On the category of M-fuzzy Q-modules

Abstract

In this paper we consider the category M-FQMod of M-fuzzy (left) Q-modules over a given quantale Q. We introduce and investigate the notion of M-fuzzy Q-submodule of a left Q-module. Also, the notions of M-fuzzy module nuclei and conuclei on a left Q-module are defined, and the elementary properties of both are studied. The relationships between such notions and M-fuzzy Q-submodule are discussed. Finally, the relationship between the category of M-fuzzy Q-modules and the category of (L, M)-quasi-fuzzy topological spaces is discussed.

Keywords

Quantale semi quantale module nucleus and module conucleus 06A06 54A10 54A40

1 Introduction

The first lattice analogy of ring module appeared in the paper of A. Joyal and M. Tierney [10], in connection with the analysis of descent theory. In the last few years a considerable amount of works has been done on fuzzy submodules of ring modules (for example see [2 , 29]). The term quantale was introduced in [16] in connection with certain aspects of C^∗-algebras. The idea of a quantale module appeared in [1] of S. Abramsky and S. Vickers as the key notion for treatment of process semantics, generalizing the already existing concept of topological systems (see [28]) based on the logic of finite observations. Such concept was motivated by the notion of a module over a ring [4]. It replaces rings by quantales and abelian groups by complete lattices. Furthermore, Q-modules have been shown to provide a proper language for the study of roughness [18], linear logic [23], lattice-valued topological spaces [27], deductive systems and two different branches of image processing, namely fuzzy image compression and mathematical morphology [24, 25].

The purpose of this paper is to make a further contribution to the theory of quantale modules. One of our purposes is to study the notion of M-fuzzy Q-submodule of a left Q-module. As we all know, the nuclei and the conuclei play an important role in quantale and Q-module theories, because they determine the quotients and subobjects in their respective categories. In this paper, one of our other purposes is to define and study the notions M-fuzzy nuclei and conuclei on a Q-module. Some of their basic properties will be given in this paper and some of their applications will be left for future study.

The paper is organized as follows. Sections 2 is preliminary and most of the results presented are known. Section 3, is devoted to define and study the concept of M-fuzzy Q-submodule of a left Q-module. In Sections 4 and 5 the concept of an M-fuzzy module nuclei and conuclei are introduced and investigated, respectively. The relationship between them and the M-fuzzy Q-submodule is discussed.

Finally, in section 6, we set up a dual adjunction between the category R (L, M)-QFTop of enriched (L, M)-quasi-fuzzy topological spaces and the category M-FQMod of M-fuzzy Q-submodules.

2 Preliminaries

By a ⋁-semilattice we mean is a partially ordered set having arbitrary ⋁. A ⋁-semilattice homomorphism is a map preserving arbitrary ⋁.

Definition 2.1. [20, 21] A semi-quantale (L, ≤ , ⋁ , ⊗) is a ⋁-semilattice (L, ≤ , ⋁) equipped with binary operation ⊗ : L × L ⟶ L, with no additional assumptions, called a tensor product.

Definition 2.2. A semi-quantale L = (L, ≤ , ⊗) is called:

a unital semi-quantale [20] if ⊗ has an identity element e ∈ L called the unit. If the unit e of the groupoid (L, ⊗) coincides with the top element ⊤ of L, then a unital semi-quantale is called a strictly two-sided semi-quantale.

a quantale [22] if the multiplication ⊗ is associative and satisfies $a \otimes (\underset{i \in I}{⋁} b_{i}) = \underset{i \in I}{⋁} (a \otimes b_{i})$ and $(\underset{i \in I}{⋁} b_{i}) \otimes a = \underset{i \in I}{⋁} (b_{i} \otimes a)$ for all a ∈ L, {b} _i∈I ⊆ L.

A semi-quantale morphism [20] h from a semi-quantale L = (L, ≤ , ⊗) to an other semi-quantale M = (M, ≤ , ⊙) is a map h : L ⟶ M preserving the tensor product and the arbitrary joins, i.e., the next conditions are satisfied for all a, b ∈ L and {a_i : i ∈ I} ⊆ L:

h (a ⊗ b) = h (a) ⊙ h (b);

$h (\underset{i \in I}{⋁} a_{i}) = \underset{i \in I}{⋁} h (a_{i})$ .

If a semi-quantale morphism h : L ⟶ M additionally preserves the top (resp., unit) element, i.e., h (⊤ _L) = ⊤ _M (resp., h (e_L) = e_M), then it is said to be strong (resp., unital). The category SQuant comprises all semi-quantales together with semi-quantale morphisms. The non-full subcategory UnSQuant of SQuant comprises all unital semi-quantales and all unital semi-quantale morphisms [20]. Quant (resp., UnQuant) is the full subcategory of SQuant(resp., UnSQuant), which has as objects all quantales (resp., unital quantales).

Let X be a non-empty set and L ∈ |SQuant|. An L-fuzzy subset (or L-subset) of X is a mapping A : X ⟶ L. The family of all L-fuzzy subsets on X will be denoted by L^X. The smallest element and the largest element in L^X are denoted by $\underline{⊥}$ and $\underline{⊤}$ , respectively. The algebraic and lattice-theoretic structures can be extended from the semi-quantale (L, ≤ , ⋁ , ⊗) to L^X point wisely: f ≤ g ⇔ ∀ x ∈ X : f (x) ≤ g (x),

(f ⊗ g) (x) = f (x) ⊗ g (x) , x ∈ X.

Then L^X is again a semi-quantale with respect to the multiplication ⊗ and the arbitrary sups. In the case L is unital with unit e, then L^X becomes a unital semi-quantale with the unit $\underline{e}$ . For an ordinary mapping f : X ⟶ Y, one can define the mappings $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ and $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ by $f_{L}^{\to} (A) (y) = ⋁ {A (x) : x \in X, f (x) = y}$ for every A ∈ L^X and every y ∈ Y, $f_{L}^{\leftarrow} (B) = B \circ f$ for every B ∈ L^Y, respectively. For more details, we refer to [19, 20].

Definition 2.3. [5] Let (L, ≤ , ⊗) , (M, ≤ , ⊙) ∈ |SQuant|.

An M-fuzzy semi-quantale on L is a map μ : L ⟶ M satisfying the following conditions: For all a, b ∈ L and {a_j|j ∈ J} ⊆ L,

μ (a) ⊙ μ (b) ≤ μ (a ⊗ b),

⋀_j∈Jμ (a_j) ≤ μ (⋁ _j∈Ja_j).

An M-fuzzy semi-quantale μ is called strong if μ (⊤ _L) = ⊤ _M.

In case (L, ≤ , ⊗) is a unital semi-quantale with the unit e_L, an M-fuzzy semi-quantale μ is called unital if μ (e_L) = ⊤ _M.

Definition 2.4. [5] Let (L, ≤ , ⊗) , (M, ≤ , ⊙) ∈ |SQuant|, and X be a non-empty set.

A map τ : L^X ⟶ M is called an (L, M)-quasi-fuzzy topology on X iff τ is an M-fuzzy semi-quantale on L^X, i.e., the next conditions are satisfied for all f, g ∈ L^X and {f_j|j ∈ J} ⊆ L^X :

τ (f) ⊙ τ (g) ≤ τ (f ⊗ g),

⋀_j∈Jτ (f_j) ≤ τ (⋁ _j∈Jf_j).

An (L, M)-quasi-fuzzy topology is strong iff $τ (\underline{⊤}) = ⊤_{M}$ .

If L is a unital semi-quantale with unit e. An (L, M)-quasi-fuzzy topology is then called an (L, M)-fuzzy topology iff $τ (\underline{e}) = ⊤_{M}$ .

The pair (X, τ) is called an (L, M)-quasi-fuzzy (resp., strong (L, M)-quasi-fuzzy, (L, M)-fuzzy) topological space if τ is an (L, M)-quasi-fuzzy (resp., strong (L, M)-quasi-fuzzy, (L, M)-fuzzy) topology on X.

An (L, M)-quasi-fuzzy (resp., strong (L, M)-quasi-fuzzy, (L, M)-fuzzy) topology τ on X [9] is called enriched iff τ satisfies the subsequent axiom τ (λ) ≤ τ (α ⊗ λ), ∀α ∈ L, λ ∈ L^X.

In this case, the pair (X, τ) is called an enriched (L, M)-quasi-fuzzy (resp., strong (L, M)-quasi-fuzzy, (L, M)-fuzzy) topological space.

Definition 2.5. [5] Let (L, ≤ , ⊗) , (M, ≤ , ⊙) ∈ |SQuant|.

(L, M)-QFTop denotes a category whose objects are all (L, M)-quasi-fuzzy topological spaces, and whose morphisms from (X, τ) to (Y, ν) are all functions f : X ⟶ Y such that $ν \leq τ \circ f_{L}^{\leftarrow}$ . Composition and identities in (L, M)-QFTop are the same as in Set.

(L, M)-SQFTop is a full subcategory of (L, M)-QFTop consisting of all strong (L, M)-quasi-fuzzy topological spaces.

For a unital semi-quantale L, (L, M)-FTop is a full subcategory of(L, M)-QFTop consisting of all (L, M)-fuzzy topological spaces.

The category of all enriched (L, M)-quasi-fuzzy (resp., strong (L, M)-quasi-fuzzy, (L, M)-fuzzy) topological spaces is denoted by R (L, M)-QFTop (resp., R (L, M)-SQFTop, R (L, M)-FTop).

In case where L is a strictly two-sided semi-quantale, (L, M)-SQFTop coincides with (L, M)-FTop in the sense of Kubiak and Šostak [12].

Definition 2.6. [9] Let R : L^X → M be a map and let τ _R is the coarsest (L, M)-fuzzy topology on X provided with the property $R (λ) \leq τ (λ) for all λ \in L^{X} .$ (1) A map R : L^X → M is called an (L, M)-fuzzy subbase of an (L, M)-fuzzy topology τ iff τ = τ_R.

Definition 2.7. [17] Let (Q, ≤ , ⊗) ∈ |Quant|. A left module over Q (shortly a (left) Q-module) is a ⋁-semilattice (A, ≤), together with a module action ∗ : Q × A → A satisfying:

(a ⊗ b) ∗ d = a ∗ (b ∗ d).

(⋁ S) ∗ d = ⋁ {s ∗ d : s ∈ S}.

a ∗ ⋁ X = ⋁ {a ∗ x : x ∈ X}.

for all a, b ∈ Q, d ∈ A, S ⊆ Q, X ⊆ A.

For the sake of shortness, through this paper, Q-module, means left Q-module. Some properties of Q-modules are considered in [27]. Right Q-modules are defined in a similar way, the only difference being that the action is an operation ∗ : A × Q → A, satisfying the obvious set of conditions.

Every (left) Q-module (A, ≤ , ∗) has two maps, which are induced by its binary operation ∗ and which are defined by ↠: Q × A → A, ⇝ : A × A → Q such that for all a, b ∈ A, q ∈ Q, respectively. These operations have the standard properties of poset adjunctions [[8], Sections 03] or (order preserving) Galois connections [6], for example, $q * a \leq b \Leftrightarrow a \leq q ↠ b \Leftrightarrow q \leq b ⇝ a .$ (2)

Let (A, ≤ , ∗) and (B, ≤ , ∗) be Q-modules. A mapping f : (A, ≤ , ∗) → (B, ≤ , ∗) is a Q-module morphism if

$f (⋁ S) = ⋁ f (S)$ for every S ⊆ A,

f (q ∗ a) = q ∗ f (a) for all q ∈ Q, a ∈ A.

Example 2.8. [11] Let X be a non-empty set and (L, ≤ , ⊗) ∈ |Quant|. A pair (L^X, ∗) is an L-module, where ∗ : L × L^X → L^X is a map define by (a ∗ λ) (x) = a ⊗ λ (x), ∀a ∈ L, λ ∈ L^X, x ∈ X.

In the case where L = ([0, 1] , ≤ , ∧), the pair ([0, 1] ^X, ∗) is called [0, 1]-module.

Lemma 2.9. [7] The following properties hold for every Q-module (A, ≤ , ∗), q ∈ Q and a, b ∈ A:

(a ⇝ b) ∗ b ≤ a,

q ∗ (q ↠a) ≤ a,

a ≤ q ↠(q ∗ a) and q ≤ (q ∗ a) ⇝ a,

(q ↠a) ⇝ b = q ↠(a ⇝ b),

[(a ⇝ b) ∗ b] ⇝ b = a ⇝ b,

a ≤ b ⇒ c ⇝ b ≤ c ⇝ a,

3 M-fuzzy Q-submodules

By analogy with the concept of fuzzy submodules of ring modules, we decided to introduce and study the concept of fuzzy submodules of quantale modules.

Definition 3.1. Let Q ∈ |Quant| and (A, ≤ , ∗) is a left (resp., right) Q-module.

An M-fuzzy left (resp., right) Q-submodule of A is a map μ : A → M satisfying the following conditions: For all a ∈ A, q ∈ Q and {S_i : i ∈ I} ⊆ A.

μ (⋁ _i∈IS_i) ≥ ⋀ _i∈Iμ (S_i),

μ (q ∗ a) ≥ μ (a) (resp., μ (a∗ q) ≥ μ (a)).

The pair (A, μ) is called an M-fuzzy left (resp., right) Q-module of (A, ≤ , ∗). An M-fuzzy Q-module homomorphisms from (A₁, μ₁) to (A₂, μ₂) are all Q-module homomorphisms h : A₁ → A₂ such that h (μ₁) ≤ μ₂.

An M-fuzzy left (resp., right) Q-submodule μ is called strong if μ (⊤ _A) = ⊤ _M. In this case the pair (A, μ) is called a strong M-fuzzy left (resp., right) Q-module.

In case A is a unital left (resp., right) Q-module with the unit e_A, an M-fuzzy left (resp., right) Q-submodule μ is called unital if μ (e_A) = ⊤ _M. In this case, the pair (A, μ) is called a unital M-fuzzy left (resp., right) Q-module.

Definition 3.2. Let Q ∈ |Quant| and (A, ≤ , ∗) is a left (resp., right) Q-module.

M-FQMod (resp., M-FModQ) denotes a category whose objects are all M-fuzzy left (resp., right) Q-submodules of (A, ≤ , ∗) and whose morphisms are M-fuzzy Q-module homomorphisms from (A₁, μ₁) to (A₂, μ₂).

M-SFQMod (resp., M-SFModQ) is a full subcategory of M-FQMod (resp., M-FModQ) consisting of all strong M-fuzzy left (resp., right) Q-modules.

For a unital Q-module (A, ≤ , ∗), M-UFQMod (resp., M-UFModQ) is a full subcategory of M-FQMod (resp., M-FModQ) consisting of all unital M-fuzzy left (resp., right) Q-modules.

Example 3.3. Let X be a non-empty set and (L, ≤ , ⊗) ∈ |Quant|. Since (L^X, ∗) is an L-module (see Example 2.8), then according to Definition 2.4, we have that:

Any enriched (L, M)-quasi-fuzzy topology τ : L^X → M is an M-fuzzy L-submodule of L^X.

Any enriched strong (L, M)-quasi-fuzzy topology τ : L^X → M is a strong M-fuzzy L-submodule of L^X.

If (L, ≤ , ⊗) is unital, then any enriched (L, M)-fuzzy topology τ : L^X → M is a unital M-fuzzy L-submodule of L^X.

Example 3.4. Let (Q, ≤ , ⊗) ∈ |Quant|. A pair (Q, ∗) is a Q-module, where ∗ : Q × Q → Q is a map define by q ∗ λ = q ⊗ λ, ∀q ∈ Q, λ ∈ Q. An M-fuzzy semi-quantale μ : Q → M is an M-fuzzy Q-submodule of Q if μ (q ∗ λ) ≥ μ (λ).

Proposition 3.5. For (Q, ≤ , ⊗) ∈ |Quant|, let (A, ≤ , ∗) be a Q-module and μ₁, μ₂ are M-fuzzy Q-submodules of (A, ≤ , ∗). The intersection μ₁ ∩ μ₂ is an M-fuzzy Q-submodule of (A, ≤ , ∗).

Proof. Let a ∈ A, q ∈ Q and {S_i : i ∈ I} ⊆ A.

(μ₁ ∩ μ₂) (⋁ _iS_i) = μ₁ (⋁ _iS_i) ∧ μ₂ (⋁ _iS_i), ≥ ⋀ _iμ₁ (S_i) ∧ ⋀ _iμ₂ (S_i) = ⋀ _i (μ₁ ∩ μ₂) (S_i). Hence, (μ₁ ∩ μ₂) (⋁ _iS_i) ≥ ⋀_i (μ₁ ∩ μ₂) (S_i).

(μ₁ ∩ μ₂) (q ∗ a) = μ₁ (q ∗ a) ∧ μ₂ (q ∗ a), ≥ μ₁ (a) ∧ μ₂ (a) = (μ₁ ∩ μ₂) (a).

So, (μ₁ ∩ μ₂) (q ∗ a) ≥ (μ₁ ∩ μ₂) (a).

□

Corollary 3.6. Let {μ_i : i = 1, 2, . . . , n} be a family of M-fuzzy Q-submodules of A. Then so is ∩_{i∈I_n}μ_i.

Proposition 3.7. For (M, ≤ , ⊗) ∈ | Quant | with (M, ≤) be a completely distributive lattice, let h : A ⟶ B be a Q-module homomorphism from (A, ≤ , ∗) into (B, ≤ , ∗). If μ is an M-fuzzy Q-submodule of (A, ≤ , ∗), then $h_{M}^{\to} (μ)$ is an M-fuzzy Q-submodule of (B, ≤ , ∗), where $h_{M}^{\to} : M^{A} \to M^{B}$ is defined by $h_{M}^{\to} (μ) (b) = ⋁ {μ (a) : μ \in M^{A}, b \in B, h (a) = b}$ .

Proof.

Let {b_j : j ∈ J } ⊆ B. By the complete distributivity of (M, ≤) , we may write

$⋀_{j \in J} h_{M}^{\to} (μ) (b_{j}) = \underset{j \in J}{⋀} (\underset{a \in h^{- 1} (b_{j})}{⋁} μ (a))$ $= \underset{k \in \prod_{j \in J} h^{- 1} (b_{j})}{⋁} (\underset{j \in J}{⋀} μ (k (j)))$ $\leq \underset{k \in \prod_{j \in J} h^{- 1} (b_{j})}{⋁} (μ (\underset{j \in J}{⋁} k (j)))$ . Furthermore, for every $k \in \prod_{j \in J} h^{- 1} (b_{j})$ , since h (k (j)) = b_j for every j ∈ J, we easily get $\underset{j \in J}{⋁} k (j) \in h^{- 1} (\underset{j \in J}{⋁} b_{j})$ , which gives $\underset{k \in \prod_{j \in J} h^{- 1} (b_{j})}{⋁} (μ (\underset{j \in J}{⋁} k (j))) \leq \underset{z \in h^{- 1} (\underset{j \in J}{⋁} b_{j})}{⋁} μ (z) = h_{M}^{\to} (μ) (\underset{j \in J}{⋁} b_{j}) .$ Therefore, it follows that $\underset{j \in J}{⋀} h_{M}^{\to} (μ) (b_{j}) \leq h_{M}^{\to} (μ) (\underset{j \in J}{⋁} b_{j})$ .

Let b ∈ B and q ∈ Q, $h_{M}^{\to} (μ) (b) = \underset{b = h (a)}{⋁} {μ (a) : a \in A}$

$\leq \underset{q * b = q * h (a)}{⋁} {μ (q * a)}$

$= \underset{q * b = h (q * a)}{⋁} {μ (q * a)}$

$= \underset{q * b = h (z)}{⋁} {μ (z)}$

$= h_{M}^{\to} (μ) (q * b)$ Therefore $h_{M}^{\to} (μ)$ is an M-fuzzy Q-submodule of (B, ≤ , ∗).

□

Proposition 3.8. Let h : A ⟶ B be a Q-module homomorphism from (A, ≤ , ∗) into (B, ≤ , ∗). If ν is an M-fuzzy Q-submodule of (B, ≤ , ∗), then $h_{M}^{\leftarrow} (ν)$ is an M-fuzzy Q-submodule of (A, ≤ , ∗).

Proof.

Let {a_i : i ∈ I} ⊆ A. Then $h_{M}^{\leftarrow} (ν) (⋁_{i \in I} a_{i}) = ν (h (⋁_{i \in I} a_{i}))$

= ν (⋁ _i∈Ih (a_i)) ≥ ⋀ _i∈Iν (h (a_i))

$= ⋀_{i \in I} h_{M}^{\leftarrow} (ν) (a_{i})$ .

Let a ∈ A and q ∈ Q, $h_{M}^{\leftarrow} (ν) (q * a) = ν (h (q * a))$

= ν (q ∗ h (a))

≥ν (h (a))

$= h_{M}^{\leftarrow} (ν) (a)$ .

Hence

h_{M}^{\leftarrow} (ν)

is an M-fuzzy Q-submodule of A.□

4 M-fuzzy module nuclei

In this section, we shall introduce the concept of M-fuzzy module nucleus, and we shall investigate and characterize some basic properties of it. The relationships between such notion and M-fuzzy Q-modules will be discussed.

Definition 4.1. Let (Q, ≤ , ⊗) ∈ |Quant|, (M, ≤ , ⊙) ∈ |SQuant| and (A, ≤ , ∗) be a (left) Q-module. A mapping j : A × M → A is called an M-fuzzy module nucleus on A if it satisfies the following conditions: For all q ∈ Q, a, b ∈ A and α, β ∈ M,

a≤ j (a, α) ;

j (a, α) ≤ j (b, β), if a ≤ b, and α ≤ β.

q ∗ j (a, α) ≤ j (q ∗ a, α).

j (a, α) = j (j (a, α) , α).

Example 4.2. Let Q = ([0, 1] , ≤ , ∧), M = ((0, 1], ≤, ∧) and X = {a, b, c}. By Example 2.8, we know that [0, 1] ^X is a Q-module. Let j : Q^X × M ⟶ Q^X by an operator, defined by j (A, α) (a) = ⋁ _b∈X (R (a, b) ∧ A (b)),

where R : X × X ⟶ [0, 1] is a Q-relation defined as follows:

R	a	b	c
a	0.5	0.4	0.3
b	0.3	0.8	0.1
b	0.5	0.5	0.6

and A ∈ [0, 1] ^X defined by A (a) =0.1, A (b) =0.7, A (c) =0.3 and α = 0, 5.

We show that j is an M-fuzzy module nucleus as follows:

j (A, 0.5) (a) =0.4 > 0.1 = A (a), j (A, 0.5) (b) =0.7 = A (b) and j (A, 0.5) (c) =0.5 > 0.3 = A (c).

If A ≤ B, and α ≤ β, for B : X → [0, 1] with B (a) =0.5, B (b) =0.9, B (c) =0.6 and β = 0, 7.

j (A, 0.5) (a) =0.4 < 0.5 = j (B, 0.7) (a),

j (A, 0.5) (b) =0.7 < 0.8 = j (B, 0.7) (b),

j (A, 0.5) (c) =0.5 < 0.6 = j (B, 0.7) (c).

For q = 0.2, we have

q ∧ j (A, 0.5) (a) =0.2 = j (q ∧ A, 0.5) (a),

q ∧ j (A, 0.5) (b) =0.2 = j (q ∧ A, 0.5) (b),

q ∧ j (A, 0.5) (c) =0.2 = j (q ∧ A, 0.5) (c).

j (A, 0.5) (a) =0.4 = j (j (A, 0.5) , 0.5) (a),

j (A, 0.5) (b) =0.7 = j (j (A, 0.5) , 0.5) (b),

j (A, 0.5) (c) =0.5 = j (j (A, 0.5) , 0.5) (c).

Proposition 4.3. Let (Q, ≤ , ⊗) ∈ |Quant|, (A, ≤ , ∗) be a (left) Q-module and μ : A → M be an M-fuzzy Q-submodule of (A, ≤ , ∗). A mapping $j_{μ} : A \times M \to A$ defined by the equality $j_{μ} (a, α) = \land {x \in A : x \geq a, μ (x) \geq α}, \forall a \in A, α \in M,$ is an M-fuzzy module nucleus on A.

Proof. Let μ : A → M be an M-fuzzy Q-submodule of A. To prove that the map $j_{μ} : A \times M \to A$ defined by the equality $j_{μ} (a, α) = \land {x \in A : x \geq a, μ (x) \geq α}, \forall a \in A, α \in M,$ is an M-fuzzy module nucleus on A, we will prove the conditions (J1 - J4)

By definition of j_μ, we have j_μ (a, α) = ∧ { x ∈ A: x ≥ a, μ (x) ≥ α} ≥ a .

So, j_μ (a, α) ≥ a .

If a ≤ b and α ≤ β, then

j_μ (b, β) = ∧ {x ∈ A: x ≥ b, μ (x) ≥ β }, ≥ ⋀ {x ∈ A : x ≥ b ≥ a, μ (x) ≥ β ≥ α} , = j_μ (a, α).

Hence, j_μ (a, α) ≤ j_μ (b, β).

$\begin{array}{l} j_{μ} (q * a, α) = \land {x \in A : x \geq q * a, μ (x) \geq α} \\ \geq q * \land {x \in A : x \geq a, μ (x) \geq α} \\ = q * j_{μ} (a, α) . \end{array}$

Hence, $j_{μ} (q * a, α) \geq q * j_{μ} (a, α) .$

Since j_μ (a, α)∈ A and j_μ (j_μ (a, α), α)=∧ { x ∈ A: x ≥ j_μ (a, α), μ (x) ≥ α}, we have that μ (x) ≥ μ (j_μ (a, α)) ≥ α. Then, putting x = j_μ (a, α), we have j_μ (j_μ (a, α), α)=∧ j_μ (a, α) and this implies j_μ (a, α)≥ j_μ (j_μ (a, α), α). Also, from (j₁), we have that j_μ (a, α)≤ j_μ (j_μ (a, α), α). Then the equality holds.

□

Proposition 4.4. Let (Q, ≤ , ⊗) ∈ |Quant|, (A, ≤ , ∗) be a (left) Q-module and given an M-fuzzy module nucleus j : A × M ⟶ A. An M-fuzzy set $μ_{j} \in M^{A}$ which defined by $μ_{g} (a) = ⋁ {a \in M : g (a, α) \geq a, a \in A},$ is an M-fuzzy Q-submodule of (A, ≤ , ∗).

Proof. Let j : A × M ⟶ A be an M-fuzzy module nuclei on (A, ≤ , ∗). We need to show that μ_j is an M-fuzzy Q-submodule of (A, ≤ , ∗). To this end.

For a family of {S_i : i ∈ I} ⊆ A, we have μ_j (⋁ _i∈IS_i) = ⋁ {α ∈ M : j (⋁ _i∈IS_i, α) ≥ ⋁ _i∈IS_i}, = ⋁ {α ∈ M : ⋁ _i∈Ij (S_i, α) ≥ ⋁ _i∈IS_i}, $\geq \land_{i \in I} \lor {a \in M : j (S_{i}, a) \geq S_{i}}$ = ∧ _i∈Iμ_j (S_i).

Then, μ_j (⋁ _i∈IS_i) ≥ ∧ _i∈Iμ_j (S_i).

For a ∈ A, q ∈ Q and α ∈ M. μ_j (q ∗ a) = ⋁ {α ∈ M : j (q ∗ a, α) ≥ q ∗ a}, = ⋁ {α ∈ M : q ∗ j (a, α) ≥ q ∗ a}, ≥ ⋁ {α ∈ M : j (a, α) ≥ a}, = μ_j (a).

Then, μ_j (q ∗ a) ≥ μ_j (a).

□

Lemma 4.5. Let (Q, ≤ , ⊗) ∈ |Quant| and (A, ≤ , ∗) be a (left) Q-module. An M-fuzzy module nucleus j : A × M → A, on (A, ≤ , ∗), satisfies the following conditions: For all a, b ∈ A, q ∈ Q and α ∈ M.

j (q ∗ j (a, α) , α) ≤ j (q ∗ a, α).

j (a, α) ⇝ b = j (a, α) ⇝ j (b, α).

j (q ↠a, α) ≤ q ↠j (a, α).

Proof.

Let j : A × M → A be an M-fuzzy module nucleus on (A, ≤ , ∗), then q ∗ j (a, α) ≤ j (q ∗ a, α), and so we have j (q ∗ j (a, α) , α) ≤ j (j (q ∗ a, α) , α) = j (q ∗ a, α) , hence, j (q ∗ j (a, α) , α) ≤ j (q ∗ a, α).

Let a, b ∈ A, α ∈ M, the inequality j (a, α) ⇝ j (b, α) ≤ j (a, α) ⇝ b follows from the fact that b ≤ j (b, α) by Lemma 2.9(6). For the reverse inequality, we have [j (a, α) ⇝ b] ∗ j (b, α) ≤ j ([j (a, α) ⇝ b] ∗ b, α) ≤ j (j (a, α) , α) = j (a, α). By Lemma 2.9(1) and the monotonicity of j, we have j (a, α) ⇝ b ≤ j (a, α) ⇝ j (b, α). Hence, j (a, α) ⇝ b = j (a, α) ⇝ j (b, α).

Let a ∈ A and q ∈ Q, we have q ∗ j (q ↠a, α) ≤ j (q ∗ (q ↠a) , α) ≤ j (a, α) by Lemma 2.9(2). So, j (q ↠a, α) ≤ q ↠j (a, α).

□

Lemma 4.6. Let (Q, ≤ , ⊗) ∈ |UnQuant| and (A, ≤ , ∗) be a unital (left) Q-module. An M-fuzzy module nucleus j : A × M → A on (A, ≤ , ∗), satisfies the following conditions:

j(b,α)⇝ j(a,α) =j(b,α) ⇝ a for all a, b ∈ A, α ∈ M.

j(a,β)⇝ j(a,α) =j(a,β)⇝ a for all a ∈ A, α, β ∈ M.

Proof. (1) We have, for all q ∈ Q, q ≤ j (b, α) ⇝ a iff q ∗ a ≤ j (b, α) iff j(q * a,α)≤ j(j(b,α),α) =j(b,α) iff q ∗ j (a, α) ≤ j (b, α) iff q ≤ j (b, α) ⇝ j (a, α). So, j(b,α) ⇝ a=j(b,α)⇝ j(a,α).

(2) Can be proved similarly. □

Proposition 4.7. Let (Q, ≤ , ⊗) ∈ |UnQuant| and (A, ≤ , ∗) be a unital (left) Q-module. For all a, b ∈ A, α, β ∈ M, a mapping j: A × M → A with j(b,α)⇝ j(a,α) =j(b,α) ⇝ a, or j(a,β)⇝ j(a,α) =j(a,β)⇝ a,

is an M-fuzzy module nucleus on (A, ≤ , ∗).

Proof. For all a, b ∈ A, α, β ∈ M, suppose that j(b,α)⇝ j(a,α) =j(b,α) ⇝ a or j(a,β)⇝ j(a,α) =j(a,β)⇝ a. By the unital assumption e_Q, we have that:

e_Q ∗ j (a, α) ≤ j (a, α) ⇔ e_Q ≤ j (a, α) ⇝ j (a, α) ≤ j (a, α) ⇝ a ⇔ e_Q ∗ a ≤ j (a, α) ⇔ a ≤ j (a, α).

If a ≤ b and α ≤ β, then e_Q ∗ a ≤ b ≤ j (b, β) ⇔ e_Q ≤ j (b, β) ⇝ a ⇔ e_Q ≤ j (b, β) ⇝ j (a, α) ⇔ e_Q ∗ j (a, α) ≤ j (b, β) ⇔ j (a, α) ≤ j (b, β).

For q ∈ Q and a ∈ A. From (1), we have:

q ∗ a ≤ j (q ∗ a, α) ⇔ q ≤ j (q ∗ a, α) ⇝ a ⇔ q ≤ j (q ∗ a, α) ⇝ j (a, α)

⇔ q ∗ j (a, α) ≤ j (q ∗ a, α).

Now, we show that j is idempotent, to this end e_Q ∗ j (a, α) ≤ j (a, α) ⇔ e_Q ≤ j (a, α) ⇝ j (a, α)

⇔ e_Q ≤ j (a, α) ⇝ j (j (a, α) , α) ⇔ e_Q ∗ j (j (a, α) , α) ≤ j (a, α) by Eq (1) ⇔ j (j (a, α) , α) ≤ j (a, α) ≤ j (j (a, α) , α), i.e., j(a,α)= j(j(a,α),α).

So it follows that j is a an M-fuzzy module nucleus on (A, ≤ , ∗).□

5 M-fuzzy module conuclei

In this section, we shall introduce and study the concept of an M-fuzzy Q-module conucleus. The relationships between such notion and M-fuzzy Q-modules will be discussed.

Definition 5.1. Let (Q, ≤ , ⊗) ∈ |Quant|, (M, ≤ , ⊙) ∈ |SQuant| and (A, ≤ , ∗) be a (left) Q-module. A mapping g : A × M → A is called an M-fuzzy module conucleus on (A, ≤ , ∗) if it satisfies the following conditions: For all q ∈ Q, a, b ∈ A and α, β ∈ M,

g (a, α) ≤ a

g (a, α) ≤ g (b, β) whenever a ≤ b, β ≤ α.

q ∗ g (a, α) ≤ g (q ∗ a, α).

g (a, α) = g (g (a, α) , α).

Proposition 5.2. Let (Q, ≤ , ⊗) ∈ |Quant|, (A, ≤ , ∗) be a (left) Q-module and μ : A ⟶ M be an M-fuzzy submodule of (A, ≤ , ∗). A mapping g_μ : A × M → A defined by the equality g_μ (a, α) = ⋁ {x ∈ A : x ≤ a, μ (x) ≥ α} ∀a ∈ A and α ∈ M, is an M-fuzzy module conucleus on (A, ≤ , ∗).

Proof. Let μ : A → M be an M-fuzzy Q-submodule of (A, ≤ , ∗). To prove that the map g_μ : A × M → A defined by the equality g_μ (a, α) = ⋁ {x ∈ A : x ≤ a, μ (x) ≥ α} ∀a ∈ A and α ∈ M, is an M-fuzzy module conucleus on (A, ≤ , ∗), we will prove the conditions (g₁ - g₄)

By definition of g_μ, we have

g_μ (a, α) = ⋁ {x ∈ A : x ≤ a, μ (x) ≥ α} ≤ a.

Then, g_μ (a, α) ≤ a.

For a, b ∈ A and α, β ∈ M with a ≤ b, β ≤ α, we have

g_μ (a, α) = ⋁ {x ∈ A : x ≤ a, μ (x) ≥ α} ∀a ∈ A, α ∈ M, ≤ ⋁ {x ∈ A : x ≤ b, μ (x) ≥ β} ∀b ∈ A, β ∈ M, = g_μ (b, β).

So, g_μ (a, α) ≤ g_μ (b, β).

g_μ (q ∗ a, α) = ⋁ {x ∈ A : x ≤ q ∗ a, μ (x) ≥ α} ≥ q ∗ ⋁ {x ∈ A : x ≤ a, μ (x) ≥ α} = q ∗ g_μ (a, α) .

Hence, g_μ (q ∗ a, α) ≥ q ∗ g_μ (a, α) .

Since g_μ (a, α) ∈ A and g_μ (g_μ (a, α) , α) = ⋁ {x ∈ A : x ≤ g (a, α) , μ (x) ≥ α}, we have that μ (g_μ (a, α)) ≥ μ (x) ≥ α. Then, putting x = g_μ (a, α), we have g_μ (g_μ (a, α) , α) = ⋁ g_μ (a, α) and this implies g_μ (a, α) ≤ g_μ (g_μ (a, α) , α). Also, from (g₁), we have that g_μ (g_μ (a, α) , α) ≤ g_μ (a, α). Then the equality holds.

□

Proposition 5.3. For (Q, ≤ , ⊗) ∈ |Quant|, (A, ≤ , ∗) be a (left) Q-module and given an M-fuzzy module conucleus g : A × M ⟶ A, an M-fuzzy set μ_g ∈ M^A which defined by μ_g (a) = ⋁ {α ∈ M : g (a, α) ≥ a, a ∈ A}, is an M-fuzzy Q-submodule on (A, ≤ , ∗).

Proof. Let g : A × M ⟶ A be an M-fuzzy module conuclei on (A, ≤ , ∗). We need to show that μ_g is an M-fuzzy Q-submodule on (A, ≤ , ∗). To this end.

For a family of {S_i : i ∈ I} ⊆ A, we have μ_g (⋁ _i∈IS_i) = ⋁ {α ∈ M : g (⋁ _i∈IS_i, α) ≥ ⋁ _i∈IS_i}, = ⋁ {α ∈ M : ⋁ _i∈Ig (S_i, α) ≥ ⋁ _i∈IS_i}, ≥ ∧ _i∈I ⋁ {α ∈ M : g (S_i, α) ≥ S_i}

= ∧ _i∈Iμ_g (S_i). Then, μ_g (⋁ _i∈IS_i) ≥ ∧ _i∈Iμ_g (S_i).

For a ∈ A, q ∈ Q and α ∈ M. μ_g (q ∗ a) = ⋁ {α ∈ M : g (q ∗ a, α) ≥ q ∗ a}, = ⋁ {α ∈ M : q ∗ g (a, α) ≥ q ∗ a}, ≥ ⋁ {α ∈ M : g (a, α) ≥ a}, = μ_g (a).

Then, μ_g (q ∗ a) ≥ μ_g (a).□

6 Relations between M-FQMod and R (L, M) -QFTop

In this section, we aim to set up a dual adjunction between the category R (L, M)-QFTop (resp., R (L, M)-SQFTop of enriched (L, M)-quasi-(resp., strong (L, M)-quasi-) fuzzy topological spaces and the category M-FQMod (resp., M-SFQMod) of M-fuzzy (resp., strong M-fuzzy) Q-modules.

As given in Example 3.3, can define the functor S : R (L, M)-QFTop→M-FQMod^op by $S (X, T) = (L^{X}, T)$ and $S (f : (X, T) \to (Y, V)) = (f_{L}^{\leftarrow})^{op} : (L^{X}, T) \to (L^{Y}, V)$ .

We now construct a functor T:M-FQMod^op → R (L, M)-QFTop.

For (M, ≤ , ⊙) ∈ |Quant|. If h : (A₁, μ₁) → (A₂, μ₂) is an M-FQMod morphism, then, by Proposition 3.7, we have that $h_{M}^{\to} (μ_{1})$ is an M-fuzzy Q-submodule of A₂, where $h_{M}^{\to} : M^{A_{1}} \to M^{A_{2}}$ is defined by $h_{M}^{\to} (μ_{1}) (y) = ⋁ {μ_{1} (x) : μ_{1} \in M^{A_{1}}$ , y ∈ A₂, h (x) = y}. Let L ∈ |Quant|. For every Q-module A, denote by St (A) the set of all Q-module morphisms from A to L, i.e.,St (A) = {h : A → L : h ∈ |Q - Mod|}, and by 〈A〉, we denote the evaluation map 〈A〉 : A → L^St(A), i.e., 〈A〉 (a) (h) = h (a), ∀a ∈ A, h ∈ St (A)

Lemma 6.1. 〈A〉 : A → L^St(A) is a Q-module morphism. Proof.

Let {a_i : i ∈ I} ⊆ A, then

〈A〉 (⋁ _i∈Ia_i) (h) = h (⋁ _i∈Ia_i)

≥ ⋀ _i∈Ih (a_i) = ⋀ _i∈I〈A〉 (a_i) (h).

Let q ∈ Q, a ∈ A, then 〈A〉 (q ∗ a) (h) = h (q ∗ a) = q ∗ h (a) = q ∗ 〈A〉 (a) (h).□

As a consequence of Proposition 3.7 and the above lemma, we have the following result:

Lemma 6.2. For an M-fuzzy Q-submodule μ ∈ M^A of a Q-module (A, ≤ , ∗) and the Q-module morphism 〈A〉 : A → L^St(A), we have that $〈 A 〉_{M}^{\to} (μ)$ is an M-fuzzy Q-submodule of L^St(A).

According to Definition 2.6 and the above lemma, we use the map $〈 A 〉_{M}^{\to} (μ) : L^{St (A)} \to M$ as an (L, M)-fuzzy subbase for an (L, M)-quasi-fuzzy topology $T_{(A, μ)} : L^{St (A)} \to M$ . The pair $(St (A), T_{(A, μ)})$ is an enriched (L, M)-quasi-fuzzy topological space.

Lemma 6.3. T : M - FQMod^op → R (L, M)-QFTop defined by T (ϕ : (A₁, μ₁) ⟶ (A₂, μ₂)) = T (ϕ) : T (A₁, μ₁) ⟶ T (A₂, μ₂) is a functor, where $T (A, μ) = (St (A), T_{(A, μ)})$ and T (ϕ) : St (A₁) ⟶ St (A₂) is a function, define by T (ϕ) (h) = h ∘ ϕ^op, ∀h ∈ St (A₁).

Proof. As a consequence of the above statements, one can easily note that T is an object function from M-FQMod^op to R (L, M)-QFTop and it preserves composition and identities. Now, we only need to show that T is a morphism function, i.e. for every M-FQMod^op-morphism ϕ : (A₁, μ₁) ⟶ (A₂, μ₂), T (ϕ) : T (A₁, μ₁) ⟶ T (A₂, μ₂) is an R(L,M)-QFTop-morphism, thus the inequality $T_{(A_{2}, μ_{2})} (K) \leq T_{(A_{1}, μ_{1})} (T_{L}^{\leftarrow} (K))$ (3) holds for every K ∈ L^St(A₂). To prove Eq (3), we first show that for every a₂ ∈ A₂ with K = 〈A₂〉 (a₂) $T_{L}^{\leftarrow} (K) = 〈 A_{1} 〉 (ϕ^{op} (a_{2})) .$ (4) Given a₂ ∈ A₂ with K = 〈A₂〉 (a₂), then we have h₂ ∈ St (A₂) and K (h₂) = 〈A₂〉 (a₂) (h₂) = h₂ (a₂). Since T (ϕ) (h₁) = h₁ ∘ ϕ^op ∈ St (A₂), for every h₁ ∈ St (A₁) and by the definition of $T (ϕ)_{L}^{\leftarrow}$ , we obtain Eq (3) from the equality: $[T (ϕ)_{L}^{\leftarrow} (K)] (h_{1}) = K (T (ϕ) (h_{1}))$ = [T (ϕ) (h₁)] (a₂) = h₁ ∘ ϕ^op (a₂) = h₁ (ϕ^op (a₂)) = [〈A₁〉 (ϕ^op (a₂))] (h₁). On the other hand, let ϕ^op : (A₂, μ₂) ⟶ (A₁, μ₁), is an M-FQMod-morphism, we have μ₂ (a₂) ≤ μ₁ (ϕ^op (a₂)). This inequality together with Eq (3) allow us to put down $T_{(A_{2}, μ_{2})} (K) = ⋁ μ_{2} (a_{2})$ , for K = 〈A₂〉 (a₂) ≤ ⋁ μ₂ (a₂), for $T (ϕ)_{L}^{\leftarrow} (K) = 〈 A_{1} 〉 (ϕ^{op} (a_{2}))$ ≤ ⋁ μ₁ (ϕ^op (a₂)), for $T (ϕ)_{L}^{\leftarrow} (K) = 〈 A_{1} 〉 (ϕ^{op} (a_{2}))$ ≤ ⋁ μ₁ (a₁), for $T (ϕ)_{L}^{\leftarrow} (K) = 〈 A_{1} 〉 (a_{1})$ $= T_{(A_{1}, μ_{1})} (T (ϕ)_{L}^{\leftarrow} (K))$ .□

Proposition 6.4. S is left adjoint to T.

Proof. For all $(X, T) \in R (L, M)$ -QFTop, the function $η_{(X, T)} : X \to St (L^{X})$ define by $η_{(X, T)} (x) = π_{x}$ ; ∀x ∈ X where π_x : L^X → M is the x-th projection function.

It is easily to see for every $(X, T) \in R (L, M)$ -QFTop and every (A, μ) ∈ M - FQMod,

$η_{(X, T)} : (X, T) \to T S (X, T)$ and

ɛ_(A,μ) = 〈A〉^op : ST (A, μ) → (A, μ).

are, respectively, an R (L, M)-QFTop-morphism and an M-FQMod^op-morphism. In addition to,

$η = (η_{(X, T)}) : {id}_{R (L, M) - QFTop} \to T S$ and

ɛ = (ɛ_(A,μ)) = : ST → id_M-FQMod^op.

are natural transformations making (η, ɛ) : S ⊣ T :M-FQMod^op → R (L, M)-QFTop

an adjoint situation, and hence S ⊣ T.

7 Conclusion

This paper is focused at studying fuzziness within the context of the Q-module. In this paper, we have studied the notion of M-fuzzy Q-submodule of a left Q-module. The notions of M-fuzzy module nuclei and conuclei on a left Q-module were introdused and studied. Furthermore, we have discussed the relationships between such notions and M-fuzzy Q-submodule. Finally, the relationship between the category of M-fuzzy Q-modules and the category of (L, M)-quasi-fuzzy topological spaces is discussed. The application of roughness and intuitionistic fuzzy roughness can be easily observed from [18 , 31].

In a future work, we intend to used the results of this paper to propose some applications of L-fuzzy rough sets within the context of the Q-module.

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