Parameterized degree of semi-precompactness in the fuzzy soft universe

Abstract

The present study is devoted to describe the semi-precompactness degree of an L-fuzzy soft set in an L -fuzzy (E, K)-soft supra topological space, where L is a complete DeMorgan algebra. For this purpose, we first extend the concepts of preopen and semipreopen operators to the fuzzy soft universe and then by using these operators, we define the parameterized degree of semi-precompactness of an L-fuzzy soft set in an L-fuzzy (E, K)-soft supra topological space. We examine the relations between compactness degree and semi-precompactness degree of an L-fuzzy soft set. Moreover, we obtain several properties and some other characterizations of the proposed notion.

Keywords

Fuzzy soft set fuzzy soft topology preopen and semipreopen operator compactness degree

1. Introduction

Since Molodtsov [25] proposed soft set theory to overcome some of the difficulties involving the parametrization process in handling uncertainties, many researchers have applied soft set theory in different directions. Especially, the definition of fuzzy soft set given by Maji et al. [23], has gained acceleration to the improvement of the investigations of proposed theories in several aspects [1 , 37–40].

Compactness is one of the most significant concepts in general topology since it is sort of a topological generalization of finiteness and on the other hand, it is essential to generalize theorems known in real analysis. If you have some object, then compactness allows you to extend results that you know are true for all finite subobjects to the object itself. A very closely related example is the compactness theorem in propositional logic: an infinite collection of sentences is consistent if every finite subcollection is consistent.

The notion of compactness has been generalized to L-topological space by many authors [5 , 34]. In these approaches of compactness, fuzzy set does not have a degree of compactness except the empty set and the whole space. Lowen and Lowen [19] considered compactness of I-topological spaces in a matter of degree. Then Sostak [36] introduced the compactness degree of an L-fuzzy set in the case L = I . Later Shi et al. [18] observed the fuzzy compactness degree in L-fuzzy topological spaces and Çetkin et al. [9] introduced the parameterized compactness degree in the parameterized fuzzy soft topological spaces. On the other hand, the stronger and the weaker forms of compactness play very important role in (fuzzy) topology. Gantner et al. [11] defined α-compactness, Lowen [20, 21] defined strong fuzzy compactness and ultra-fuzzy compactness, Hanafy [15] introduced β-compactness. Shi [35] presented the notion of semicompactness based on the his L-fuzzy semiopen and L-fuzzy preopen operators. Later, Ghareeb and Shi [13] defined fuzzy SP-compactness degree of an L-fuzzy set.

Inspired by the idea of Molodtsov, we thought that the parametrization tool is not only necessary for the sets but also the structures based on these sets. According to this idea, we published some seminal papers [4 , 8–10]. In this study, as a continuation of these works, we decided to apply the parametrization tool to the notion of semi-precompactness degree given by Ghareeb and Shi [13]. This paper is arranged in the following manner. In section 2, we recall some elementary definitions which are the backbones of the main section and also give the definitions of soft supra L-topology and fuzzy soft supra topology. In section 3, we propose the compactness in a soft L-topology and compactness degree of a fuzzy soft set in a fuzzy soft supra topology. We also extend the preopen and semipreopen operators given by Ghareeb [12] to the fuzzy soft universe. In section 4, we first define the parameterized degree of semi-precompactness of an L-fuzzy soft set in a fuzzy soft topology. We show that if a fuzzy soft set is semi-precompact, then it is also compact. We prove that the union of semi-precompact fuzzy soft sets is semi-precompact, and the intersection of a semi-precompact fuzzy soft set f and semi-preclosed fuzzy soft set g, is semi-precompact, too. Additionally, we show that semi-precompactness is preserved under semi-preirresolute functions and the semi-precompactness under semi-precontinuous functions imply the compactness. In the last section, we mention some other characterizations of semi-precompactness.

2. Preliminaries

Throughout this paper, X refers to a nonempty initial universe, E, K denotes the arbitrary nonempty sets viewed on the sets of parameters and L = (L, ∨, ∧, ′) denotes a complete DeMorgan algebra with the smallest element 0_L and the largest element 1_L . With the underlying lattice L, a mapping A : X → L is said to be an L-fuzzy set on X and by L ^X, we denote the family of all L-fuzzy sets on X .

Let a, b, c be elements in L . An element a in L is said to be coprime if a ≤ b ∨ c implies that a ≤ b or a ≤ c . The set of all coprime elements of L is denoted by c (L), and the set of all prime elements is denoted by p (L) . We say a is way below (wedge below) b, in symbols, a ⪡ b (a &z.ltri; b) or b ⪢ a (b &z.rtri; a), if for every directed (arbitrary) subset D ⊆ L, ∨D ≥ b implies a ≤ d for some d ∈ D . Clearly if a ∈ L is coprime, then a ⪡ b if and only if a &z.ltri; b . A complete lattice L iis completely distributive if and only if b = ⋁ {a ∈ L ∣ a &z.ltri; b} for each b ∈ L . For any b ∈ L, define Θ (b) = ⋁ {a ∈ L ∣ a &z.ltri; b}, the greatest minimal family and Ω (b), the greatest maximal family. The wedge below operation in a completely distributive lattice has an interpolation property, this means a &z.ltri; b implies there exists c ∈ L such that a &z.ltri; c &z.ltri; b . Details for lattices can be found in [14].

The binary operation ↦ in the complete DeMorgan algebra L is given by

α ↦ β = ⋁ {γ ∈ L ∣ α ∧ γ ≤ β}.

For all α, β, γ, δ ∈ L and {α _i}, {β _i} ⊆ L, the following properties are satisfied:

(1) (α ↦ β) ≥ γ iff α ∧ γ ≤ β .

(2) α ↦ β = 1_L iff α ≤ β.

(3) α ↦ (⋀ _i β _i) = ⋀ _i (α ↦ β _i) .

(4) (⋁ _i α _i) ↦ β = ⋀ _i (α _i ↦ β) .

(5) (α ↦ γ) ∧ (γ ↦ β) ≤ α ↦ β .

(6) α ≤ βimpliesγ ↦ α ≤ γ ↦ β .

(7) α ≤ βimpliesβ ↦ γ ≤ α ↦ γ .

(8) (α ↦ β) ∧ (γ ↦ δ) ≤ (α ∧ γ) ↦ (β ∧ δ) .

The parameterized version of an L-fuzzy set is called an L-fuzzy soft set and it is defined as follows.

Definition 2.1. [31] f is called an L-fuzzy soft set on X, where f is a mapping from E into L ^X. This means that f _e : = f (e) : X → L, is an L-fuzzy set on X, for each parameter e ∈ E .

The family of all L-fuzzy soft sets on X is denoted by (L ^X) ^E .

Definition 2.2. [27, 31] Let f and g be two L-fuzzy soft sets on X, then

(1) we say that f is an L-fuzzy soft subset of g and write f ⊑ g if f _e ≤ g _e, for each e ∈ E. f and g are called equal if f ⊑ g and g ⊑ f.

(2) the union of f and g is an L-fuzzy soft set h = f ⊔ g, where h _e = f _e ∨ g _e, for each e ∈ E.

(3) the intersection of f and g on X is an L-fuzzy soft set h = f ⊓ g, where h _e = f _e ∧ g _e, for each e ∈ E.

(4) the complement of an L-fuzzy soft set f is denoted by f′, where f′ : E → L ^X is a mapping given by $f_{e}^{'} = (f_{e})^{'}$ , for each e ∈ E. Clearly (f′) ′ = f.

Definition 2.3. [31] (1) (Null L-fuzzy soft set) An L-fuzzy soft set f on X is called a null L-fuzzy soft set and denoted by $\tilde{0}$ , if f _e (x) =0, for each e ∈ E, x ∈ X.

(2) (Absolute L-fuzzy soft set) An L-fuzzy soft set f on X is called an absolute L-fuzzy soft set and denoted by $\tilde{1}$ , if f _e (x) =1, for each e ∈ E, x ∈ X. Clearly $(\tilde{1})^{'} = \tilde{0}$ and ${\tilde{0}}^{'} = \tilde{1}$ .

Proposition 2.1. Let f, g, f _i, g _i ∈ (L ^X) ^E, for all i ∈ Δ, where Δ is an index set. Then the following properties are satisfied.

f ⊓ (⨆ _i∈Δ g _i) = ⨆ _i∈Δ (f ⊓ g _i) and f ⊔ (epsfboxG :/Tex/IOSPRESS/IFS/0 -181830/IF01 . eps _i∈Δ g _i) = epsfboxG :/Tex/IOSPRESS/IFS/0 -181830/IF01 . eps _i∈Δ (f ⊔ g _i)

${({G : / Tex / IOSPRESS / IFS / 0 - 181830 / IF 01 . eps}_{i \in Δ} f_{i})}^{'} = ⨆_{i \in Δ} f_{i}^{'}$ , ${(⨆_{i \in Δ} f_{i})}^{'} = {G : / Tex / IOSPRESS / IFS / 0 - 181830 / IF 01 . eps}_{i \in Δ} f_{i}^{'}$ .

Notation. The fuzzy soft inclusion $[\tilde{⊑}] : (L^{X})^{E} \times (L^{X})^{E} \to L$ is defined by the following equality

$[f \tilde{⊑} g] = ⋀_{x \in X} ⋀_{e \in E} (f_{e}^{'} (x) \lor g_{e} (x)) .$

Definition 2.4. [17] Let φ : X ₁ → X ₂ and ψ : E ₁ → E ₂ be two functions, where E ₁ and E ₂ are parameter sets for the crisp sets X ₁ and X ₂, respectively. Then the pair φ _ψ is called an L-fuzzy soft mapping from X ₁ to X ₂. Let f and g be two L-fuzzy soft sets over X ₁ and X ₂, respectively.

(1) The image of f under the L-fuzzy soft mapping φ _ψ, denoted by φ _ψ (f), is an L-fuzzy soft set on X ₂ defined by for all k ∈ E ₂, y ∈ X ₂,

$φ_{ψ} (f)_{k} (y) = {\begin{matrix} ⋁_{φ (x) = y} ⋁_{ψ (a) = k} f_{a} (x), & if x \in φ^{- 1} (y), a \in ψ^{- 1} (k) \\ 0, & otherwise . \end{matrix}$

(2) The pre-image of g under the L-fuzzy soft mapping φ _ψ, denoted by $φ_{ψ}^{- 1} (g)$ , is an L-fuzzy soft set on X ₁ defined by for all e ∈ E ₁, x ∈ X ₁,

$φ_{ψ}^{- 1} (g)_{e} (x) = g_{ψ (e)} (φ (x))$ .

If φ and ψ is injective (surjective), then φ _ψ is said to be injective (surjective).

Definition 2.6. [7] Let x ∈ X and α : E → c (L) be a function. Then the L-fuzzy soft set defined as follows is called an L-fuzzy soft point, and denoted by x ^α. For all e ∈ E and y ∈ X,

$x_{e}^{α} (y) = {\begin{matrix} α (e), & if y = x, \\ 0_{L}, & otherwise \end{matrix}$

An L-fuzzy soft point x ^α is said to belongs to an L-fuzzy soft set f and denoted by x ^α ∈ f if α (e) ≤ f _e (x), for each e ∈ E .

The set of all nonzero coprime elements of (L ^X) ^E is denoted by c ((L ^X) ^E). It is noted that c ((L ^X) ^E) is exactly the set of all L-fuzzy soft points.

Definition 2.7. A parameterized family $T = {T_{k}}_{k \in K}$ of $T_{k} \subseteq (L^{X})^{E}$ which satisfies the following properties for each k ∈ K is called an (E, K)-soft supra L-topology on X:

(S1) $\tilde{0}, \tilde{1} \in T_{k} .$

(S2) If ${f_{i}}_{i \in Γ} \subseteq T_{k}$ , then $⨆_{i \in Γ} f_{i} \in T_{k}$ .

If additionally it provides the following

(S3) If $f, g \in T_{k}$ , then $f ⊓ g \in T_{k} .$

then $T$ is called an (E, K)-soft L-topology on X [6]. The pair $(X, T)$ is called an (E, K)-soft L-topological space. f is called open according to k ∈ K if $f \in T_{k},$ and it is called closed according to k ∈ K if $f^{'} \in T_{k}$ .

Definition 2.8. A mapping τ : K → L ^{(L
^X)^E} is called an L-fuzzy (E, K)-soft supra topology on X if it satisfies the following conditions for each k ∈ K.

(O1) $τ_{k} (\tilde{0}) = τ_{k} (\tilde{1}) = 1_{L} .$

(O2) τ _k (⨆ _i∈Δ f _i) ≥ ⋀ _i∈Δ τ _k (f _i), for all f _i ∈ (L ^X) ^E, i ∈ Δ .

If additionally τ satisfies the following for all k ∈ K,

(O3) τ _k (f ⊓ g) ≥ τ _k (f) ∧ τ _k (g), for all f, g ∈ (L ^X) ^E .

Then τ is called an L-fuzzy (E, K)-soft topology on X [4]. The pair (X, τ) is called an L-fuzzy (E, K)-soft topological space. The value τ _k (f) is interpreted as the degree of openness of an L-fuzzy soft set f with respect to the parameter k ∈ K . The parameterized gradation of closedness is $τ_{k}^{*} (f) = τ_{k} (f^{'}) .$

Let τ ¹ and τ ² be L-fuzzy (E, K)-soft topologies on X . We say that τ ¹ is finer than τ ² (τ ² is coarser than τ ¹), denoted by τ ² ≤ τ ¹, if $τ_{k}^{2} (f) \leq τ_{k}^{1} (f)$ for each k ∈ K and f ∈ (L ^X) ^E .

Example 2.1. [9] Let

L = {(0, 0), (1, 1)} ∪ {(a, 0), (0, b), (a, a) ∣ a, b ∈ (0, 1)} . " ≤ " is defined as follows:

(m, b) ≤ (n, d) if and only if m ≤ n and b ≤ d.

Define an order reversing involution ^′ : L → L is as follows: for each x, y ∈ (0, 1), (x, 0) ′ = (1 - x, 0), (0, y) ′ = (0, 1 - y), (x, x) ′ = (1 - x, 1 - x) and (1, 1) ′ = (0, 0) . Then (L, ≤, ′) is a complete DeMorgan algebra. Let X = {x, y}, E = (0, 0.5] and f _e (x) = f _e (y) = (e, 0), g _e (x) = g _e (y) = (0, e) and h _e (x) = h _e (y) = (e, e) for each e ∈ E. Define a mapping τ : E → L ^{(L
^X)^E} as follows:

$τ_{e} (u) = {\begin{matrix} (1, 1), & if u \in {\tilde{0}, \tilde{1}, h}; \\ (e, 0), & if u = f; \\ (0, e), & if u = g; \\ (0, 0), & otherwise . \end{matrix}$

Then τ is an L-fuzzy (E, E)-soft topology on X .

Note 2.9. It is clear that if $T = {T_{k}}_{k \in K}$ is an (E, K)-soft L-topology on X, then the mapping τ : K → L ^{(L
^X)^E} defined by $τ (k) : = τ_{k} = χ_{T_{k}}$ is an (E, K)-soft L-topology on X, where

$χ_{T_{k}} (h) = {\begin{matrix} 1_{L}, & if h \in T_{k}, \\ 0_{L}, & if h \notin T_{k} . \end{matrix}$

3. Compactness degree in the fuzzy soft universe

In this section, we present the compactness of an L-fuzzy soft set in an (E, K)-soft L-topology and the compactness degree of an L-fuzzy soft set in an L-fuzzy (E, K)-soft supra topology. We propose the definitions of preopen and semi-preopen operators in the fuzzy soft universe. We also introduce semi-preirresolute and semi-precontinuous functions between fuzzy soft topological spaces.

For $U \subseteq (L^{X})^{E},$ $2^{U}$ denotes the finite subcollection of (L ^X) ^E .

Definition 3.1. Let $T$ be an (E, K)-soft L-topology on X and h ∈ (L ^X) ^E. Then h is said to be compact in (X, T) if each open k-cover $U_{k}$ of h has a finite subfamily $V_{k}$ which is an open k-cover of h, i.e.,

$⋀_{e \in E} ⋀_{y \in X} (h_{e}^{'} (y) \lor ⋁_{f \in U_{k}} f_{e} (y))$

$\leq ⋁_{V_{k} \in 2^{U_{k}}} ⋀_{e \in E} ⋀_{y \in X} (h_{e}^{'} (y) \lor ⋁_{f \in V_{k}} f_{e} (y)),$

for short

$[h \tilde{⊑} ⋁ U_{k}] \leq ⋁_{V_{k} \in 2^{U_{k}}} [h \tilde{⊑} ⋁ V_{k}] .$

Here we mean by an open k-cover $U_{k}$ that is a cover such that $U_{k} \subseteq T_{k} .$

Definition 3.2. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space and h ∈ (L ^X) ^E. Then the degree of parameterized compactness com _τ (k, h) of h is defined by

${com}_{τ} (k, h) = ⋀_{U \subseteq (L^{X})^{E}} {(⋀_{g \in U} τ_{k} (g) \land ⋀_{e \in E} ⋀_{x \in X} (h_{e}^{'} (x) \lor ⋁_{g \in U} g_{e} (x))) \mapsto ⋁_{V \in 2^{U}} ⋀_{e \in E} ⋀_{x \in X} (h_{e}^{'} (x) \lor ⋁_{g \in V} g_{e} (x))}$ ,

for short

com _τ (k, h) =

$⋀_{U \subseteq (L^{X})^{E}} {(⋀_{g \in U} τ_{k} (g) \land [h \tilde{⊑} ⋁ U]) \mapsto ⋁_{V \in 2^{U}} [h \tilde{⊑} ⋁ V]} .$

The L-fuzzy soft set h is said to be compact with respect to k iff com _τ (k, h) =1_L . The compactness degree of h ∈ (L ^X) ^E in the whole supra space (X, τ) is computed by the value ⋀_k∈K com _τ (k, h) . So, h is said to be compact in the whole supra space (X, τ) if and only if the degree ⋀_k∈K com _τ (k, h) =1_L .

Definition 3.3. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and h ∈ (L ^X) ^E . Then the mapping τ ^p : K → L ^{(L
^X)^E} which is defined as follows is called a fuzzy soft preopen operator: for each k ∈ K,

$τ_{k}^{p} (h) = ⋀_{x^{α} ⪡ h} ⋁_{x^{α} ⪡ g} {τ_{k} (g) \land ⋀_{y^{β} ⪡ g} ⋀_{y^{β} f ⊒ h} (τ_{k} (f^{'}))^{'}} .$

The value $τ_{k}^{p} (h)$ can be regarded as the degree to which h is preopen with respect to the parameter k .

Definition 3.4. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and h ∈ (L ^X) ^E . Then the mapping τ ^sp : K → L ^{(L
^X)^E} defined as follows, is called a fuzzy soft semi-preopen operator if for all k ∈ K,

$τ_{k}^{sp} (h) = ⋁_{g ⊑ h} {τ_{k}^{p} (g) \land ⋀_{x^{α} ⪡ h} ⋀_{x^{α} f ⊒ g} (τ_{k} (f^{'}))^{'}} .$

Here the values $τ_{k}^{sp} (h)$ and $τ_{k}^{sp} (h^{'})$ can be regarded as the parameterized semi-preopenness degree and the parameterized semi-preclosedness degree of h with respect to k, respectively.

Lemma 3.1. If (X, τ) is an L-fuzzy (E, K)-soft supra topological space, then semi-preopenness operator τ ^sp is also an L-fuzzy (E, K)-soft supra topology on X .

Definition 3.5. Let (X ₁, τ ¹) be an L-fuzzy (E ₁, K ₁)-soft topological space and (X ₂, τ ²) be an L-fuzzy (E ₂, K ₂)-soft topological space. Let φ : X ₁ → X ₂, ψ : E ₁ → E ₂ and η : K ₁ → K ₂ be functions.

Then we say

(1) φ _ψ,η is semi-preirresolute function if for all k ∈ K ₁ and h ∈ (L ^{X
₂}) ^{E
₂},

$(τ^{1})_{k}^{sp} (φ_{ψ}^{- 1} (h)) \geq (τ^{2})_{η (k)}^{sp} (h) .$

(2) φ _ψ,η is semi-precontinuous function if for all k ∈ K ₁ and h ∈ (L ^{X
₂}) ^{E
₂},

$(τ^{1})_{k}^{sp} (φ_{ψ}^{- 1} (h)) \geq τ_{η (k)}^{2} (h) .$

4. Semi-precompactness degree in the fuzzy soft universe

In this section, we introduce the parameterized degree of semi-precompactness of an L-fuzzy soft set in a fuzzy soft supra topological spaces. We denote that the notion of semi-precompactness is stronger than compactness. We also extend the well known properties in general topology to the fuzzy soft universe by considering the parametrization tool, and we examine some structural properties of the given concept.

Definition 4.1. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space and g ∈ (L ^X) ^E . Define an operator Spcom : K × (L ^X) ^E → L as follows:

Spcom (k, g) =

$⋀_{U \subseteq (L^{X})^{E}} {(⋀_{h \in U} τ_{k}^{sp} (h) \land ⋀_{e \in E} ⋀_{x \in X} (g_{e}^{'} (x) \lor ⋁_{h \in U} h_{e} (x)))$ }

For short,

Spcom (k, g) = $⋀_{U \subseteq (L^{X})^{E}} {(⋀_{h \in U} τ_{k}^{sp} (h) \land [g \tilde{⊑} ⋁ U]) \mapsto ⋁_{V \in 2^{U}} [g \tilde{⊑} ⋁ V]} .$

The value Spcom (k, g) is called the parameterized semi-precompactness degree of g with respect to k. The semi-precompactness degree of g ∈ (L ^X) ^E in the whole space (X, τ) is computed by the value Spcom (g) = ⋀ _k∈K Spcom (k, g). Then we deduce that, an L-fuzzy soft set g is said to be semi-precompact w.r.t. k iff Spcom (k, g) =1_L, and g is said to be semi-precompact in the whole space (X, τ) iff ⋀_k∈K Spcom (k, g) =1_L.

In addition, if $⋀_{k \in K} Spcom (k, \tilde{1}) = 1_{L},$ then (X, τ) is called semi-precompact.

Remark 4.2. If (X, τ) is an L-fuzzy (E, K)-soft topological space, since the operator τ ^sp is an L-fuzzy (E, K)-soft supra topology, then for each g ∈ (L ^X) ^E and k ∈ K, the following equality is satisfied, ${com}_{τ^{sp}} (k, g) = Spcom (k, g) .$ Example 4.1. Let $X = E = ℝ$ be the set of real numbers, K = {k ₁, k ₂} and L = [0, 1]. Now define a fuzzy soft set f : E → L ^X as follows: where $ℚ$ is the set of rational numbers, $f_{e} (x) = {\begin{matrix} 1, & if x \in (- \infty, e) \cap ℚ, \\ 0.5, & if x \notin (- \infty, e) \cap ℚ . \end{matrix}$

Now define an L-fuzzy (E, K)-soft supra topology τ : K → L ^{(L
^X)^E} as follows:

$τ_{k_{1}} (h) = {\begin{matrix} 1, & if h \in {\tilde{0}, \tilde{1}}, \\ 0.5, & if h = f, \\ 0, & otherwise \end{matrix}$

$τ_{k_{2}} (h) = {\begin{matrix} 1, & if h \in {\tilde{0}, \tilde{1}}, \\ 0, & otherwise . \end{matrix}$

Then we obtain the corresponding semi-preopen operators by

$τ_{k_{1}}^{sp} (h) = {\begin{matrix} 1, & if h \in {\tilde{0}, \tilde{1}}, \\ 0.5, & if h = f, \\ 0, & otherwise \end{matrix}$

$τ_{k_{2}}^{sp} (h) = {\begin{matrix} 1, & if h \in {\tilde{0}, \tilde{1}}, \\ 0, & otherwise . \end{matrix}$

By applying Definition 4.1, we obtain that $Spcom (k, \tilde{1}) = 1, \forall k \in K$ and so, (X, τ) is semi-precompact. Moreover, Spcom (k ₁, f) =0.5 and Spcom (k ₂, f) =1.

Theorem 4.1. Let $T = {T_{k} ∣ k \in K}$ be an (E, K)-soft supra L-topology on X and h ∈ (L ^X) ^E. Then h is semi-precompact fuzzy soft set if and only if ${Spcom}_{χ_{T}} (k, h) = 1_{L} .$

Theorem 4.2. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and g ∈ (L ^X) ^E . Then the inequality Spcom (k, g) ≤ com _τ (k, g) is valid for each k ∈ K.

Lemma 4.1. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space and k ∈ K, g ∈ (L ^X) ^E . Then Spcom (k, g) ≥ α if and only if for each $U \in (L^{X})^{E}$ , $τ_{k}^{sp} (U) \land [g \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [g \tilde{⊑} ⋁ V]$

Proof. Let α ∈ L, g ∈ (L ^X) ^E and k ∈ K arbitrarily be given. Hence,

Spcom (k, g) ≥ α if and only if $⋀_{U \subseteq (L^{X})^{E}} (τ_{k}^{sp} (U) \mapsto ([g \tilde{⊑} \lor U] \mapsto ⋁_{V \in 2^{U}} [g \tilde{⊑} \lor V])) \geq α$ ⇔ $τ_{k}^{sp} (U) \mapsto ([g \tilde{⊑} \lor U] \mapsto ⋁_{V \in 2^{U}} [g \tilde{⊑} \lor V])) \geq α$ ⇔ $(τ_{k}^{sp} (U) \land [g \tilde{⊑} \lor U]) \mapsto ⋁_{V \in 2^{U}} [g \tilde{⊑} \lor V]) \geq α$ ⇔ $τ_{k}^{sp} (U) \land [g \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [g \tilde{⊑} ⋁ V]$ , for each $U \subseteq (L^{X})^{E}$ .

Theorem 4.3. For an L-fuzzy (E, K)-soft supra topological space (X, τ), k ∈ K and h ∈ (L ^X) ^E, we have Spcom (k, h) ≥ α if and only if

$⋁_{f \in P} (τ^{*})_{k}^{sp} (f^{'}) \lor (⋁_{x \in X} ⋁_{e \in E} (h_{e} (x) \land ⋀_{f \in P} f_{e} (x))) \lor α^{'} \geq ⋀_{R \in 2^{P}} ⋁_{x \in X} ⋁_{e \in E} (h_{e} (x) \land ⋀_{f \in R} f_{e} (x)),$ for all $P \subseteq (L^{X})^{E} .$

Proof. The proof is clear from Lemma 4.1, complement operator and the definition of semi-preclosedness.

Theorem 4.4. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space, k ∈ K and f ∈ (L ^X) ^E. Then the following is satisfied.

$Spcom (k, f) = ⋁ {α \in L ∣ ⋀_{g \in U} τ_{k}^{sp} (g) \land [f \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X})^{E}} .$

Proof. From Lemma 4.1, one knows that Spcom (k, f) is the upper bound of the following set

${α \in L ∣ ⋀_{g \in U} τ_{k}^{sp} (g) \land [f \tilde{⊑} \lor U] \land α \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} \lor V], \forall U \subseteq (L^{X})^{E}}$ .

By using the definition of semi-precompactness, for each $U \subseteq (L^{X})^{E}$ , the following is obtained.

Spcom (k, f) $\leq ⋀_{g \in U} τ_{k}^{sp} (g) \mapsto ([f \tilde{⊑} \lor U] \mapsto ⋁_{V \in 2^{U}} [f \tilde{⊑} \lor V])$ $= (⋀_{g \in 2^{U}} τ_{k}^{sp} (g) \land [f \tilde{⊑} \lor U]) \mapsto ⋁_{V \in 2^{U}} [f \tilde{⊑} \lor V]$ .

Now taking into consideration the properties of the operation " ↦ ", one can get

$⋀_{g \in U} τ_{k}^{sp} (g) \land [f \tilde{⊑} \lor U] \land Spcom (k, f) \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} \lor V]$

and therefore,

$Spcom (k, f) \in {α \in L ∣ ⋀_{g \in U} τ_{k}^{sp} (g) \land [f \tilde{⊑} \lor U] \land α \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} \lor V], \forall U \subseteq (L^{X})^{E}}$ .

So,

$Spcom (k, f) \leq ⋁ {α \in L ∣ ⋀_{g \in U} τ_{k}^{sp} (g) \land [f \tilde{⊑} \lor U] \land α \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} \lor V], \forall U \subseteq (L^{X})^{E}}$

Hence, the desired equality is proved.

Theorem 4.5. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space and f, g ∈ (L ^X) ^E . Then for each k ∈ K, $Spcom (k, f ⊔ g) \geq Spcom (k, f) \land Spcom (k, g) .$

Proof. Let the hypothesis be valid. Now we obtain the result as follows:

$Spcom (k, f ⊔ g) = ⋁ {α ∣ ⋀_{h \in U} \land [f ⊔ g \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [f ⊔ g \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X})^{E}}$ $= ⋁ {α ∣ ⋀_{h \in U} τ_{k}^{sp} (h) \land [f \tilde{⊑} ⋁ U] \land [g \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} ⋁ V] \land [g \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X})^{E}}$ $\geq ⋁ {α \in L ∣ ⋀_{h \in U} τ_{k}^{sp} (h) \land [f \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X})^{E}}$ $\land ⋁ {α \in L ∣ ⋀_{h \in U} τ_{k}^{sp} (h) \land [g \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [g \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X})^{E}}$ = Spcom (k, f) ∧ Spcom (k, g).

Theorem 4.6. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space and f, g ∈ (L ^X) ^E . Then the following inequality is valid for each k ∈ K, $Spcom (k, f ⊓ g) \geq Spcom (k, f) \land τ_{k}^{sp} (g^{'}) .$

Proof. One can prove the claim in a following way: $Spcom (k, f ⊓ g) = ⋁ {α \in L ∣ ⋀_{h \in U} τ_{k}^{sp} (h) \land [f ⊓ g \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [f ⊓ g \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X})^{E}}$ $= ⋁ {α \in L ∣ ⋀_{h \in U} τ_{k}^{sp} (h) \land [f \tilde{⊑} g^{'} ⊔ ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [f \tilde{⊑} g^{'} ⊔ ⋁ V], \forall U \subseteq (L^{X})^{E}}$ $\geq Spcom (k, f) \land τ_{k}^{sp} (g^{'}) .$

Corollary 4.1. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space and h ∈ (L ^X) ^E . Then the following result is obvious for each k ∈ K, $Spcom (k, h) \geq Spcom (k, \tilde{1}) \land τ_{k}^{sp} (h^{'}) .$

Theorem 4.7. Let (X, τ ¹) and (X, τ ²) be two L-fuzzy (E, K)-soft topological spaces s.t τ ¹ ≤ τ ². Then for any k ∈ K and g ∈ (L ^X) ^E, we have Spcom _{τ
²} (k, g) ≤ Spcom _{τ
¹} (k, g) .

Proof. It is clear from Definition 4.1.

Theorem 4.8. Let φ _ψ,η : (X ₁, τ ¹) → (X ₂, τ ²) be a semi-preirresolute function between L-fuzzy (E ₁, K ₁)-soft and L-fuzzy (E ₂, K ₂)-soft supra topological spaces. Then for each k ∈ K ₁ and g ∈ (L ^{X
₁}) ^{E
₁}, we have ${Spcom}_{τ^{1}} (k, g) \leq {Spcom}_{τ^{2}} (η (k), φ_{ψ} (g)) .$

Proof. Let k ∈ K ₁ and g be a fuzzy soft set on X ₁ with the parameter set E ₁. Then one gets

${Spcom}_{τ^{2}} (η (k), φ_{ψ} (g)) = ⋁ {α ∣ ⋀_{h \in U} (τ^{2})_{η (k)}^{sp} (h) \land [φ_{ψ} (g) \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [φ_{ψ} (g) \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X_{2}})^{E_{2}}}$ $\geq ⋁ {α ∣ ⋀_{h \in U} (τ^{1})_{k}^{sp} (φ_{ψ}^{- 1} (h)) \land [g \tilde{⊑} ⋁ φ_{ψ}^{- 1} (U)] \land α \leq ⋁_{V \in 2^{U}} [g \tilde{⊑} ⋁ φ_{ψ}^{- 1} (V)], \forall U \subseteq (L^{X_{2}})^{E_{2}}}$ $\geq ⋁ {α ∣ ⋀_{f \in P} (τ^{1})_{k}^{sp} (f) \land [g \tilde{⊑} ⋁ P] \land α \leq ⋁_{R \in 2^{P}} [g \tilde{⊑} ⋁ R], \forall P \subseteq (L^{X_{1}})^{E_{1}}}$ = Spcom _{τ
¹} (k, g).

This completes the proof.

Theorem 4.9. Let φ _ψ,η : (X ₁, τ ¹) → (X ₂, τ ²) be a semi-precontinuous function between L-fuzzy (E ₁, K ₁)-soft and L-fuzzy (E ₂, K ₂)-soft supra topological spaces. Then for each k ∈ K ₁ and g ∈ (L ^{X
₁}) ^{E
₁}, we have ${Spcom}_{τ^{1}} (k, g) \leq {com}_{τ^{2}} (η (k), φ_{ψ} (g)) .$

Proof. Let k ∈ K ₁ and g be a fuzzy soft set on X ₁ with the parameter set E ₁. Then one gets

${com}_{τ^{2}} (η (k), φ_{ψ} (g)) = ⋁ {α \in L ∣ ⋀_{h \in U} τ_{η (k)}^{2} (h) \land [φ_{ψ} (g) \tilde{⊑} ⋁ U] \land α \leq ⋁_{V \in 2^{U}} [φ_{ψ} (g) \tilde{⊑} ⋁ V], \forall U \subseteq (L^{X_{2}})^{E_{2}}}$ $\geq ⋁ {α ∣ ⋀_{h \in U} (τ^{1})_{k}^{sp} (φ_{ψ}^{- 1} (h)) \land [g \tilde{⊑} ⋁ φ_{ψ}^{- 1} (U)] \land α \leq ⋁_{V \in 2^{U}} [g \tilde{⊑} ⋁ φ_{ψ}^{- 1} (V)], \forall U \subseteq (L^{X_{2}})^{E_{2}}}$ $\geq ⋁ {⋀_{f \in P} (τ^{1})_{k}^{sp} (f) \land [g \tilde{⊑} ⋁ P] \land α \leq$ $⋁_{R \in 2^{P}} [g \tilde{⊑} ⋁ R], \forall P \in (L^{X_{1}})^{E_{1}}}$ = Spcom _{τ
¹} (k, g).

This completes the proof.

5. Other characterizations of semi-precompactness

Definition 5.1. Let (X, τ) be an L-fuzzy (E, K)-soft topological space, α ∈ L \ {0_L} and f ∈ (L ^X) ^E . Then the subfamily $U \subseteq (L^{X})^{E}$ is said to be

(1) an "α-shading of f" if for any x ∈ X and e ∈ E, it follows that $f_{e}^{'} (x) \lor ⋁_{g \in U} g_{e} (x) ≰ α .$

(2) a "strong α-shading of f" if $[f \tilde{⊑} ⋁ U] ≰ α .$

(3) an "α-remote family of f" if for any x ∈ X and e ∈ E, it follows that $f_{e} (x) \land ⋁_{h \in U} h_{e} (x) α .$

(4) a "strong α-remote family of f" if $⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋁_{h \in U} h_{e} (x)) α .$

(5) a "Q _α-cover of f" if for any x ∈ X and e ∈ E with f _e (x) ≰ α′, it follows that $⋁_{g \in U} g_{e} (x) \geq α .$

(6) a "Θ _α-cover of f" if for any x ∈ X and e ∈ E, it follows that $α \in Θ (f_{e}^{'} (x) \lor ⋁_{g \in U} g_{e} (x)) .$

(7) a "strong Θ _α-cover of f" if $α \in Θ ([f \tilde{⊑} ⋁ U]) .$

It is noted that if for α ∈ c (L), $U$ is a Q _α-cover of f iff $α \leq [f \tilde{⊑} ⋁ U] .$ It is obvious that a strong Θ _α-cover of f is a Θ _α-cover of f.

Theorem 5.1. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space, f ∈ (L ^X) ^E and α ∈ L \ {0_L}. Then the following statements are equivalent

(1) Spcom (k, f) ≥ α .

(2) For each β ∈ p (L), βnotgeqα, ever strong β-shading $U$ of f with $⋀_{h \in U} τ_{k}^{sp} (h) ≰ β$ has a finite subcollection $V$ which is a strong β-shading of f .

(3) For each β ∈ p (L), βnotgeqα, ever strong β-shading $U$ of f with $⋀_{h \in U} τ_{k}^{sp} (h) ≰ β$ has a finite subcollection $V$ and γ ∈ Ω ^* (β) where $V$ is a γ-shading of f .

(4) For each β ∈ p (L), βnotgeqα, ever strong β-shading $U$ of f with $⋀_{h \in U} τ_{k}^{sp} (h) ≰ β$ has a finite subcollection $V$ and γ ∈ Ω ^* (β) where $V$ is a strong γ-shading of f .

(5) For each β ∈ c (L), β ≰ α′, every strong β-remote family $P$ of f with $⋁_{h \in P} (τ^{*})_{k}^{sp} (h) ≰ β^{'}$ has a finite subcollection $R$ which is a strong β-remote family of f .

(6) For each β ∈ c (L), β ≰ α′, every strong β-remote family $P$ of f with $⋁_{h \in P} (τ^{*})_{k}^{sp} (h) ≰ β^{'}$ has a finite subcollection $R$ and γ ∈ Θ ^* (β) where $R$ is a γ-remote family of f .

(7) For each β ∈ c (L), β ≰ α′, every strong β-remote family $P$ of f with $⋁_{h \in P} (τ^{*})_{k}^{sp} (h) ≰ β^{'}$ there is a finite subcollection $R$ of $P$ and γ ∈ Θ ^* (β) where $R$ is a strong γ-remote family of f .

(8) For each β ≤ α, γ ∈ Θ (β), β, γ ≠ 0, every Q _β-cover $U \subseteq (τ k)_{β}$ of f has a finite subfamily $V$ which is a Q _γ-cover of f .

(9) For each β ≤ α, γ ∈ Θ (β), β, γ ≠ 0, every Q _β-cover $U \subseteq (τ k)_{β}$ of f has a finite subfamily $V$ which is a strong Θ _β-cover of f .

(10) For each β ≤ α, γ ∈ Θ (β), β, γ ≠ 0, every Q _β-cover $U \subseteq (τ k)_{β}$ of f has a finite subfamily $V$ which is a Θ _γ-cover of f .

(11) For each β ≤ α, γ ∈ Θ (β), β, γ ≠ 0, every strong Θ _β-cover $U \subseteq (τ k)_{β}$ of f has a finite subfamily $V$ which is a Q _γ-cover of f .

(12) For each β ≤ α, γ ∈ Θ (β), β, γ ≠ 0, every strong Θ _β-cover $U \subseteq (τ kitscx)_{β}$ of f has a finite subfamily $V$ which is a strong Θ _γ-cover of f .

(13) For each β ≤ α, γ ∈ Θ (β), β, γ ≠ 0, every strong Θ _β-cover $U \subseteq (τ k)_{β}$ of f has a finite subfamily $V$ which is a Θ _γ-cover of f .

Theorem 5.2. Let (X, τ) be an L-fuzzy (E, K)-soft supra topological space, f ∈ (L ^X) ^E and α ∈ L \ {0_L}. If Θ (γ ∧ δ) = Θ (γ) ∧ Θ (δ), for each γ, δ ∈ L, then the following statements are equivalent.

(1) Spcom (k, f) ≥ α .

(2) For each β ∈ Θ (α), β ≠ 0, every strong Θ _β-cover $U$ of f with $β \in Θ (τ_{k}^{sp} (U))$ has a finite subcollection $V$ which is a Q _β-cover of f .

(3) For each β ∈ Θ (α), β ≠ 0, every strong Θ _β-cover $U$ of f with $β \in Θ (τ_{k}^{sp} (U))$ has a finite subcollection $V$ which is a strong Θ _β-cover of f .

(4) For each β ∈ Θ (α), β ≠ 0, every strong Θ _β-cover $U$ of f with $β \in Θ (τ_{k}^{sp} (U))$ has a finite subcollection $V$ which is a Θ _β-cover of f .

6. Conclusion

To the best of our knowledge, the tool of soft set theory is a new efficacious technique to dispose uncertainties and it focuses on the parametrization. For this reason, we opt to handle this tool in theoretical aspect and to deal with the parameterized version of semi-precompactness, which is a one of the stronger form of compactness and important topological concept.

It is known that, not only theoretics but also applications of these theoretics takes very important place in our life. Especially, it is worth noting that decision making in an imprecise environment has been showing more and more role in real world applications. To reflect this role, Zhan et al. [16 , 41–48] published some interesting papers and applied softness and roughness to the decision making problems well. For further research, we hope to impose our results obtained in this study and also in [9] to the decision making problems in the light of the Prof. J. Zhan’s papers.

Footnotes

Acknowledgements

The authors wish to thank the associate editor and the referees for their valuable suggestions.

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