Alexandrov L -quasi-G-filters and Alexandrov L -fuzzy pre-uniformities

Abstract

This paper presents the idea of Alexandrov L-quasi-G-filter space and examines its relationship with L-fuzzy relations. It is proved that every Alexandrov L-quasi-G-filter induces an L-fuzzy relation and vice-versa. By identifying certain functors, the study has brought out the connections between the categories of Alexandrov L-quasi-G-filter spaces and Alexandrov L-fuzzy pre-uniform spaces. Further, the study has explored and thereby establishes the scope of Alexandrov L-quasi-G-filter spaces in mathematical modeling and decision-making processes.

Keywords

Lattice category functor fuzzy relation

1 Introduction

Several authors have investigated the idea of fuzzy uniformity and presented various kinds of fuzzy uniformities [4 , 25]. In [17], Kim et al. explored L-uniformizable spaces and looked into the connection between L-fuzzy quasi-uniformity and L-fuzzy cotopology. The concept of fuzzy proximities was introduced by Katsaras [14] in 1979, and he also examined the links among fuzzy proximities, fuzzy topologies, and fuzzy uniformities in [15]. Again, Kim et al. presented the idea of Alexandrov L-fuzzy pre-uniform spaces in [16] and investigated the relationships across Alexandrov L-fuzzy topological spaces, L-lower approximation spaces and L-upper approximation spaces.

It was in 1979 that Lowen [21] introduced the idea of filters in I^X. Subsequently, Höhle and Šostak [5] developed the notion of L-filters and stratified L-filters on a complete quasimonoidal lattice. In 2012, Yao [24] introduced the notion of LM-filters and defined LM-fuzzy generalized convergence spaces. Later, Jäger [7] introduced the concept of stratified LM-filters as a generalization of stratified L-filters.

Owing much to the application potential in mathematical modeling and decision-making, the authors introduced the notion of LM-G-filters as a generalization of LM-filters in [8] and studied catalyzed LM-G-filter spaces in [9]. In [10] and [11], the authors inquired into the categorical connections of L-G-filters with L-filters, L-interior operators, L-fuzzy preproximities, L-fuzzy grills, L-closure operators and L-fuzzy cotopologies. The concepts of weak and strong LM-G-filter spaces were introduced in [12]. Afterwards, images of LM-G-filter spaces and LM-G-filterbases induced by certain functions have been investigated in [13].

This paper introduces the concept of Alexandrov L-quasi-G-filter spaces and investigates the categorical properties of these spaces. In particular, the study attempts to answer the following questions:

Are the notions of Alexandrov L-quasi-G-filters and L-G-filters completely independent?

How the new concept is adapted to the existing theory?

Does there exist any connection between Alexandrov L-quasi-G-filters and Alexandrov L-fuzzy pre-uniformities?

What is the application potential of Alexandrov L-quasi-G-filter spaces?

The paper is organized as follows: Section 1 gives an introduction to the study. Section 2 provides certain definitions and known results required for the subsequent development of the study. The idea of Alexandrov L-quasi-G-filters is introduced in section 3. This section answers question (i) above and shows that the notion of Alexandrov L-quasi-G-filter is independent from that of L-G-filters. Besides, as every L-fuzzy relation gives rise to an Alexandrov L-quasi-G-filter and vice-versa, the new concept enriches the existing theory in a productive way.

The study undertaken in section 4 with a view to answer question (iii) above identifies functors between the categories of Alexandrov L-quasi-G-filter spaces and Alexandrov L-fuzzy pre-uniform spaces. As a result, we raise an open question too. Section 5 reveals the relevance and application potential of the newly introduced concept of Alexandrov L-quasi-G-filter spaces in the field of mathematical modeling and optimal decision-making.

2 Preliminaries

Definition 2.1. [2] An algebra (L, ∨ , ∧ , ⊙ , → , 0, 1) is a complete residuated lattice if it satisfies the following properties:

(L, ≤ , ∨ , ∧ , 0, 1) is a complete lattice with universal upper bound 1 and the universal lower bound 0;

(L, ⊙ , 1) is a commutative monoid;

p ⊙ q ≤ r if and only if p ≤ q → r for p, q, r ∈ L.

Unless otherwise specified, in this paper L stands for a complete residuated lattice with an order reversing involution ∗ defined by p^∗ = p → 0, satisfying the law of double negation i.e. p^** = p and p ⊕ q is defined by p ⊕ q = p^∗ → q.

Remark 2.2. Complete locally finite BL-algebra and complete locally finite MV-algebra are complete residuated lattices with an order reversing involution ∗ satisfying the law of double negation.

Lemma 2.3. [2 , 23] Let L be a complete residuated lattice satisfying law of double negation. Then for each p, q, r, p_i, q_i, s ∈ L, the following properties hold.

1 → p = p, 0 ⊙ p = 0 and p ≤ q if and only if p → q = 1.

If q ≤ r, then p ⊙ q ≤ p ⊙ r, p ⊕ q ≤ p ⊕ r, p → q ≤ p → r and r → p ≤ q → p.

p ⊙ q ≤ p, q, p ⊙ q ≤ p ∧ q, p ⊕ q ≥ p, q, p ⊕ q ≥ p ∨ q.

p ⊙ p^* = 0, 1^* = 0, 0^* = 1.

$(⋀_{i \in Γ} q_{i})^{*} = ⋁_{i \in Γ} q_{i}^{*}, (⋁_{i \in Γ} q_{i})^{*} = ⋀_{i \in Γ} q_{i}^{*} .$

p → (⋀ _i∈Γq_i) = ⋀ _i∈Γ (p → q_i) and (⋁ _i∈Γp_i) → q = ⋀ _i∈Γ (p_i → q).

p ⊙ (⋀ _i∈Γq_i) ≤ ⋀ _i∈Γ (p ⊙ q_i) and p ⊕ (⋁ _i∈Γq_i) ≥ ⋁ _i∈Γ (p ⊕ q_i).

p ⊙ (⋁ _i∈Γq_i) = ⋁ _i∈Γ (p ⊙ q_i) and (⋀ _i∈Γp_i) ⊕ q = ⋀ _i∈Γ (p_i ⊕ q).

p ⊙ q = (p → q^*) ^*, p → q = q^* → p^* and (p ⊕ q) ^* = p^* ⊙ q^*.

(p → q) ⊙ (r → s) ≤ (p ⊕ r) → (q ⊕ s) and (p → q) ⊙ (r → s) ≤ (p ⊙ r) → (q ⊙ s).

(p ⊙ q) → r = p → (q → r) = q → (p → r) .

Notation 2.4. In this paper X and Y stand for non-empty ordinary sets. All algebraic operations on L can be extended pointwise to L^X as A ≤ B if and only if A (x) ≤ B (x) and (A ⊙ B) (x) = A (x) ⊙ B (x) for all x ∈ X. For all β ∈ L, (β → A) (x) = β → A (x) , (β ⊙ A) (x) = β ⊙ A (x) and β_X ∈ L^X is defined by β_X (x) = β for all x ∈ X. Similarly all algebraic operations on L can be extended pointwise to L^X×X. ⊤_x ∈ L^X, ⊤_(x,z) ∈ L^X×X are defined by $⊤_{x} (z) = {\begin{matrix} 1 & if z = x, \\ 0 & otherwise \end{matrix} and$ $⊤_{(x, z)} (r, s) = {\begin{matrix} 1 & if (r, s) = (x, z), \\ 0 & otherwise . \end{matrix}$

Definition 2.5. [20] Let g : X → Y be an ordinary mapping. The L-fuzzy mapping g^→ : L^X → L^Y is defined by g^→ (A) (y) = ⋁ {A (x) |x ∈ X, g (x) = y}, ∀A ∈ L^X, ∀y ∈ Y. The L-fuzzy reverse mapping g^← : L^Y → L^X is defined by g^← (B) (x) = B (g (x)), ∀B ∈ L^Y, ∀x ∈ X.

Similarly, the L-fuzzy mapping (g × g) ^→ : L^X×X → L^Y×Y is defined by (g × g) ^→ (u) (y₁, y₂) = ⋁ {u (x₁, x₂) |x₁, x₂ ∈ X, g (x₁) = y₁, g (x₂) = y₂}, ∀u ∈ L^X×X, ∀y₁, y₂ ∈ Y. The L-fuzzy reverse mapping (g × g) ^← : L^Y×Y → L^X×X is defined by g^← (v) (x₁, x₂) = v (g (x₁) , g (x₂)), ∀v ∈ L^Y×Y, ∀x₁, x₂ ∈ X.

Theorem 2.6. [20] Let f : X → Y and g : Y → Z be an ordinary mappings. Then,

g^→f^→ = (gf) ^→.

f^←g^← = (gf) ^←.

Note: For the notions of category theory, the readers can refer [1].

Definition 2.7. [19] A map R : X × X → L is called an L-fuzzy relation on X. An L-fuzzy relation is called reflexive if R (x, x) =1 for all x ∈ X.

Definition 2.8. [19] Let R_X and R_Y be L-fuzzy relations on X and Y, respectively. A map g : (X, R_X) → (Y, R_Y) is called an order preserving map if R_X (x, z) ≤ R_Y (g (x) , g (z)) for all x, z ∈ X.

Lemma 2.9. [2] Let a binary map S : L^X × L^X → L be defined by S (P, Q) = ⋀ _x∈X (P (x) → Q (x)). Then, for each P, Q, R, Q_j ∈ L^X the following properties hold.

S (P, Q) ⊙ S (Q, R) ≤ S (P, R).

If P ≤ Q, then S (R, P) ≤ S (R, Q) and S (P, R) ≥ S (Q, R).

S (P, β → Q) = β → S (P, Q).

S (P, ⋀ _j∈JQ_j) = ⋀ _j∈JS (P, Q_j).

For the rest of the paper, S represents the map defined in the above Lemma.

Definition 2.10. [8] An L-G-filter on a set X is defined to be a mapping G : L^X → L satisfying the following conditions:

G (1_X) =1;

For every A, B ∈ L^X such that A ≤ B, G (A) ≤ G (B);

For every A, B ∈ L^X, G (A ⊙ B) ≥ G (A) ⊙ G (B).

The pair (X, G) is called an L-G-filter space.

Definition 2.11. [16] An Alexandrov L-fuzzy pre-uniformity is a map $V : L^{X \times X} \to L$ with the following properties:

(AU1) There exists w ∈ L^X×X with $V (w) = 1$ ;

(AU2) If w ≤ v, then $V (w) \leq V (v)$ ;

(AU3) $V (⋀_{j \in J} w_{j}) = ⋀_{j \in J} V (w_{j})$ for each arbitrary family {w_j ∈ L^X×X|j ∈ J};

(AU4) $V (w) \leq ⋀_{x \in X} w (x, x)$ ;

(AU5) $V (β \to w) = β \to V (w)$ for each β ∈ L and w ∈ L^X×X.

The pair $(X, V)$ is called an Alexandrov L-fuzzy pre-uniform space.

Definition 2.12. Given two Alexandrov L-fuzzy pre-uniform spaces, $(X, V_{1})$ and $(Y, V_{2})$ , a map $g : (X, V_{1}) \to (Y, V_{2})$ is said to be L-fuzzy uniformly continuous map if $V_{2} (w) \leq V_{1} ((g \times g)^{\leftarrow} (w))$ for each w ∈ L^Y×Y.

Notation 2.13. In this paper, L-R denotes the category of L-fuzzy relations with order preserving maps as morphisms and AL-PU denotes the category of Alexandrov L-fuzzy pre-uniform spaces with uniformly continuous maps as morphisms.

3 Alexandrov L-quasi-G-filter spaces and L-fuzzy relations

This section introduces the notion of Alexandrov L-quasi-G-filter spaces, investigates the relationship between Alexandrov L-G-filters and L-fuzzy relations and identifies functors between the categories of Alexandrov L-quasi-G-filters and L-fuzzy relations. It is proved that every Alexandrov L-quasi-G-filter induces an L-fuzzy relation and vice-versa.

Definition 3.1. A map G : L^X → L is called an Alexandrov L-quasi-G-filter if the following properties hold.

(AG1) G (1_X) =1;

(AG2) G (⋀ _j∈JA_j) = ⋀ _j∈JG (A_j) for each arbitrary family {A_j ∈ L^X|j ∈ J};

(AG3) G (β → A) = β → G (A) for every β ∈ L and A ∈ L^X.

The pair (X, G) is called an Alexandrov L-quasi-G-filter space. In addition, if G satisfies G (A ⊙ B) ≥ G (A) ⊙ G (B) for all A, B ∈ L^X, then G becomes an L-G-filter.

Remark 3.2. Every Alexandrov L-quasi-G-filter need not be an L-G-filter and vice-versa. For example, let X = {x, z}, L = ([0, 1] , ⊙ , → , * , 0, 1) be the complete residuated lattice with p ⊙ q = max {0, p + q - 1} , p → q = min {1 - p + q, 1} and p^* = 1 - p. Then G : L^X → L defined by G (A) = (0.5 → A (x)) ∧ (0.7 → A (z)) for all A ∈ L^X is an Alexandrov L-quasi-G-filter on X. For A ∈ L^X defined by A (x) =0.6, A (z) =0.4 and B ∈ L^X defined by B (x) =0.8, B (z) =0.9, A ⊙ B ∈ L^X is obtained as (A ⊙ B) (x) =0.4, (A ⊙ B) (z) =0.3. By definition of G, G (A) =0.7, G (B) =1 and G (A ⊙ B) =0.6. Therefore, G (A ⊙ B) ngeqG (A) ⊙ G (B). Hence G is not an L-G-filter.

Let $\hat{G} : L^{X} \to L$ be defined by

$\hat{G} (A) = {\begin{matrix} 1 & if A = 1_{X}, \\ 0.7 & if A > B, \\ 0.4 & otherwise \end{matrix}$ where B ∈ L^X is defined by B (x) =0.4, B (z) =0.8. Clearly $\hat{G}$ is an L-G-filter on X. For A₁ ∈ L^X defined by A₁ (x) =0.4, A₁ (z) =0.9 and A₂ ∈ L^X defined by A₂ (x) =0.5, A₂ (z) =0.8, A₁ ∧ A₂ = B. But $\hat{G} (A_{1}) = \hat{G} (A_{2}) = 0.7$ and $\hat{G} (B) = 0.4$ . For β = 0.9, (β → B) ∈ L^X is given by (β → B) (x) =0.5, (β → B) (z) =0.9. Therefore, by definition of $\hat{G}$ , $\hat{G} ((β \to B)) = 0.7$ and $β \to \hat{G} (B) = 0.5$ . Hence $\hat{G} ((β \to B)) \neq β \to \hat{G} (B)$ . Therefore, the conditions (AG2) and (AG3) are not satisfied by $\hat{G}$ and hence $\hat{G}$ is not an Alexandrov L-quasi-G-filter on X.

Definition 3.3. Let (X, G₁) and (Y, G₂) be Alexandrov L-quasi-G-filter spaces. A map g : (X, G₁) → (Y, G₂) is called an Alexandrov L-quasi-G-filter map if G₂ (B) ≤ G₁ (g^← (B)) for each B ∈ L^Y. A map g : (X, G₁) → (Y, G₂) is called an Alexandrov L-quasi-G-filter preserving map if G₁ (A) ≤ G₂ (g^→ (A)) for each A ∈ L^X.

Remark 3.4. By Theorem 2.6, it is easy to observe that composition of Alexandrov L-quasi-G-filter maps is an Alexandrov L-quasi-G-filter map and composition of Alexandrov L-quasi-G-filter preserving maps is an Alexandrov L-quasi-G-filter preserving map.

Remark 3.5. Let (X, G₁), (Y, G₂) and (Z, G₃) be Alexandrov L-quasi-G-filter spaces. Let f : (X, G₁) → (Y, G₂) and g : (Y, G₂) → (Z, G₃) be Alexandrov L-quasi-G-filter maps. Since the composition of ordinary mappings is associative, the composition of Alexandrov L-quasi-G-filter maps is associative by Theorem 2.6. The identity map id_X : (X, G₁) → (X, G₁) is an Alexandrov L-quasi-G-filter map. Therefore, Alexandrov L-quasi-G-filter spaces with Alexandrov L-quasi-G-filter maps as morphisms is a category.

Notation 3.6. Let AL-QG denote the category of Alexandrov L-quasi-G-filter spaces with Alexandrov L-quasi-G-filter maps as morphisms.

The following theorem associates an Alexandrov L-quasi-G-filter with a given L-fuzzy relation and thereby derives a functor from L-R to AL-QG.

Theorem 3.7. Let R ∈ L^X×X be an L-fuzzy relation on X. Define the mapping G_R : L^X → L by G_R (A) = ⋀ _x,z∈X (R (x, z) → A (x) ) for all A ∈ L^X. Then,

G_R is an Alexandrov L-quasi-G-filter on X;

If g : (X, R_X) → (Y, R_Y) is an order preserving map, then g : (X, G_{R
_X}) → (Y, G_{R
_Y}) is an Alexandrov L-quasi-G-filter map.

Proof.

We prove only (ii.). For all B ∈ L^Y, $\begin{matrix} G_{R_{Y}} (B) & = & ⋀_{y, t \in Y} (R_{Y} (y, t) \to B (y)) \\ \leq & ⋀_{x, z \in X} (R_{Y} (g (x), g (z)) \to B (g (x))) \\ \leq & ⋀_{x, z \in X} (R_{X} (x, z) \to g^{\leftarrow} (B) (x)) \\ = & G_{R_{X}} (g^{\leftarrow} (B)) . \end{matrix}$

Corollary 3.8. Let ζ : L-R→ AL-QG be defined by ζ ((X, R)) = (X, G_R). Then ζ is a functor from L-R to AL-QG.

Example 3.9. Let X = {x, z} and L = ([0, 1] , ⊙ , → , * , 0, 1) be the lattice defined in Remark 3.2. Let the L-fuzzy relation on X, R : X × X → L be defined by Table 1. Then by previous theorem, the Alexandrov L-quasi-G-filter on X, G_R : L^X → L is given by G_R (A) = (0.4 → A (x)) ∧ (0.9 → A (z)) for all A ∈ L^X.

Table 1
L-fuzzy relation R on X = {x, z}

R x z

x 0.2 0.4

z 0.9 0.7

R	x	z
x	0.2	0.4
z	0.9	0.7

Conversely, an Alexandrov L-quasi-G-filter space gives rise to an L-fuzzy relation as shown in the following theorem and establishes a functor from AL-QG to L-R.

Theorem 3.10. Let G : L^X → L be an Alexandrov L-quasi-G-filter. Then,

$G (A) = ⋀_{x \in X} (G (⊤_{x}^{*}) \oplus A (x))$ for all A ∈ L^X;

R_G ∈ L^X×X defined by

R_G (x, z) = ⋀ _A∈L^X ((G (A) ⊙ A (x)) → A (z) ) for each x, z ∈ X is a reflexive L-fuzzy relation on X;

If g : (X, G₁) → (Y, G₂) is an Alexandrov L-quasi-G-filter map, then g : (X, R_G
₁) → (Y, R_G
₂) is an order preserving map.

Proof. Since $A = ⋀_{x \in X} (⊤_{x}^{*} \oplus A (x))$ , $G (A) = ⋀_{x \in X} (G (⊤_{x}^{*}) \oplus A (x))$ for all A ∈ L^X. Proof of (ii.) is trivial. For all x, z ∈ X, $\begin{matrix} R_{G_{2}} (g (x), g (z)) \\ = ⋀_{B \in L^{Y}} ((G_{2} (B) ⊙ B (g (x))) \to B (g (z))) \\ \geq ⋀_{B \in L^{Y}} ((G_{1} (g^{\leftarrow} (B)) ⊙ g^{\leftarrow} (B) (x)) \to g^{\leftarrow} (B) (z)) \\ \geq ⋀_{A \in L^{X}} ((G_{1} (A) ⊙ A (x)) \to A (z)) \\ = R_{G_{1}} (x, z) . \end{matrix}$

Corollary 3.11. Let η : AL-QG→L-R be defined by η ((X, G)) = (X, R_G). Then η is a functor from AL-QG to L-R.

Example 3.12. Let X = {x, z} and L = ({0, 1} , ⊙ , → , * , 0, 1) be the complete residuated lattice with p ⊙ q = min {p, q} , p → q = max {1 - p, q} and p^* = 1 - p. Let the Alexandrov L-quasi-G-filter on X, G : L^X → L be defined by Table 2. Then by previous theorem, the reflexive L-fuzzy relation on X, R_G : X × X → L is given by R_G (x, x) = R_G (z, z) = R_G (z, x) =1 and R_G (x, z) =0.

Table 2

Alexandrov L-quasi-G-filter G on X = {x, z}

	x	z	G (A_j)
A ₁	1	1	1
A ₂	1	0	1
A ₃	0	1	0
A ₄	0	0	0

4 Alexandrov L-quasi-G-filter spaces and Alexandrov L-fuzzy pre-uniform spaces

In this section the relationship between Alexandrov L-quasi-G-filter spaces and Alexandrov L-fuzzy pre-uniform spaces is investigated in detail.

The following theorem derives an Alexandrov L-quasi-G-filter from a given Alexandrov L-fuzzy pre-uniformity.

Theorem 4.1. Let $V : L^{X \times X} \to L$ be an Alexandrov L-fuzzy pre-uniformity on X. Then G : L^X → L defined by $G (A) = ⋀_{w \in L^{X \times X}} (S (\hat{w}, A) \oplus V (w))$ for all A ∈ L^X, where $\hat{w} \in L^{X}$ is given by $\hat{w} (x) = ⋀_{z \in X} w (x, z)$ is an Alexandrov L-quasi-G-filter on X.

Proof. (AG1) is obvious.

(AG2) For each family {A_j ∈ L^X|j ∈ J}, $\begin{matrix} G (⋀_{j \in J} A_{j}) & = & ⋀_{w \in L^{X \times X}} (S (\hat{w}, ⋀_{j \in J} A_{j}) \oplus V (w)) \\ = & ⋀_{w \in L^{X \times X}} ((⋀_{j \in J} S (\hat{w}, A_{j})) \oplus V (w)) \\ = & ⋀_{w \in L^{X \times X}} ((⋀_{j \in J} S (\hat{w}, A_{j}) \oplus V (w))) \\ = & ⋀_{j \in J} ⋀_{w \in L^{X \times X}} (S (\hat{w}, A_{j}) \oplus V (w)) \\ = & ⋀_{j \in J} G (A_{j}) . \end{matrix}$

[(AG3)] For all A ∈ L^X and β ∈ L, $\begin{matrix} G (β \to A) \\ = ⋀_{w \in L^{X \times X}} (S (\hat{w}, β \to A) \oplus V (w)) \\ = ⋀_{w \in L^{X \times X}} ((β \to S (\hat{w}, A)) \oplus V (w)) \\ = ⋀_{w \in L^{X \times X}} (β \to (S (\hat{w}, A) \oplus V (w))) \\ (since (β \to p) \oplus q = β \to (p \oplus q)) \\ = β \to ⋀_{w \in L^{X \times X}} (S (\hat{w}, A) \oplus V (w)) \\ = β \to G (A) . \end{matrix}$

Example 4.2. let X = {x, z} and L = ({0, 1} , ⊙ , → , * , 0, 1) be the lattice mentioned in Example 3.12. Let the Alexandrov L-fuzzy pre-uniformity on X, $V : L^{X \times X} \to L$ be defined by Table 3. Then by previous theorem the Alexandrov L-quasi-G-filter on X, G : L^X → L is obtained as $G (A) = S ({\hat{w}}_{3}, A) = A (z)$ for all A ∈ L^X.

Table 3
Alexandrov L-fuzzy pre-uniformity $V$ on X = {x, z}

w ₁ w ₂ w ₃ w ₄ w ₅ w ₆ w ₇ w ₈ w ₉ w ₁₀ w ₁₁ w ₁₂ w ₁₃ w ₁₄ w ₁₅ w ₁₆

(x, x) 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0

(x, z) 1 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0

(z, x) 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0

(z, z) 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0

$V (w_{j})$ 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0

	w ₁	w ₂	w ₃	w ₄	w ₅	w ₆	w ₇	w ₈	w ₉	w ₁₀	w ₁₁	w ₁₂	w ₁₃	w ₁₄	w ₁₅
(x, x)	1	1	0	1	1	1	1	1	0	0	0	1	0	0	0
(x, z)	1	0	1	1	1	1	0	0	0	1	1	0	1	0	0
(z, x)	1	1	1	0	1	0	1	0	1	1	0	0	0	1	0
(z, z)	1	1	1	1	0	0	0	1	1	0	1	0	0	0	1
$V (w_{j})$	1	1	0	1	1	1	1	1	0	0	0	1	0	0	0

In addition to the above, there are many other ways to obtain Alexandrov L-quasi-G-filters from a given Alexandrov L-fuzzy pre-uniformity. The following lemma is very much useful in deriving two such Alexandrov L-quasi-G-filters as seen in Theorems 4.4 and 4.6.

Lemma 4.3. For every A, B ∈ L^X, define w^[A,B] ∈ L^X×X by w^[A,B] (x, z) = A (x) ⊙ B (z) for all x, z ∈ X. Then for all A, B, C, D, A_i, B_i ∈ L^X, the following properties hold.

w^[1_X,1_X] = 1_X×X.

w^[A_X,0_X] = w^[0_X,A_X] = 0_X×X.

If A ≤ C, B ≤ D, then w^[A,B] ≤ w^[C,D].

w^[A,B] ∘ w^[B,C] ≤ w^[A,C] where u ∘ w is defined by u ∘ w (x, y) = ⋁ _z∈Xu (x, z) ⊙ w (z, y) for all u, w ∈ L^X×X.

w^{[∨_j∈JA_j,B]} = ⋁ _j∈Jw^[A_j,B] and

w^{[A,∨_j∈JB_i]} = ⋁ _j∈Jw^[A,B_i].

w^{[∧_j∈JA_j,B]} ≤ ⋀ _j∈Jw^[A_j,B] and

w^{[A,∧_j∈JB_i]} ≤ ⋀ _j∈Jw^[A,B_i].

w^[β⊙A,B] = β ⊙ w^[A,B] and w^[A,β⊙B] = β ⊙ w^[A,B].

w^[β→A,B] ≤ β → w^[A,B] and w^[A,β→B] ≤ β → w^[A,B].

For any map g : X → Y, B₁, B₂ ∈ L^Y,

w^{[g^←(B₁),g^←(B₂)]} = (g × g) ^← (w^[B₁,B₂]).

w^[β→A,1_X] = β → w^[A,1_X].

w^{[∧_j∈JA_j,1_X]} = ⋀ _j∈Jw^[A_j,1_X] and

w^{[1_X,∧_j∈JB_i]} = ⋀ _j∈Jw^[1_X,B_i].

Proof. We only prove (iv .) and (viii .) as the rest are trivial. $\begin{matrix} (iv .) & w^{[A, B]} \circ w^{[B, C]} (x, y) \\ = ⋁_{z \in X} (w^{[A, B]} (x, z) ⊙ w^{[B, C]} (z, y)) \\ = ⋁_{z \in X} (A (x) ⊙ B (z) ⊙ B (z) ⊙ C (y)) \\ \leq A (x) ⊙ C (y) \\ = w^{[A, C]} (x, y) . \end{matrix}$ $\begin{matrix} (viii .) & w^{[β \to A, B]} (x, z) \\ = (β \to A (x)) ⊙ B (z) \\ \leq β \to (A (x) ⊙ B (z)) \\ = (β \to w^{[A, B]}) (x, z) . \end{matrix}$ Similarly we get w^[A,β→B] ≤ β → w^[A,B]. □

An Alexandrov L-fuzzy pre-uniformity together with an L-fuzzy relation gives rise to an Alexandrov L-quasi-G-filter as shown below:

Theorem 4.4. Let $V : L^{X \times X} \to L$ be an Alexandrov L-fuzzy pre-uniformity, R : X × X → L be an L-fuzzy relation and G : L^X → L be defined by $G (A) = ⋀_{x, z \in X} (R (x, z) \to V (w^{[A, 1_{X}]}))$ for all A ∈ L^X. Then,

G is an Alexandrov L-quasi-G-filter on X;

If $g : (X, V_{1}) \to (Y, V_{2})$ is an L-fuzzy uniformly continuous map and g : (X, R₁) → (Y, R₂) is an order preserving map, then g : (X, G₁) → (Y, G₂) is an Alexandrov L-quasi-G-filter map.

Proof.

(AG1) is obvious.

(AG2) For each family {A_j ∈ L^X|j ∈ J}, $\begin{matrix} G (⋀_{j \in J} A_{j}) \\ = ⋀_{x, z \in X} (R (x, z) \to V (w^{[\land_{j \in J} A_{j}, 1_{X}]})) \\ = ⋀_{x, z \in X} (R (x, z) \to (⋀_{j \in J} V (w^{[A_{j}, 1_{X}]}))) \\ = ⋀_{x, z \in X} ⋀_{j \in J} (R (x, z) \to V (w^{[A_{j}, 1_{X}]})) \\ = ⋀_{j \in J} G (A_{j}) . \end{matrix}$

[(AG3)] For all A ∈ L^X and β ∈ L, $\begin{matrix} G (β \to A) \\ = ⋀_{x, z \in X} (R (x, z) \to V (w^{[β \to A, 1_{X}]})) \\ = ⋀_{x, z \in X} (R (x, z) \to (β \to V (w^{[A, 1_{X}]}))) \\ = ⋀_{x, z \in X} (β \to (R (x, z) \to V (w^{[A, 1_{X}]}))) \\ = β \to ⋀_{x, z \in X} (R (x, z) \to V (w^{[A, 1_{X}]})) \\ = β \to G (A) . \end{matrix}$

For all B ∈ L^Y, $\begin{matrix} G_{1} (g^{\leftarrow} (B)) \\ = ⋀_{x, z \in X} (R_{1} (x, z) \to V_{1} (w^{[g^{\leftarrow} (B), 1_{X}]})) \\ = ⋀_{x, z \in X} (R_{1} (x, z) \to V_{1} ((g \times g)^{\leftarrow} (w^{[B, 1_{Y}]}))) \\ \geq ⋀_{x, z \in X} (R_{2} (g (x), g (z)) \to V_{2} (w^{[B, 1_{Y}]})) \\ \geq ⋀_{y, t \in Y} (R_{2} (y, t) \to V_{2} (w^{[B, 1_{Y}]})) \\ = G_{2} (B) . \end{matrix}$ □

Example 4.5. Let X = {x, z}, L = ([0, 1] , ⊙ , → , * , 0, 1) be the lattice mentioned in Remark 3.2 and R be the L-fuzzy relation defined by Table 1. Let the Alexandrov L-fuzzy pre-uniformity on X, $V : L^{X \times X} \to L$ be defined by $V (w) = ⋀_{x, z \in X} w (x, z)$ for all w ∈ L^X×X. Then by previous theorem, the Alexandrov L-quasi-G-filter on X, G : L^X → L is obtained as G (A) =0.9 → (A (x) ∧ A (z)) for all A ∈ L^X.

The following theorem establishes a functor from AL-PU to AL-QG.

Theorem 4.6. Let $V : L^{X \times X} \to L$ be an Alexandrov L-fuzzy pre-uniformity and $ψ_{β} (V) : L^{X} \to L$ be defined by $ψ_{β} (V) (A) = β \to V (w^{[A, 1_{X}]})$ for all A ∈ L^X. Then,

$ψ_{β} (V)$ is an Alexandrov L-quasi-G-filter on X;

If $g : (X, V_{1}) \to (Y, V_{2})$ is an L-fuzzy uniformly continuous map, then $g : (X, ψ_{β} (V_{1})) \to (Y, ψ_{β} (V_{2}))$ is an Alexandrov L-quasi-G-filter map.

Proof.

From (x.) and (xi.) of Lemma 4.3, we get $ψ_{β} (V)$ is an Alexandrov L-quasi-G-filter on X.

Let $ψ_{β} (V) = G$ . For all B ∈ L^Y, $\begin{matrix} G_{2} (B) & = & β \to V_{2} (w^{[B, 1_{Y}]}) \\ \leq & β \to V_{1} ((g \times g)^{\leftarrow} (w^{[B, 1_{Y}]})) \\ = & β \to V_{1} (w^{[g^{\leftarrow} (B), 1_{X}]}) \\ = & G_{1} (g^{\leftarrow} (B)) . \end{matrix}$ □

Corollary 4.7. ψ_β is a functor from AL-PU to AL-QG.

Example 4.8. Let X = {x, z} and L = ([0, 1] , ⊙ , → , * , 0, 1) be the lattice mentioned in Remark 3.2. Let the Alexandrov L-fuzzy pre-uniformity on X, $V : L^{X \times X} \to L$ be defined by $V (w) = ⋀_{x, z \in X} w (x, z)$ for all w ∈ L^X×X. Then by previous theorem, for β = 0.3, the Alexandrov L-quasi-G-filter on X, $ψ_{0.3} (V) : L^{X} \to L$ is obtained as $ψ_{0.3} (V) (A) = 0.3 \to (A (x) \land A (z))$ for all A ∈ L^X.

The following theorem associates an Alexandrov L-quasi-G-filter with a given Alexandrov L-fuzzy pre-uniformity.

Theorem 4.9. Let $V : L^{X \times X} \to L$ be an Alexandrov L-fuzzy pre-uniformity. Define the map $G_{V} : L^{X} \to L$ by $G_{V} (A) = ⋀_{x, z \in X} (V^{*} (⊤_{(x, z)}^{*}) \to A (z))$ for every A ∈ L^X. Then,

$G_{V}$ is an Alexandrov L-quasi-G-filter on X;

If $V (⊤_{r, r}^{*}) = 0$ , then $G_{V}$ is an L-G-filter;

If $g : (X, V_{1}) \to (Y, V_{2})$ is an L-fuzzy uniformly continuous map, then $g : (X, G_{V_{1}}) \to (Y, G_{V_{2}})$ is an Alexandrov L-quasi-G-filter map.

Proof. We prove only (ii.) and (iii.).

(ii .) Since $V (⊤_{r, r}^{*}) = 0$ , $⋁_{r \in X} (V^{*} (⊤_{x, z}^{*}) ⊙ V^{*} (⊤_{z, r}^{*})) \geq V^{*} (⊤_{x, z}^{*}) ⊙ V^{*} (⊤_{z, z}^{*}) = V^{*} (⊤_{x, z}^{*})$ . Therefore, for all A, B ∈ L^X,

$\begin{matrix} G_{V} (A) ⊙ G_{V} (B) \\ = ⋀_{x, z \in X} (V^{*} (⊤_{(x, z)}^{*}) \to A (z)) ⊙ \\ ⋀_{r, s \in X} (V^{*} (⊤_{(r, s)}^{*}) \to B (s)) \\ \leq ⋀_{x, z, r, s \in X} ((V^{*} (⊤_{(x, z)}^{*}) \to A (z)) ⊙ \\ (V^{*} (⊤_{(r, s)}^{*}) \to B (s))) \\ \leq ⋀_{x, z, r, s \in X} ((V^{*} (⊤_{(x, z)}^{*}) ⊙ V^{*} (⊤_{(r, s)}^{*})) \to \\ (A (z) ⊙ B (s))) \\ \leq ⋀_{x, z, r \in X} ((V^{*} (⊤_{(x, z)}^{*}) ⊙ V^{*} (⊤_{(z, r)}^{*})) \to \\ (A ⊙ B) (z)) \\ = ⋀_{x, r \in X} ((⋁_{r \in X} (V^{*} (⊤_{(x, z)}^{*}) ⊙ V^{*} (⊤_{(z, r)}^{*}))) \\ \to (A ⊙ B) (z)) \\ \leq ⋀_{x, r \in X} (V^{*} (⊤_{(x, z)}^{*}) \to (A ⊙ B) (z)) \\ = G_{V} (A ⊙ B) . \end{matrix}$

[(iii .)] For all B ∈ L^Y, $\begin{matrix} G_{V_{2}} (B) \\ = ⋀_{y, t \in Y} ({V_{2}}^{*} (⊤_{(y, t)}^{*}) \to B (t)) \\ = ⋀_{x, z \in X} ({V_{2}}^{*} (⊤_{(g (x), g (z))}^{*}) \to B (g (z))) \\ \leq ⋀_{x, z \in X} ({V_{1}}^{*} ((g \times g)^{\leftarrow} (⊤_{(g (x), g (z))}^{*})) \to g^{\leftarrow} (B) (z)) \\ \leq ⋀_{x, z \in X} ({V_{1}}^{*} (⊤_{(x, z)}^{*}) \to g^{\leftarrow} (B) (z)) \\ (since (g \times g)^{\leftarrow} (⊤_{(g (x), g (z))}) \geq ⊤_{(x, z)}) \\ = G_{V_{1}} (g^{\leftarrow} (B)) . \end{matrix}$ □

Corollary 4.10. Let $G :$ AL-PU → AL-QG be defined by $G ((X, V)) = (X, G_{V})$ . Then $G$ is a functor from AL-PU to AL-QG.

Example 4.11. Let X = {x, z}, L = ([0, 1] , ⊙ , → , * , 0, 1) be the lattice given in Remark 3.2. Let the Alexandrov L-fuzzy pre-uniformity on X, $V : L^{X \times X} \to L$ be defined by $V (w) = ⋀_{x, z \in X} w (x, z)$ for all w ∈ L^X×X. Then the Alexandrov L-quasi-G-filter on X, $G_{V} : L^{X} \to L$ is obtained as $G_{V} (A) = A (x) \land A (z)$ for all A ∈ L^X.

The study conducted so far has derived Alexandrov L-quasi-G-filters from a given Alexandrov L-fuzzy pre-uniformity in four different ways. As revealed from examples, all these Alexandrov L-quasi-G-filters are different. In the following theorem we explore the converse direction and obtain Alexandrov L-fuzzy pre-uniformity from a given Alexandrov L-quasi-G-filter.

Theorem 4.12. Let G : L^X → L be an Alexandrov L-quasi-G-filter, $V_{G} : L^{X \times X} \to L$ be defined by $V_{G} (w) = ⋀_{x, z \in X} ((G^{*} (z_{⊤}^{*}) \oplus ⊤_{z} (x)) \to w (x, z))$ for all w ∈ L^X×X. Then,

$V_{G}$ is an Alexandrov L-fuzzy pre-uniformity on X;

$V_{G} (w) \geq ⋁ {V_{G} (v) ⊙ V_{G} (u) | v \circ u \leq w}$ ;

$V_{G} (⊤_{r, s}^{*}) = {\begin{matrix} 0 & if r = s \\ G (⊤_{s}^{*}) & otherwise \end{matrix}$

If g : (X, G₁) → (Y, G₂) is an Alexandrov L-quasi-G-filter map, then $g : (X, V_{G_{1}}) \to (Y, V_{G_{2}})$ is an L-fuzzy uniformly continuous map.

Proof. We prove (i.),(ii.) and (iv.).

It is easy to prove (AU1) , (AU2) and (AU4).

(AU3) For each family {w_j ∈ L^X×X|j ∈ J}, $\begin{matrix} V_{G} (⋀_{j \in J} w_{j}) \\ = ⋀_{x, z \in X} ((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to (⋀_{j \in J} w_{j}) (x, z)) \\ = ⋀_{x, z \in X} ⋀_{j \in J} ((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to w_{j} (x, z)) \\ = ⋀_{j \in J} V_{G} (w_{j}) . \end{matrix}$

(AU5) For all w ∈ L^X×X, β ∈ L, $\begin{matrix} V_{G} (β \to w) \\ = ⋀_{x, z \in X} ((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to (β \to w) (x, z)) \\ = ⋀_{x, z \in X} β \to ((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to w (x, z)) \\ = β \to V_{G} (w) . \end{matrix}$

For v, u ∈ L^Y×Y such that v ∘ u ≤ w, $\begin{matrix} V_{G} (v) ⊙ V_{G} (u) \\ = ⋀_{x, z \in X} ((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to v (x, z)) ⊙ \\ ⋀_{r, s \in X} ((G^{*} (⊤_{s}^{*}) \oplus ⊤_{s} (r)) \to u (r, s)) \\ \leq ⋀_{x, z, r, s \in X} (((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to v (x, z)) ⊙ \\ ((G^{*} (⊤_{s}^{*}) \oplus ⊤_{s} (r)) \to u (r, s))) \\ \leq ⋀_{x, z, r, s \in X} (((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) ⊙ (G^{*} (⊤_{s}^{*}) \oplus ⊤_{s} (r))) \\ \to (v (x, z) ⊙ u (r, s))) \\ \leq ⋀_{x, z, r \in X} (((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) ⊙ (G^{*} (⊤_{r}^{*}) \oplus ⊤_{r} (z))) \\ \to (v (x, z) ⊙ u (z, r))) \\ \leq ⋀_{x, r \in X} (⋁_{z \in X} ((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) ⊙ (G^{*} (⊤_{r}^{*}) \oplus ⊤_{r} (z))) \\ \to ⋁_{z \in X} (v (x, z) ⊙ u (z, r))) \\ \leq ⋀_{x, r \in X} ((G^{*} (⊤_{r}^{*}) \oplus ⊤_{r} (x)) \to w (x, r)) \\ = V_{G} (w) . \end{matrix}$

(iv .) For all v ∈ L^Y×Y, $\begin{matrix} V_{G_{2}} (v) \\ = ⋀_{y, t \in Y} (({G_{2}}^{*} (⊤_{t}^{*}) \oplus ⊤_{t} (y)) \to v (y, t)) \\ \leq ⋀_{x, z \in X} (({G_{2}}^{*} (⊤_{g (z)}^{*}) \oplus ⊤_{g (z)} (g (x))) \to \\ v (g (x), g (z))) \\ \leq ⋀_{x, z \in X} (({G_{1}}^{*} (g^{\leftarrow} (⊤_{g (z)}^{*})) \oplus ⊤_{z} (x)) \to \\ (g \times g)^{\leftarrow} (v) (x, z)) \\ \leq ⋀_{x, z \in X} (({G_{1}}^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to (g \times g)^{\leftarrow} (v) (x, z)) \\ (since g^{\leftarrow} (⊤_{g (z)}) \geq ⊤_{z}) \\ = V_{G_{1}} ((g \times g)^{\leftarrow} (v)) . \end{matrix}$ □

Corollary 4.13. Let $U :$ AL-QG→ AL-PU be defined by $U ((X, G)) = (X, V_{G})$ . Then $U$ is a functor from AL-QG to AL-PU.

Example 4.14. Let X = {x, z}, L = ([0, 1] , ⊙ , → , * , 0, 1) be the lattice mentioned in Remark 3.2. Let the Alexandrov L-quasi-G-filter on X, G : L^X → L be defined by G (A) = A (x) ∧ A (z) for all A ∈ L^X. Then the Alexandrov L-fuzzy pre-uniformity on X, $V_{G} : L^{X \times X} \to L$ is obtained as $V_{G} (w) = ⋀_{x, z \in X} w (x, z)$ for all w ∈ L^X×X.

The following two theorems indeed play a vital role in establishing the relationship between the functors defined in Corollary 4.10 and Corollary 4.13.

Theorem 4.15. Let G : L^X → L be an Alexandrov L-quasi-G-filter. Then $G_{V_{G}} \leq G$ .

Proof.

Since $V_{G} (⊤_{(x, z)}^{*}) = ⋀_{r, s \in X} ((G^{*} (⊤_{s}^{*}) \oplus ⊤_{s} (r)) \to (⊤_{(x, z)}^{*}) (r, s)) = {(G^{*} (⊤_{z} *) \oplus ⊤_{z} (x))}^{*}$ , for all B ∈ L^X, $\begin{matrix} G_{V_{G}} (B) & = & ⋀_{x, z \in X} ({V_{G}}^{*} (⊤_{(x, z)}^{*}) \to B (z)) \\ = & ⋀_{x, z \in X} ((G^{*} (⊤_{z}^{*}) \oplus ⊤_{z} (x)) \to B (z)) \\ = & ⋀_{x, z \in X} (B^{*} (z) \to (G (⊤_{z}^{*}) ⊙ ⊤_{z}^{*} (x))) \\ \leq & ⋀_{x, z \in X, x \neq z} (B^{*} (z) \to (G (⊤_{z}^{*}) ⊙ ⊤_{z}^{*} (x))) \\ = & ⋀_{z \in X} (B^{*} (z) \to G (⊤_{z}^{*})) \\ = & ⋀_{z \in X} G (B^{*} (z) \to ⊤_{z}^{*}) \\ = & G (⋀_{z \in X} (B^{*} (z) \to ⊤_{z}^{*})) \\ = & G (B) . \end{matrix}$ □

Corollary 4.16. $G \circ U \leq {id}_{A L - QG}$ where id_AL-QG is the identity functor in AL-QG.

Theorem 4.17. Let $V : L^{X \times X} \to L$ be an Alexandrov L-fuzzy pre-uniformity. Then $V_{G_{V}} \leq V$ .

Proof. Since $G_{V} (⊤_{z}^{*}) = ⋀_{r, s \in X} (V^{*} (⊤_{(r, s)}^{*}) \to ⊤_{z}^{*} (s)) = ⋀_{r \in X} V (⊤_{(r, z)}^{*})$ , for all w ∈ L^X×X, $\begin{matrix} V_{G_{V}} (w) \\ = ⋀_{x, z \in X} (([⋁_{r \in X} V^{*} (⊤_{(r, z)}^{*})] \oplus ⊤_{z} (x)) \to w (x, z)) \\ \leq ⋀_{x, z \in X} ((V^{*} (⊤_{(x, z)}^{*}) \oplus ⊤_{z} (x)) \to w (x, z)) \\ \leq ⋀_{x, z \in X} (V^{*} (⊤_{(x, z)}^{*}) \to w (x, z)) \\ = ⋀_{x, z \in X} (w^{*} (x, z) \to V (⊤_{(x, z)}^{*})) \\ = ⋀_{x, z \in X} V (w^{*} (x, z) \to ⊤_{(x, z)}^{*}) \\ = V (⋀_{x, z \in X} (w^{*} (x, z) \to ⊤_{(x, z)}^{*})) \\ = V (w) . \end{matrix}$ □

Corollary 4.18. $U \circ G \leq {id}_{A L - PU}$ where id_AL-PU is the identity functor in AL-PU.

From Example 4.11 and Example 4.14, it is observed that $G_{V_{G}} = G$ and $V_{G_{V}} = V$ . Does these equalities hold always? We leave it as an open question.

Question 4.19. Let $G$ and $U$ be the functors defined in Corollary 4.10 and Corollary 4.13 respectively. Do $U \circ G$ and $G \circ U$ correspond to the identity functor in AL-PU and AL-QG respectively?

5 An application

Let X = {d₁, d₂, d₃, d₄} be a set of four vitamin deficiency diseases, Y = {c₁, c₂, c₃, c₄, c₅, c₆, c₇, c₈} be a set of eight countries and let Z = {d₁, d₂, d₃}. Let L = ([0, 1] , ⊙ , → , * , 0, 1) be the complete residuated lattice with p ⊙ q = max {0, p + q - 1} , p → q = min {1 - p + q, 1} and p^* = 1 - p. Consider the fuzzy information F ∈ [0, 1] ^X×Y where F (d_i, c_j) indicates the intensity of the vitamin deficiency disease d_i in country c_j as given in Table 4.

Table 4
Fuzzy information F

F c ₁ c ₂ c ₃ c ₄ c ₅ c ₆ c ₇ c ₈

d ₁ 0.9 0.9 0.7 0.8 0.8 0.5 0.3 0.7

d ₂ 0.3 0.2 0.3 0.5 0.4 0.2 0.1 0.5

d ₃ 0.5 0.4 0.4 0.2 0.6 0.3 0.1 0.4

d ₄ 0 0 0.1 0 0 0.2 0 0

F	c ₁	c ₂	c ₃	c ₄	c ₅	c ₆	c ₇	c ₈
d ₁	0.9	0.9	0.7	0.8	0.8	0.5	0.3	0.7
d ₂	0.3	0.2	0.3	0.5	0.4	0.2	0.1	0.5
d ₃	0.5	0.4	0.4	0.2	0.6	0.3	0.1	0.4
d ₄	0	0	0.1	0	0	0.2	0	0

This shows that the deficiency disease d₁ is frequent in all the eight countries whereas d₄ is rarely spotted.

Let $R : X \times X \to [0, 1]$ be a fuzzy relation where $R (d_{i}, d_{j})$ denotes the degree to which a person affected by deficiency disease d_i is prone to the deficiency disease d_j as shown in Table 5. It is clear that a person diagnosed with a deficiency disease d_i ; i ∈ {1, 2, 3, 4} is more susceptible to deficiency disease d₁ and that is why d₁ is more predominant in various countries, as evident from Table 4.

Table 5

Fuzzy relation $R$

$R$	d ₁	d ₂	d ₃	d ₄
d ₁	0.9	0.4	0.3	0.2
d ₂	0.6	0.3	0.5	0.1
d ₃	0.5	0.2	0.4	0.1
d ₄	0.8	0.6	0.7	0.4

Our objective is to design a mathematical model for financial support for the production of protein powders that are effective in curing the deficiency diseases d₁, d₂ and d₃ with more emphasis on protein powder that can cure the most extensive disease d₁. Let each A ∈ [0, 1] ^Z represents a protein powder where A (d_i) gives the degree to which protein powder A can cure deficiency disease d_i. The extent of financial assistance required for the production of protein powder A is determined by the Alexandrov L-quasi-G-filter, G : [0, 1] ^Z → [0, 1] where $G (A) = ⋀_{i \in {1, 2, 3}} (R (d_{i}, d_{1}) \to A (d_{i}))$ . It should be noted that G (0_Z) =0.1.

As the production of the protein powder that is effective in treating the most prevalent disease d₁ needs to be prioritized, the term $R (d_{i}, d_{1})$ in the expression of G (A) is justified. Since G is an Alexandrov L-quasi-G-filter, when we consider a collection of protein powders {A_i|i ∈ I}, the minimum of the costs for the production of each A_i is equal to the cost for the production of the protein powder ⋀_i∈IA_i, the protein powder which is the most effective among the set of all protein powders that are less effective than each one in the collection {A_i|i ∈ I}.

From Table 4, it is obvious that the disease d₄ is not very common. However, since the values of $R (d_{4}, d_{i}); i \in {1, 2, 3}$ are all nonzero, a person affected by the rarely spotted disease d₄ is prone to other diseases d₁, d₂ and d₃ to a great extent. Hence, even if a particular protein powder is not effective in curing either d₁, d₂ or d₃, its production matters if it is effective in curing d₄. For instance, extension of the zero function 0_Z : Z → [0, 1], P : X → [0, 1] where P (d₄) =0.8 cures the disease d₄ though it fails to cure d₁, d₂ and d₃. As a result, the non zero value that is assigned to 0_Z by G is validated by its contribution to the production of protein powder P.

Consequently, given a budget, we may select the best Alexandrov L-quasi-G-filter modeling the situation inorder to distribute the fund effectively for the production of protein powders for curing deficiency diseases.

6 Conclusion

The study has introduced the concept of Alexandrov L-quasi-G-filter spaces and investigated the categorical properties of these spaces. It is verified that Alexandrov L-quasi-G-filter spaces form a category with Alexandrov L-quasi-G-filter maps as morphisms. The connection between L-fuzzy relations and Alexandrov L-quasi-G-filters has been illustrated. It is proved that every L-fuzzy relation on a set X induces an Alexandrov L-quasi-G-filter on X and vice-versa.

The solid bond that exists between the categories of Alexandrov L-quasi-G-filters and Alexandrov L-fuzzy pre-uniformities is established by identifying certain functors between them. Recognizing more such functors is a part of our future study which is basic for illustrating more strikingly the structure of the category of Alexandrov L-quasi-G-filter spaces.

Finally, the study has demonstrated the application potential of the notion of Alexandrov L-quasi-G-filter space in mathematical modeling, particularly its scope in situations requiring optimal decision-making.

Acknowledgements

The authors are thankful to the reviewers for their valuable comments and constructive suggestions which improved the presentation of the paper. The first author wishes to thank CSIR, India for giving financial support under the Senior Research Fellowship awarded by order No. 08/528(0004)/2019-EMR-1 dated 08/04/2021.

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Alexandrov L -quasi-G-filters and Alexandrov L -fuzzy pre-uniformities

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Alexandrov L-quasi-G-filter spaces and L-fuzzy relations

Table 1 L-fuzzy relation R on X = {x, z} R x z x 0.2 0.4 z 0.9 0.7

Table 4 Fuzzy information F F c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 d 1 0.9 0.9 0.7 0.8 0.8 0.5 0.3 0.7 d 2 0.3 0.2 0.3 0.5 0.4 0.2 0.1 0.5 d 3 0.5 0.4 0.4 0.2 0.6 0.3 0.1 0.4 d 4 0 0 0.1 0 0 0.2 0 0

Acknowledgements

References

Table 1
L-fuzzy relation R on X = {x, z}

R x z

x 0.2 0.4

z 0.9 0.7

Table 4
Fuzzy information F

F c ₁ c ₂ c ₃ c ₄ c ₅ c ₆ c ₇ c ₈

d ₁ 0.9 0.9 0.7 0.8 0.8 0.5 0.3 0.7

d ₂ 0.3 0.2 0.3 0.5 0.4 0.2 0.1 0.5

d ₃ 0.5 0.4 0.4 0.2 0.6 0.3 0.1 0.4

d ₄ 0 0 0.1 0 0 0.2 0 0