ μ (y) → μ (z) : x ≤ z ≤ y, x, y, z ∈ G}

Remark 4.2. It is obvious that CD (μ) =⊤ if and only if μ is an L-fuzzy convex subgroup.

By Lemma 2.10, it is easy to obtain the following lemmas.

Lemma 4.3. Let G be an ordered group and μ ∈ L^G. Then a ⩽ CD (μ) if and only if m (μ) ≥ a and μ (x) ∧ μ (y) ∧ a ≤ μ (z) for all x, y, z ∈ G with x ≤ z ≤ y.

Lemma 4.4 Let G be an ordered group and μ ∈ L^G. Then CD (μ) = ⋁ {a ∈ L : μ (s) ∧ μ (t) ∧ a ≤ μ (st^-1) , μ (x) ∧ μ (y) ∧ a ≤ μ (z) , s, t, x, y, z ∈ G, x ≤ z ≤ y}.

Lemma 4.5 Let G be an ordered group and μ ∈ L^G. Then CD (μ) = m (μ) ∧ ⋁ {a ∈ L : μ (x) ∧ μ (y) ∧ a ≤ μ (z) , x, y, z ∈ G, x ≤ z ≤ y}

In what follows, some characterizations of L-convex subgroup degree will be presented.

Theorem 4.6. Let G be an ordered group and μ ∈ L^G. Then CD (μ) = ⋁ {a ∈ L ∣ ∀ b ⩽ a, μ_[b] is a convex subgroup}.

Proof. Assume that a ∈ L, let m (μ) ≥ a and let μ (x) ∧ μ (y) ∧ a ≤ μ (z) for x, y, z ∈ G with x ≤ z ≤ y. By Theorem 2.11, we know μ_[b] is a subgroup on G for all b ⩽ a. Now we prove that μ_[b] is convex. Let x, y ∈ μ_[b] and x ≤ z ≤ y, then μ (x) ≥ b, μ (y) ≥ b. By a ≥ b, we have μ (z) ≥ μ (x) ∧ μ (y) ∧ a ≥ b, i.e., z ∈ μ_[b]. Hence μ_[b] is a convex subgroup. By Lemma 4.5, we know CD (μ) ≤ ⋁ {a ∈ L ∣ ∀ b ⩽ a, μ_[b]is a convex subgroup}.

Conversely, suppose that μ_[b] is convex subgroup for any b ⩽ a. By Theorem 2.11, we know m (μ) ≥ a. We need to prove μ (x) ∧ μ (y) ∧ a ≤ μ (z) for all x, y, z ∈ G with x ≤ z ≤ y. Suppose that b ∈ L with b ⩽ μ (x) ∧ μ (y) ∧ a, then x, y ∈ μ_[b] and a ≥ b. Since μ_[b]is a convex subgroup, it follows that z ∈ μ_[b], i.e., μ (z) ⩾ b. By the arbitrariness of b, we can obtain μ (x) ∧ μ (y) ∧ a ≤ μ (z). By Lemma 4.5, we know CD (μ) ≥ ⋁ {a ∈ L ∣ ∀ b ⩽ a, μ_[b]is a convex subgroup}. □

Theorem 4.7. Let G be an ordered group and μ ∈ L^G. Then CD (μ) = ⋁ {a ∈ L ∣ ∀ b ∈ P (L) , a ≰ b, μ^(b) is a convex subgroup}.

Proof. Assume that a ∈ L, let m (μ) ≥ a and let μ (x) ∧ μ (y) ∧ a ≤ μ (z) for x, y, z ∈ G with x ≤ z ≤ y. By Theorem 2.11, we know μ^(b) is a subgroup on G for all b ∈ P (L) with a ≰ b. Now we prove that μ^(b) is convex. Let x, y ∈ μ^(b) and x ≤ z ≤ y, then μ (x) ≰ b, μ (y) ≰ b, we have μ (x) ∧ μ (y) ∧ a ≰ b. Since μ (z) ≥ μ (x) ∧ μ (y) ∧ a, it follows that μ (z) ≰ b i.e., z ∈ μ^(b). Hence μ^(b) is a convex subgroup. By Lemma 4.5, we know CD (μ) ≤ ⋁ {a ∈ L ∣ ∀ b ∈ P (L) , a ≰ b, μ^(b)is a convex subgroup}.

Conversely, suppose that μ^(b) is a convex subgroup for any b ∈ P (L) with a ≰ b. By Theorem 2.11, we know m (μ) ≥ a. We need to prove μ (x) ∧ μ (y) ∧ a ≤ μ (z) for x, y, z ∈ G with x ≤ z ≤ y. Suppose that b ∈ P (L) with μ (x) ∧ μ (y) ∧ a ≰ b, then x, y ∈ μ^(b) and a ≰ b. Since μ^(b)isaconvex for all a ≰ b, it follows that z ∈ μ^(b), i.e., μ (z) ≰ b. By the arbitrariness of b, we can obtain μ (x) ∧ μ (y) ∧ a ≤ μ (z). By Lemma 4.5, we know CD (μ) ≥ ⋁ {a ∈ L ∣ ∀ b ⩽ a, μ^(b) is a convex subgroup}. □

Theorem 4.8. Let G be an ordered group and μ ∈ L^G. Then CD (μ) = ⋁ {a ∈ L ∣ ∀ b ∉ α (a) , μ^[b] is a convex subgroup}.

Proof. Assume that a ∈ L, let m (μ) ≥ a and let μ (x) ∧ μ (y) ∧ a ≤ μ (z) for x, y, z ∈ G with x ≤ z ≤ y. By Theorem 2.11, we know μ^[b] is a subgroup on G for all b ∉ α (a). Now we prove that μ^[b] is convex. Let x, y ∈ μ^[b] and x ≤ z ≤ y, then b ∉ α (μ (x)) , b ∉ α (μ (y)). By b ∉ α (a), we have b ∉ α (μ (x)) ∪ α (μ (y)) ∪ α (a). By α (μ (x)) ∪ α (μ (y)) ∪ α (a) = α (μ (x) ∧ μ (y) ∧ a), we obtain b ∉ α (μ (x) ∧ μ (y) ∧ a). Since μ (z) ≥ μ (x) ∧ μ (y) ∧ a, it follows that b ∉ α (μ (z)), i.e., z ∈ μ^[b]. Hence μ^[b] is a convex subgroup on G. By Lemma 4.5, we know CD (μ) ≤ ⋁ {a ∈ L ∣ ∀ b ∉ α (a) , μ^[b] is a convex subgroup}.

Conversely, suppose that μ^[b] is a convex subgroup for all b ∉ α (a). By Theorem 2.11, we know m (μ) ≥ a. We need to prove μ (x) ∧ μ (y) ∧ a ≤ μ (z) for x, y, z ∈ G with x ≤ z ≤ y. suppose that b ∈ L with b ∉ α (μ (x) ∧ μ (y) ∧ a), By α (μ (x) ∧ μ (y) ∧ a) = α (μ (x)) ∪ α (μ (y)) ∪ α (a), we obtain b ∉ α (μ (x)) ∪ α (μ (y)) ∪ α (a). Then b ∉ α (μ (x)), b ∉ α (μ (y)), b ∉ α (a). So x, y ∈ μ^[b] and b ∉ α (a). Since μ^[b] is a convex subgroup, it follows that z ∈ μ^[b], i.e., b ∉ α (μ (z)). Since α is an order-reversing mapping, it follows that μ (x) ∧ μ (y) ∧ a ≤ μ (z). By Lemma 4.5, we know CD (μ) ≥ ⋁ {a ∈ L ∣ ∀ b ∉ α (a) , μ^[b] is a convex subgroup}. □

Theorem 4.9. Let G be an ordered group and μ ∈ L^G. If β (a ∧ b) = β (a) ∩ β (b) for all a, b ∈ L, then CD (μ) = ⋁ {a ∈ L ∣ ∀ b ∈ β (a) , μ_(b) is a convex subgroup}.

Proof. Assume that a ∈ L, let m (μ) ≥ a and let μ (x) ∧ μ (y) ∧ a ≤ μ (z) for x, y, z ∈ G with x ≤ z ≤ y. By Theorem 2.12, we know μ_(b) is a subgroup for all b ∈ β (a). Now we prove that μ_(b) is convex. Let x, y ∈ μ_(b) with x ≤ z ≤ y, then b ∈ β (μ (x)) , b ∈ β (μ (y)) and b ∈ β (a), we have b ∈ β (μ (x)) ∩ β (μ (y)) ∩ β (a). By β (μ (x)) ∩ β (μ (y)) ∩ β (a) = β (μ (x) ∧ μ (y) ∧ a), we obtain b ∈ β (μ (x) ∧ μ (y) ∧ a). Since μ (x) ∧ μ (y) ∧ a ≤ μ (z), it follows that b ∈ β (μ (z)), i.e., z ∈ μ_(b). Hence μ_(b) is convex. By Lemma 4.5, we know CD (μ) ≤ ⋁ {a ∈ L ∣ ∀ b ∈ β (a) , μ_(b) is a convex subgroup}.

Conversely, assume that a ∈ L,μ_(b) is a convex subgroup for all b ∈ β (a). By Theorem \ref shixin3, we know m (μ) ≥ a. We need to prove μ (x) ∧ μ (y) ∧ a ≤ μ (z) for x, y, z ∈ G with x ≤ z ≤ y. Suppose that b ∈ L with b ∈ β (μ (x) ∧ μ (y) ∧ a). By β (μ (x)) ∩ β (μ (y)) ∩ β (a) = β (μ (x) ∧ μ (y) ∧ a), we have b ∈ β (μ (x)) ∩ β (μ (y)) ∩ β (a). then b ∈ β (μ (x)), b ∈ β (μ (y)) and b ∈ β (a). So x, y ∈ μ_(b) and b ∈ β (a). Since μ_(b) is a convex subgroup for any b ∈ β (a), it follows that z ∈ μ_(b), i.e., b ∈ β (μ (z)). By the arbitrariness of b, we have β (μ (x) ∧ μ (y) ∧ a) ⊆ β (μ (z)). Since β is an order-preserving mapping, it follows that we can obtain μ (x) ∧ μ (y) ∧ a ≤ μ (z). By Lemma 4.5, we know CD (μ) ≥ ⋁ {a ∈ L ∣ ∀ b ∈ β (a) , μ_(b) is a convex subgroup}. □

In what follows, we shall study L-fuzzy convex subgroup degrees of the order preserving image and the pre-image of an L-fuzzy subgroup.

Theorem 4.10. Let f : G ⟶ H be a group homomorphism and order preserving. Let μ ∈ L^G, λ ∈ L^H, then

${CD}_{H} (f_{L}^{\to} (μ)) \geq {CD}_{G} (μ)$ , and if f is injective, then ${CD}_{H} (f_{L}^{\to} (μ)) = {CD}_{G} (μ)$ ;

${CD}_{G} (f_{L}^{\leftarrow} (λ)) \geq {CD}_{H} (λ)$ , and if f is surjective, then ${CD}_{G} (f_{L}^{\leftarrow} (λ)) = {CD}_{H} (λ)$ .

Proof. (1) Let a ∈ L such that CD_G (μ) ≥ a, by Lemma 4.3, we have a ⩽ m (μ) and μ (x) ∧ μ (y) ∧ a ≤ μ (z) for all x, y, z ∈ G with x ≤ z ≤ y. By Theorem 2.13, we know $m (f_{L}^{\to} (μ)) \geq a$ . $\begin{matrix} ⋁ {a \in L : f_{L}^{\to} (μ) (x^{'}) \land f_{L}^{\to} (μ) (y^{'}) \land a \leq \\ f_{L}^{\to} (μ) (z^{'}) ∣ \forall x^{'}, y^{'}, z^{'} \in H, x^{'} \leq z^{'} \leq y^{'}} \\ = & ⋁ {a \in L : ⋁_{f (x) = x^{'}} μ (x) \land ⋁_{f (y) = y^{'}} μ (y) \land a \leq \\ ⋁_{f (z) = z^{'}} μ (z) ∣ \forall x^{'}, y^{'}, z^{'} \in H, x^{'} \leq z^{'} \leq y^{'}} \\ \geq & ⋁ {a \in L : μ (x) \land μ (y) \land a \leq \\ μ (z) ∣ \forall x, y, z \in G, x \leq z \leq y} . \end{matrix}$

By Lemma 4.5, we have ${CD}_{H} (f_{L}^{\to} (μ)) \geq a$ . By the arbitrariness of a, we obtain

${CD}_{H} (f_{L}^{\to} (μ)) \geq {CD}_{G} (μ)$ .

If f is injective, By Theorem 2.13, we know $m (f_{L}^{\to} (μ)) = m (μ)$ .

$\begin{matrix} ⋁ {a \in L : f_{L}^{\to} (μ) (x^{'}) \land f_{L}^{\to} (μ) (y^{'}) \land a \leq \\ f_{L}^{\to} (μ) (z^{'}) ∣ \forall x^{'}, y^{'}, z^{'} \in H, x^{'} \leq z^{'} \leq y^{'}} \\ \leq & ⋁ {a \in L : (f_{L}^{\to} (μ)) (f (x)) \land (f_{L}^{\to} (μ)) (f (y)) \land a \leq \\ (f_{L}^{\to} (μ)) (f (z)) ∣ \forall x, y, z \in G, x \leq z \leq y} \\ = & ⋁ {a \in L : (f_{L}^{\leftarrow} (f_{L}^{\to} (μ))) (x) \land (f_{L}^{\leftarrow} (f_{L}^{\to} (μ))) (y) \\ \land a \leq (f_{L}^{\leftarrow} (f_{L}^{\to} (μ))) (z) ∣ \forall x, y, z \in G, x \leq z \leq y} \\ = & ⋁ {a \in L : μ (x) \land μ (y) \land a \leq \\ μ (z) ∣ \forall x, y, z \in G, x \leq z \leq y} . \end{matrix}$

By Lemma 4.5, we have ${CD}_{H} (f_{L}^{\to} (μ)) \leq {CD}_{G} (μ)$ . Combining with the aforesaid, we have ${CD}_{H} (f_{L}^{\to} (μ)) = {CD}_{G} (μ)$ .

(2) Let a ∈ L such that CD_H (λ) ≥ a, by Definition 4.1, we have a ⩽ m (λ) and λ (x′) ∧ λ (y′) → λ (z′) for all x′, y′, z′ ∈ H such that x′ ≤ z′ ≤ y′. By Theorem 2.13, we know $m (f_{L}^{\leftarrow} (λ)) \geq a$ . Then $\begin{matrix} ⋀ {f_{L}^{\leftarrow} (λ) (x) \land f_{L}^{\leftarrow} (λ) (y) \to \\ f_{L}^{\leftarrow} (λ) (z) ∣ \forall x, y, z \in G, x \leq z \leq y} \\ = & ⋀ {λ (f (x)) \land λ (f (y)) \to λ (f (z)) ∣ f (x), \\ f (y), f (z) \in H, f (x) \leq f (z) \leq f (y)} \\ \geq & ⋀ {λ (x^{'}) \land λ (y^{'}) \to λ (z^{'}) ∣ \forall x^{'}, y^{'}, z^{'} \in H, \\ x^{'} \leq z^{'} \leq y^{'}} \end{matrix} .$

By Definition 4.1, we have ${CD}_{G} (f_{L}^{\leftarrow} (λ)) \geq a$ . By the arbitrariness of a, we obtain

${CD}_{G} (f_{L}^{\leftarrow} (λ)) \geq {CD}_{H} (λ)$ .

If f is surjective, by (1) we have that ${CD}_{H} (λ) = {CD}_{H} (f_{L}^{\to} (f_{L}^{\leftarrow} λ) \geq {CD}_{G} (f_{L}^{\leftarrow} λ)$ . Combining with the aforesaid, we have

${CD}_{G} (f_{L}^{\leftarrow} (λ)) = {CD}_{H} (λ) .$ □

Theorem 4.11. Let G and H be ordered groups, let μ ∈ L^G and λ ∈ L^H be L-fuzzy convex subgroups. Then CD (μ) ∧ CD (λ) ≤ CD (μ × λ).

Proof. By Theorem 2.14, we know m (μ) ∧ m (λ) ≤ m (μ × λ). Since μ ∈ L^G and λ ∈ L^H be L-fuzzy convex subgroups for all x = (x_G, x_H) , y = (y_G, y_H) , z = (z_G, z_H) ∈ G × H with x_G ≤ z_G ≤ y_G, x_H ≤ z_H ≤ y_H, we have $\begin{matrix} ⋀ {(μ \times λ) (x) \land (μ \times λ) (y) \to (μ \times λ) (z)} \\ = & ⋀ {(μ \times λ) (x_{G}, x_{H}) \land (μ \times λ) (y_{G}, y_{H}) \to \\ (μ \times λ) (z_{G}, z_{H})} \\ = & μ (x_{G}) \land λ (x_{H})) \land (μ (y_{G}) \land λ (y_{H})) \to \\ (μ (z_{G}) \land λ (z_{H})) \\ \geq & ⋀ {μ (x_{G} \land μ (y_{G}) ⟶ μ (z_{G})} \land \\ {λ (x_{H} \land λ (y_{H}) ⟶ λ (z_{H})} \end{matrix}$

So CD (μ) ∧ CD (λ) ≤ CD (μ × λ). □

Theorem 4.11 can be generalized to the finite case.

Lemma 4.12. Let {G_i} _1⩽i⩽n be a family of ordered groups. Let μ_i ∈ L^{G
_i} be an L-fuzzy convex subgroups on G_i for all 1 ⩽ i ⩽ n. Then ⋀_1≤i≤nCD (μ_i) ≤ CD (μ₁ × μ₂ × ⋯ × μ_n).

5 L-fuzzy convexity induced by L-fuzzy convex subgroup degree

In this section, by means of L-fuzzy convex subgroup degrees, we obtain an L-fuzzy convexity on subgroup. Furthermore, we shall analyze its corresponding L-fuzzy convexity preserving mapping and L-fuzzy convex-to-convex mapping.

For each μ ∈ L^G, CD (μ) can be naturally considered as a mapping CD: L^G ⟶ L defined by μ ⟼ CD (μ). The following theorem will show that CD is an L-fuzzy convexity on G.

Theorem 5.1. Let G be an ordered group. Then the mapping CD: L^G ⟶ L defined by μ ⟼ CD (μ) is an L-fuzzy convexity on G, which is called the L-fuzzy convexity induced by L-convex subgroup degrees on G.

Proof. (LMC1) It is straightforward that $CD (χ_{\emptyset}) = CD (χ_{G}) = ⊤ .$ (LMC2) Let {μ_i ∣ i ∈ Ω} ⊆ L^G be nonempty. To prove ⋀_i∈ΩCD (μ_i) ⩽ CD (⋀ _i∈Ωμ_i), we need to prove that a ⩽ CD (⋀ _i∈Ωμ_i) for any a ⩽ ⋀ _i∈ΩCD (μ_i). By Lemma 4.4, for all i ∈ Ω, we have μ_i (s) ∧ μ_i (t) ∧ a ⩽ μ_i (st^-1), μ_i (x) ∧ μ_i (y) ∧ a ⩽ μ_i (z) for all s, t ∈ G and x, y, z ∈ G with x ≤ z ≤ y. This implies $\begin{matrix} (⋀_{i \in Ω} μ_{i} (s)) \land (⋀_{j \in Ω} μ_{j} (t)) \land a \\ ⩽ & ⋀_{i \in Ω} (μ_{i} (s) \land μ_{i} (t) \land a) ⩽ ⋀_{i \in Ω} μ_{i} ({st}^{- 1}) \end{matrix}$ and $\begin{matrix} (⋀_{i \in Ω} μ_{i} (x)) \land (⋀_{j \in Ω} μ_{j} (y)) \land a \\ ⩽ & ⋀_{i \in Ω} (μ_{i} (x) \land μ_{i} (y) \land a) ⩽ ⋀_{i \in Ω} μ_{i} (z) . \end{matrix}$

By Lemma 4.4, we have a ⩽ CD (⋀ _i∈Ωμ_i). From the arbitrariness of a, we obtain $⋀_{i \in Ω} CD (μ_{i}) ⩽ CD (⋀_{i \in Ω} μ_{i}) .$

(LMC3) Let {μ_i ∣ i ∈ Ω} ⊆ L^G be nonempty and totally ordered. In order to prove $⋀_{i \in Ω} CD (μ_{i}) ⩽ CD (⋁_{i \in Ω} μ_{i}),$ it needs to show that a ⩽ CD (⋁ _i∈Ωμ_i) for any a ⩽ ⋀ _i∈ΩCD (μ_i). By Lemma 4.4, for all i ∈ Ω, we have μ_i (s) ∧ μ_i (t) ∧ a ⩽ μ_i (st^-1), μ_i (x) ∧ μ_i (y) ∧ a ⩽ μ_i (z) for all s, t ∈ G and x, y, z ∈ G with x ≤ z ≤ y. Let b ∈ J (L) such that b ≺ (⋁ _i∈Ωμ_i (s)) ∧ (⋁ _i∈Ωμ_i (t)) ∧ a and b ≺ (⋁ _i∈Ωμ_i (x)) ∧ (⋁ _i∈Ωμ_i (y)) ∧ a, then we have b ≺ ⋁ _i∈Ωμ_i (s), b ≺ ⋁ _i∈Ωμ_i (t), b ≺ ⋁ _i∈Ωμ_i (x), b ≺ ⋁ _i∈Ωμ_i (y) and b ⩽ a. Hence there exists some i, j, k, l ∈ Ω such that b ⩽ μ_i (s), b ⩽ μ_j (t), b ⩽ μ_k (x), b ⩽ μ_l (y) and b ⩽ a. Since {μ_i ∣ i ∈ Ω} is totally ordered, we assume μ_j ⩽ μ_i, μ_k ⩽ μ_l, it follows that b ⩽ μ_i (s) ∧ μ_i (t) ∧ a and b ⩽ μ_l (x) ∧ μ_l (y) ∧ a. By μ_i (s) ∧ μ_i (t) ∧ a ⩽ μ_i (st^-1), μ_l (x) ∧ μ_l (y) ∧ a ⩽ μ_l (z). we obtain b ⩽ μ_i (st^-1) and b ⩽ μ_l (z). Hence b ⩽ ⋁ _i∈Ωμ_i (st^-1) and b ⩽ ⋁ _l∈Ωμ_l (z). From the arbitrariness of b, we have $\begin{matrix} (⋁_{i \in Ω} μ_{i} (s)) \land (⋁_{i \in Ω} μ_{i} (t)) \land a ⩽ ⋁_{i \in Ω} μ_{i} ({st}^{- 1}) \end{matrix}$ and $\begin{matrix} (⋁_{l \in Ω} μ_{l} (x)) \land (⋁_{l \in Ω} μ_{l} (y)) \land a ⩽ ⋁_{l \in Ω} μ_{l} (z) . \end{matrix}$

Combining Lemma 4.4, we have a ⩽ CD (⋁ _i∈Ωμ_i). By the arbitrariness of a, we obtain ⋀_i∈ΩCD (μ_i) ⩽ CD (⋁ _i∈Ωμ_i).

Therefore CD is an L-fuzzy convexity on G. □

In what follows, some properties of the L-fuzzy convexity on groups are discussed.

From [32], we know scripfontL is an L-fuzzy convexity if and only if scripfontL_[a] is an L-convexity for all a ∈ L ∖ {⊥} or scripfontL^[a] is an L-convexity for all a ∈ α (⊥). Then the following theorem is obvious.

Theorem 5.2. Let G be an ordered group. Then the following conditions are equivalent:

CD is an L-fuzzy convexity induced by L-convex subgroup degrees on G ;

for each a ∈ J (L), CD_[a] is an L-convexity on G ;

for each a ∈ α (⊥), CD^[a] is an L-convexity on G .

By Theorem 4.10 (2), the following theorem is obvious.

Theorem 5.3. Let f : G ⟶ H be group homomorphism and order preserving. Let CD_G and CD_H be the L-fuzzy convexities induced by L-convex subgroup degrees on G and on H, respectively. Then f : (G, CD_G) ⟶ (H, CD_H) is an L-fuzzy convexity preserving mapping.

The following theorem will present some equivalent characterizations of L-fuzzy convexity preserving mappings.

Theorem 5.4. Let G and H be ordered groups. Let CD_G and CD_H be the L-fuzzy convexities induced by L-convex subgroup degrees on G and on H, respectively. Then for each mapping f : G ⟶ H, the following conditions are equivalent:

f : (X, CD_G) ⟶ (H, CD_H) is an L-fuzzy convexity preserving mapping;

f : (G, (CD_G) _[a]) ⟶ (H, (CD_H) _[a]) is a convexity preserving mapping for each a∈ L \ {⊥} ;

f : (G, (CD_G) _[a]) ⟶ (H, (CD_H) _[a]) is a convexity preserving mapping for each a∈ J (L) ;

f : (G, (CD_G) ^[a]) ⟶ (H, (CD_H) ^[a]) is a convexity preserving mapping for each a ∈ α (⊥) .

Proof. (1) ⇒ (2). We need to prove $f_{L}^{\leftarrow} (λ) \in ({CD}_{G})_{[a]}$ for any λ ∈ (CD_H) _[a]. Since λ ∈ (CD_H) _[a], we have a ⩽ CD_H (λ). Note that f is an L-fuzzy convexity preserving mapping, we have ${CD}_{H} (λ) ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ . Hence $a ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ . Therefore $f_{L}^{\leftarrow} (λ) \in ({CD}_{G})_{[a]}$ .

(2) ⇒ (3) is trivial.

(3) ⇒ (1). In order to prove ${CD}_{H} (λ) ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ for any λ ∈ L^H, it suffices to show that $a ⩽ {CD}_{G} ((f_{L}^{\leftarrow} (λ))$ for any a ⩽ CD_H (λ). Let a ⩽ CD_H (λ) for any a ∈ J (L), i.e., λ ∈ (CD_H) _[a]. Note that f is a convexity preserving mapping, we have $f_{L}^{\leftarrow} (λ) \in ({CD}_{G})_{[a]}$ , i.e., $a ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ . By the arbitrariness of a, we obtain ${CD}_{H} (λ) ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ .

(1) ⇒ (4). We need to prove $f_{L}^{\leftarrow} (λ) \in ({CD}_{G})^{[a]}$ for any λ ∈ (CD_H) ^[a]. Since λ ∈ (CD_H) ^[a], we have a ∉ α (CD_H (λ)). Note that f is an L-fuzzy convexity preserving mapping, we have ${CD}_{H} (λ) ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ . Since α is an order-reversing mapping, it follows that $α ({CD}_{G} (f_{L}^{\leftarrow} (λ))) \subseteq α ({CD}_{H} (λ))$ . Hence $a \notin α ({CD}_{G} (f_{L}^{\leftarrow} (λ)))$ . Therefore $f_{L}^{\leftarrow} (λ) \in ({CD}_{G})^{[a]}$ .

(4) ⇒ (1). In order to prove ${CD}_{H} (λ) ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ for any λ ∈ L^H, it suffices to show that $α ({CD}_{H} (λ)) \supseteq α ({CD}_{G} (f_{L}^{\leftarrow} (λ)))$ . Let a ∈ α (⊥) with a ∉ α (CD_H (λ)). Then λ ∈ (CD_H) ^[a]. Note that f is a convexity preserving mapping, we have $f_{L}^{\leftarrow} (λ) \in ({CD}_{G})^{[a]}$ , i.e., $a \notin α ({CD}_{G} (f_{L}^{\leftarrow} (λ)))$ .

Therefore ${CD}_{H} (λ) ⩽ {CD}_{G} (f_{L}^{\leftarrow} (λ))$ . □

By Theorem 4.10 (1), we have the following theorem.

Theorem 5.5. Let G and H be ordered groups. Let CD_G and CD_H be the L-fuzzy convexities induced by L-convex group degrees on G and on H, respectively. Let f : (X, CD_G) ⟶ (Y, CD_H) be group homomorphism and order preserving. Then f is an L-fuzzy convex-to-convex mapping.

In what follows, some equivalent characterizations of L-fuzzy convex-to-convex mappings are obtained.

Theorem 5.6. Let G and H be ordered groups. Let CD_G and CD_H be the L-fuzzy convexities induced by L-convex group degrees on G and on H, respectively. Then for each mapping f : G ⟶ H, the following conditions are equivalent:

f : (G, CD_G) ⟶ (H, CD_H) is an L-fuzzy convex-to-convex mapping;

f : (G, (CD_G) _[a]) ⟶ (H, (CD_H) _[a]) is a convex-to-convex mapping for each a∈ L \ {⊥} ;

f : (G, (CD_G) _[a]) ⟶ (H, (CD_H) _[a]) is a convex-to-convex mapping for each a∈ J (L) ;

f : (G, (CD_G) ^[a]) ⟶ (H, (CD_H) ^[a]) is a convex-to-convex mapping for each a ∈ α (⊥) .

Proof. The verifications are analogous to that of Theorem 5.4 and omitted here. □

6 Conclusions

In this paper, the concept of degree to which an L-fuzzy subgroup is an L-fuzzy convex subgroup was proposed. Thus an L-fuzzy convexity on a group is naturally constructed and some properties of this kind of L-fuzzy convexity were studied.

It is worth noting that the same thought can be applied to different algebraic systems such as rings, fields and so on. Thus L-fuzzy convexities can be induced by different algebraic systems. The above facts will be useful to help further investigations.

Footnotes

Acknowledgement

This work is supported by the National Natural Science Foundation of China (11871097) and Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048).

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