New degrees for functions in ( L,M )-fuzzy topological spaces based on ( L,M )-fuzzy semiopen and ( L,M )-fuzzy preopen operators

Abstract

In this paper, we will introduce the notions of semiopenness, semicontinuity, preopenness, precontinuity, irresolutness and preirresolutness degree of functions in (L, M)-fuzzy topological spaces based on the implication operation. Their elementary properties in topology will be extended to the setting of (L, M)-fuzzy topology through graded concepts. Further we characterize and discuss their relationships with the degree of semi(pre)-compactness, semi(pre)-connectedness, Semi - T₁, Pre - T₁, Semi - T₂, and Pre - T₂.

Keywords

semiopen function semicontinuous function irresolute function semi-homomorphism preopen function precontinuous function preirresolute function pre-homomorphism

1 Introduction

After fuzzy topology was initiated by Chang [3] and later redefined in a slightly different method by Goguen [8], many researchers interested in generalizing it. In Chang’s approach, open sets are fuzzy while the topological structure still crisp. This approach has been extended independently to a more general form by Höhle [9] and Ying [36]. In 1985, Kubiak [12] and Šostak [26] presented independently a new generalization to the concept of fuzzy topology. Since then, many researchers have been studied deeply lattice-valued topology (see, for instance, [11, 25]).

The concept fuzzy category [29] was defined six years after introducing (L, M)-fuzzy topology and its study has been pursued for many years (see [13, 31]). In fuzzy category, potential objects and morphisms have been endowed with a certain degree of the corresponding lattice. Some categories related to algebra and topology were fuzzified in [30]. The degree of continuity for functions of (L, M)-fuzzy topological spaces considered as an important example of morphism degree in fuzzy category. Recently, many topological concepts endowed with degrees. In 2014, Pang [17] presented the degree of continuity, openness, and closeness of functions in L-fuzzifying topological spaces. Afterwards, Liang and Shi [14] used the implication operation to introduce the same degrees in (L, M)-fuzzy topological spaces. Many elementary properties in general topology extended to the setting of (L, M)-fuzzy topology through graded concepts. Many spatial structures are discussed in the degree approach, such as fuzzy convergence structures [18 , 34], fuzzy order [35], and fuzzy convex structures [32, 33]. Recently, Shi [22] has developed a new approach for presenting semiopenness and preopenness in L-fuzzy topological spaces by defining L-fuzzy semiopen and L-fuzzy preopen operators based on implication operation. Furthermore, many concepts have been reformulated based on Shi’s operators. In this framework, Ghareeb [4 –7] constructed new operators based on Shi’s operators and score some kinds of continuity, connectedness, and compactness degrees.

The main goal of this paper is to introduce semiopenness, semicontinuity, preopenness, precontinuity, irresolutness and preirresolutness degree of functions in (L, M)-fuzzy topological spaces based on implication operation and Shi’s operators. Their elementary properties in topology will be extended to the setting of (L, M)-fuzzy topological spaces through graded concepts. Therefore, the function can be regard as semiopen, semicontinuous, preopen, precontinuous, irresolute and preirresolute to some degree. Their characterizations and the relationship with semi (pre)-compactness, semi (pre)-connectedness, Semi - T₁, Pre - T₁, Semi - T₂, and Pre - T₂ are introduced and discussed.

2 Preliminaries

In the sequel, both L and M refer to a completely distributive De Morgan algebras. X is a nonempty set. The smallest and largest elements in L and M are denoted by ⊥_L, ⊤_L and ⊥_M, ⊤_M, respectively. For α, β ∈ L, we say that α is wedge below β in L [20], in symbols α ≪ β, if for every subset D ⊆ L, ⋁D ≥ β implies γ ≥ α for some γ ∈ D. A complete lattice L is completely distributive if and only if β = ⋁ {α ∈ L ∣ α ≪ β} for each β ∈ L. An element α in L is called co-prime if α ≤ β ∨ γ implies α ≤ β or α ≤ γ. The set of non-zero co-prime elements in L is denoted by J (L). L^X refers to the family of all L-subsets on X, where X is a nonempty set. $2^{U}$ refers to the family of all finite subfamilies of $U \subseteq L^{X}$ . It is obvious that L^X is a completely distributive De Morgan algebra when it inherits the structure of the lattice L in a natural way, by defining ⋁, ⋀, ≤ and ^′ pointwisely. Clearly, {x_λ ∣ λ ∈ J (L)} is the family of non-zero co-primes in L^X.

If L is a completely distributive De Morgan algebra, there exists an implication operation ↦ : L × L → L as the right adjoint for the meet operation ∧ by

$α \mapsto β = ⋁ {γ \in L ∣ α \land γ \leq β} .$ Moreover, the operation ↔ defined by $α \leftrightarrow β = (α \mapsto β) \land (β \mapsto α) .$ The following lemma lists some properties of implication operation.

Lemma 2.1. [10] if (L, ⋁, ⋀) is a completely distributive lattice and ↦ is the implication operation corresponding to ∧, then for each α, β, γ ∈ L, {α_i} _i∈Ω, and {β_i} _i∈Ω ⊆ L, the following statements hold:

(α ↦ β) ≥ γ ⇔ α ∧ γ ≤ β.

α ≤ β ⇔ α ↦ β = ⊤ _L.

α ↦ (β ↦ γ) = (α ∧ β) ↦ γ.

(γ ↦ α) ∧ (α ↦ β) ≤ γ ↦ β.

γ ↦ α ≤ (α ↦ β) ↦ (γ ↦ β).

α ↦ ⋀ _i∈Ωα_i = ⋀ _i∈Ω (α ↦ α_i), hence α ↦ β ≤ α ↦ γ whenever β ≤ γ.

⋁_i∈Ωα_i ↦ β = ⋀ _i∈Ω (a_i ↦ β), hence α ↦ γ ≥ β ↦ γ whenever α ≤ β.

Definition 2.2. [10 , 26] An (L, M)-fuzzy topology on a set X is a map $T : L^{X} \to M$ such that:

$T ({\underline{⊤}}_{L^{X}}) = T ({\underline{⊥}}_{L^{X}}) = ⊤_{M}$ ;

∀ U, V ∈ L^X, $T (U \land V) \geq T (U) \land T (V)$ ;

∀ U_i ∈ L^X, i ∈ Ω, $T (\lor_{i \in Ω} U_{i}) \geq \land_{i \in Ω} T (U_{i})$ .

$T (U)$ can be interpreted as the degree to which U is an open set. $T^{*} (U) = T (U^{'})$ will be called the degree of closedness of U. The pair $(X, T)$ is called an (L, M)-fuzzy topological space. A mapping $f : (X, T_{1}) ⟶ (Y, T_{2})$ is said to be continuous with respect to (L, M)-fuzzy topologies $T_{1}$ and $T_{2}$ if $T_{1} (f \leftarrow (U)) \geq T_{2} (U)$ holds for all U ∈ L^Y, where f^← is defined by f^← (U) (x) = U (f (x)).

Definition 2.3. [1, 16] An L-subset U in an L-topological space $(X, T)$ is called:

Semiopen if there exists a $V \in T$ such that V ≤ U ≤ Cl (V). U is called semiclosed iff U′ is semiopen.

Preopen if U ≤ Int (Cl (U)). U is called pre-closed iff U′ is preopen.

Definition 2.4. Let $(X, T_{1})$ and $(Y, T_{2})$ be two L-topological spaces. A map $f : (X, T_{1}) ⟶ (Y, T_{2})$ is called:

semicontinuous [1] (resp. precontinuous [16]) if f^← (V) is semiopen (resp. preopen) in $(X, T_{1})$ for every open L-subset V in $(Y, T_{2})$ .

Irresolute [2] (resp. preirresolute [16]) if f^← (V) is semiopen (resp. preopen) in $(X, T_{1})$ for every semiopen (resp. preopen) L-subset V in $(Y, T_{2})$ .

In [24], Shi defined the concept of (L, M)-fuzzy closure operators as follows.

Definition 2.5. [24] Let $T$ be an (L, M)-fuzzy topology on X. The mapping ${Cl}^{T} : L^{X} ⟶ M^{J (L^{X})}$ defined as: ∀x_λ ∈ J (L^X), ∀U ∈ L^X, ${Cl}^{T} (U) (x_{λ}) = \underset{x_{λ \leq U_{1} \geq U}}{\land} {(T (U_{1}^{'}))}^{'}$ is called the (L, M)-fuzzy closure operator induced by $T$ .

Definition 2.6. [22] Let $T$ be an (L, M)-fuzzy topology on X. For any U ∈ L^X, define a mapping $S, P : L^{X} ⟶ M$ by $S (U) = \underset{_{V \leq U}}{\lor} {T (V) \land \underset{x_{λ} ≪ U}{\land} \underset{x_{λ} \leq W \geq V}{\land} {(T (W^{'}))}^{'}},$ and $\begin{array}{l} P (U) & = \\ \underset{x_{λ} ≪ U}{\land} \underset{x_{λ} ≪ U}{\lor} {T (V) \land \underset{y_{μ} ≪ V}{\land} \underset{y_{μ} \leq W \geq U}{\land} {(T (W^{'}))}^{'}} \end{array}$ Then $S$ and $P$ are called the (L, M)-fuzzy semiopen and (L, M)-fuzzy preopen operators induced by $T$ respectively, where $S (U)$ (resp. $P (U)$ ) can be regarded as the degree to which U is semiopen (resp. preopen) and $S^{*} (U) = S (U^{'})$ (resp. $P^{*} (U) = P (U^{'})$ ) can be regarded as the degree to which U is semiclosed (resp. preclosed).

Shi [22] combined Definition 2.5 and Definition 2.6 to state the following corollary.

Corollay 2.7. Let $T$ be an (L, M)-fuzzy topology on X. Then for any U ∈ L^X, $S (U) = \underset{V \leq U}{\lor} {T (V) \land \underset{x_{λ} ≪ U}{\land} {Cl}^{T} (V) (x_{λ})},$ and $P (U) = \underset{x_{λ} ≪ U}{\land} \underset{x_{λ} ≺ V}{\lor} {T (V) \land \underset{y_{μ} ≪ U}{\land} {Cl}^{T} (U) (y_{μ})} .$

Theorem 2.8. [22] Let $T$ be an (L, M)-fuzzy topology on X, $S$ and $P$ be the (L, M)-fuzzy semiopen and (L, M)-fuzzy preopen operators induced by $T$ , respectively. Then $T (U) \leq S (U)$ and $T (U) \leq P (U)$ for each U ∈ L^X.

Theorem 2.9. [22] Let $T$ be an (L, M)-fuzzy topology on X, $S$ and $P$ be the (L, M)-fuzzy semiopen and (L, M)-fuzzy preopen operators induced by $T$ , respectively. Then $S (\underset{i \in Ω}{\lor} U_{i}) \geq \underset{i \in Ω}{\land} S (U_{i})$ and $P (\underset{i \in Ω}{\lor} U_{i}) \geq \underset{i \in Ω}{\land} P (U_{i})$ for each subfamily {U_i : i ∈ Ω} of L^X.

Definition 2.10. [22] A mapping $f : (X, T_{1}) ⟶ (Y, T_{2})$ between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ is called:

semicontinuous if $T_{2} (V) \leq S_{1} (f^{\leftarrow} (V))$ holds for any V ∈ L^Y;

irresolute if $S_{2} (V) \leq S_{1} (f^{\leftarrow} (V))$ holds for each V ∈ L^Y.

precontinuous if $T_{2} (V) \leq P_{1} (f^{\leftarrow} (V))$ holds for each V ∈ L^Y;

preirresolute if $P_{2} (V) \leq P_{1} (f^{\leftarrow} (V))$ holds for each V ∈ L^Y.

Corollay 2.11. [22] Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a mapping between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ . Then:

f is semicontinuous if and only if $T_{2}^{*} (V) \leq S_{1}^{*} (f^{\leftarrow} (V))$ for each V ∈ L^Y;

f is irresolute if and only if $S_{2}^{*} (V) \leq S_{1}^{*} (f^{\leftarrow} (V))$ for each V ∈ L^Y.

f is precontinuous if and only if $T_{2}^{*} (V) \leq P_{1}^{*} (f^{\leftarrow} (V))$ for each V ∈ L^Y;

f is preirresolute if and only if $P_{2}^{*} (V) \leq P_{1}^{*} (f^{\leftarrow} (V))$ for each V ∈ L^Y.

Theorem 2.12. [22] Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a function between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ . Then:

If f is continuous function, then f is also semicontinuous.

If f is continuous function, then f is also precontinuous.

If f is irresolute function, then f is also semicontinuous.

If f is preirresolute function, then f is also precontinuous.

Definition 2.13. A mapping $f : (X, T_{1}) ⟶ (Y, T_{2})$ between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ is called:

semiopen if $T_{1} (U) \leq S_{2} (f^{\to} (U))$ holds for each U ∈ L^X;

preopen if $T_{1} (U) \leq P_{2} (f^{\to} (U))$ holds for each U ∈ L^X;

Definition 2.14. An (L, M)-fuzzy pre quasi-neighborhood system on X is a family $P = {P_{x_{λ}} ∣ x_{λ} \in J (L^{X})}$ of functions ${P_{x_{λ}} : L^{X ⟶ M}}$ satisfying the following conditions:

$P_{x_{λ}} ({\underline{⊥}}_{L^{X}}) = ⊥_{M}$ .

$P_{x_{λ}} (U) \neq ⊥_{M}$ , then x_λ ≰ U′.

$P_{x_{λ}} (U_{1} \land U_{2}) \leq P_{x_{λ}} (U_{1}) \land P_{x_{λ}} (U_{2})$ .

$P_{x_{λ}} (U) = \underset{x_{λ} ≰ U_{1} \geq U^{'}}{\lor} \underset{y_{μ} ≰ U_{1}^{'}}{\land} P_{y_{μ}} (U_{1})$ .

Definition 2.15. [5] An (L, M)-fuzzy preclosure operator on X is a mapping pCl : L^X ⟶ M^{J(L^X)} satisfying the following conditions:

pCl (U) (x_λ) = ⋀ _μ≪λpCl (U) (x_μ), for any x_λ ∈ J (L^X).

$pCl ({\underline{⊥}}_{L^{X}}) (x_{λ}) = ⊥_{M}$ for any x_λ ∈ J (L^X).

pCl (U) (x_λ) = ⊤ _M for any x_λ ≤ U.

for all α ∈ M_⊥, (pCl (⋁ (pCl (U)) _[α])) _[α] ⊂ (pCl (U)) _[α].

pCl (U) (x_λ) is called the degree to which x_λ belongs to the preclosure of U.

Theorem 2.16. [5] Let $P$ be the (L, M)-fuzzy preopen operator on X and let ${pCl}^{P}$ be the (L, M)-fuzzy preclosure operator induced by $P$ . Then for each x_λ ∈ J (L^X) and U ∈ L^X, $P C l^{P} (U) (x_{λ}) = \underset{_{x_{λ} ≰ U_{1} \geq U}}{\land} {(P (U_{1}^{'}))}^{'} .$

Definition 2.17. [5] Let $(X, T)$ be an (L, M)-fuzzy topological space and U ∈ L^X. Define $\begin{array}{l} p C o n (U) = \underset{\begin{matrix} U_{1}, U_{2} \in L^{X} \ {{\underline{⊥}}_{L} X}, \\ U = U_{1} \lor U_{2} \end{matrix}}{\land} \\ {\underset{x_{λ} \leq U_{1}}{\lor} p C l (U_{2}) (x_{λ}) \lor \underset{y_{λ} \leq U_{2}}{\lor} p C l (U_{1}) (y_{β})} . \end{array}$ Then pCon (U) is said to be the preconnectedness degree of U.

Theorem 2.18. [5] Let $(X, T)$ be an (L, M)-fuzzy topological space and U ∈ L^X. Then $P C o n (U) = \underset{\begin{matrix} U \land U_{1} \neq {\underline{⊥}}_{L^{X}}, \\ U \land U_{2} \neq {\underline{⊥}}_{L^{X}}, \\ U \land U_{1} \land U_{2} \neq {\underline{⊥}}_{L^{X}}, \\ U \leq U_{1} \lor U_{2} \end{matrix}}{\land} { {(P (U_{1}^{'}))}^{'} \lor (P (U_{2}^{'}))^{'}} .$

Similarly, we define an (L, M)-fuzzy semi quasi-neighborhood system and (L, M)-fuzzy semiclosure operator as follows:

Definition 2.19. An (L, M)-fuzzy semi (resp. pre)-neighborhood system on X is a family $N = {N_{x_{λ}} ∣ x_{λ} \in J (L^{X})}$ (resp. M = {M_{x
_λ} ∣ x_λ ∈ J (L^X)}) of functions ${N_{x_{λ}} : L^{X} ⟶ M}$ (resp. {M_{x
_λ} : L^X ⟶ M}) satisfying the following conditions:

$N_{x_{λ}} ({\underline{⊥}}_{L^{X}}) = ⊥_{M}$ (resp. $_{x_{λ}} ({\underline{⊥}}_{L^{X}}) = ⊥_{M}$ ).

$N_{x_{λ}} (U) \neg = ⊥_{M}$ (resp. M_{x
_λ} (U) ¬ = ⊥ _M), then x_λ ≰ U.

$N_{x_{λ}} (U_{1} \land U_{2}) \leq N_{x_{λ}} (U_{1}) \land N_{x_{λ}} (U_{2})$ (resp. M_{x
_λ} (U₁ ∧ U₂) ≤ M_{x
_λ} (U₁) ∧ M_{x
_λ} (U₂)).

$N_{x_{λ}} (U) = \underset{x_{λ} \leq U_{1} \geq U}{\lor} \underset{y_{μ} ≪ U_{1}}{\land} N_{y_{μ}} (U_{1})$ (resp. M_{x
_λ} (U) = ⋁ _{x_λ≤U₁≥U} ⋀ _{y_μ≪U₁}M_{y
_μ} (U₁)).

Definition 2.20. An (L, M)-fuzzy semi (resp. pre)-interior operator on X is a function sInt : L^X ⟶ M^{J(L^X)} (resp. pInt : L^X ⟶ M^{J(L^X)}) satisfying the following conditions:

sInt (U) (x_λ) = ⋀ _μ≪λsInt (U) (x_μ) (resp. pInt (U) (x_λ) = ⋀_μ≪λpInt (U) (x_μ)), for any x_λ ∈ J (L^X).

$sInt ({\underline{⊤}}_{L^{X}}) (x_{λ}) = ⊤_{M}$ (resp. $pInt ({\underline{⊤}}_{L^{X}}) (x_{λ}) = ⊤_{M}$ ) for any x_λ ∈ J (L^X).

sInt (U) (x_λ) = ⊥ _M (resp. pInt (U) (x_λ) = ⊥ _M) for any x_λ ≰ U.

For all α ∈ M_⊥, (sInt (U)) ^(α) ⊆ (sInt (⋁ (sInt (U)) ^(α))) ^(α) (resp. (pInt (U)) ^(α) ⊆ (pInt (⋁ (pInt (U)) ^(α))) ^(α)).

sInt (U) (x_λ) (resp. pInt (U) (x_λ)) is called the degree to which x_λ belongs to the semi (resp. pre)-interior of U.

Theorem 2.21. Let $T$ be an (L, M)-fuzzy topology on X and $N^{T} = {N_{x_{λ}}^{T} ∣ x_{λ} \in J (L^{X})}$ (resp. $^{T} = {_{x_{λ}}^{T} ∣ x_{λ} \in J (L^{X})}$ ) be the (L, M)-fuzzy semi (resp. pre)-neighborhood system induced by $T$ . Define the functions ${sInt}^{T}, {pInt}^{T} : L^{X} ⟶ M^{J (L^{X})}$ by ${sInt}^{T} (U) (x_{λ}) = N_{x_{λ}}^{T} (U)$ and ${pInt}^{T} (U) (x_{λ}) =_{x_{λ}}^{T} (U)$ . Then ${sInt}^{T}$ and ${pInt}^{T}$ are (L, M)-fuzzy semi-interior and preclosure operators on X, respectively.

Proof. Straightforward. □

Definition 2.22. An (L, M)-fuzzy semi quasi-neighborhood system on X is a family $S = {S_{x_{λ}} ∣ x_{λ} \in J (L^{X})}$ of functions ${S_{x_{λ}} : L^{X} ⟶ M}$ satisfying the following conditions:

$S_{x_{λ}} ({\underline{⊥}}_{L^{X}}) = ⊥_{M}$ .

$S_{x_{λ}} (U) \neg = ⊥_{M}$ , then x_λ ≰ U′.

$S_{x_{λ}} (U_{1} \land U_{2}) \leq S_{x_{λ}} (U_{1}) \land S_{x_{λ}} (U_{2})$ .

$S_{x_{λ}} (U) = \underset{x_{λ} ≰ U_{1} \geq U^{'}}{\lor} \underset{y_{μ} ≰ U_{1}^{'}}{\land} S_{y_{μ}} (U_{1})$ .

Definition 2.23. An (L, M)-fuzzy semiclosure operator on X is a mapping sCl : L^X ⟶ M^{J(L^X)} satisfying the following conditions:

sCl (U) (x_λ) = ⋀ _μ≪λsCl (U) (x_μ), for any x_λ ∈ J (L^X).

$sCl ({\underline{⊥}}_{L^{X}}) (x_{λ}) = ⊥_{M}$ for any x_λ ∈ J (L^X).

sCl (U) (x_λ) = ⊤ _M for any x_λ ≤ U.

for all α ∈ M_⊥, (sCl (⋁ (sCl (U)) _[α])) _[α] ⊂ (sCl (U)) _[α].

sCl (U) (x_λ) is called the degree to which x_λ belongs to the semiclosure of U.

Theorem 2.24. Let $S$ be the (L, M)-fuzzy semiopen operator on X and let ${sCl}^{S}$ be the (L, M)-fuzzy semiclosure operator induced by $S$ . Then for each x_λ ∈ J (L^X) and U ∈ L^X, $s C l^{S} (U) (x_{λ}) = \underset{x_{λ} ≰ U_{1} \geq U}{\land} {(S (U_{1}^{'}))}^{'} .$

Proof. Straightforward. □

Theorem 2.25. Let $T$ be an (L, M)-fuzzy topology on X and $S^{T} = {S_{x_{λ}} ∣ x_{λ} \in J (L^{X})}$ (resp. $P^{T} = {P_{x_{λ}} ∣ x_{λ} \in J (L^{X})}$ ) be the (L, M)-fuzzy semi quasi-neighborhood system (resp. (L, M)-fuzzy pre quasi-neighborhood system) induced by $T$ . Define the function ${sCl}^{T} : L^{X} ⟶ M^{J (L^{X})}$ (resp. ${pCl}^{T} : L^{X} ⟶ M^{J (L^{X})}$ ) by ${sCl}^{T} (U) (x_{λ}) = (S_{x_{λ}} (U^{'}))^{'}$ (resp. ${pCl}^{T} (U) (x_{λ}) = (P_{x_{λ}} (U^{'}))^{'}$ ). Then ${sCl}^{T}$ (resp. ${pCl}^{T}$ ) is an (L, M)-fuzzy semiclosure (resp. preclosure) operator on X.

Proof. Straightforward. □

Definition 2.26. Let $(X, T)$ be an (L, M)-fuzzy topological space and U ∈ L^X. Define $\begin{array}{l} s C o n (U) = & \underset{\begin{matrix} U_{1}, U_{2} \in L^{X} \ {{\underline{⊥}}_{L} X}, \\ U = U_{1} \lor U_{2} \end{matrix}}{\land} {\underset{x_{λ} \leq U_{1}}{\lor} s C l (U_{2}) (x_{λ}) \\ \lor \lor_{y_{β} \leq U_{2}} s C l (U_{1}) (y_{β})} \end{array}$

Then sCon (U) is said to be the semiconnectedness degree of U.

Theorem 2.27. Let $(X, T)$ be an (L, M)-fuzzy topological space and U ∈ L^X. Then $\begin{array}{l} s C o n (U) \\ = \underset{\begin{array}{l} U \land U_{1} \neq {\underline{⊥}}_{L} X, U \land U_{2} \neq {\underline{⊥}}_{L} X, \\ U \land U_{1} \land U_{2} \neq {\underline{⊥}}_{L} X, U \leq U_{1} \lor U_{2} \end{array}}{\land} {{(S ({U^{'}}_{1}))}^{'} \lor {(S ({U^{'}}_{2}))}^{'}} . \end{array}$

Proof. Straightforward. □

Definition 2.28. [27, 28] An L-fuzzy inclusion on X is a function $\tilde{\subset} : L^{X} \times L^{X} ⟶ L$ given by the equality $\tilde{\subset} (U_{1}, U_{2}) = \underset{x \in X}{\land} (U_{1}^{'} (x) \lor U_{2} (x)) [U_{1} \tilde{\subset} U_{2}] \tilde{\subset} (U_{1}, U_{2})$ . In the sequel, we shall write $[U_{1} \tilde{\subset} U_{2}]$ instead of $\tilde{\subset} (U_{1}, U_{2})$ .

Lemma 2.29. [23] Let f : X ⟶ Y be a function. Then for any $U \subseteq L^{X}$ , we have $\begin{array}{l} \underset{y \in Y}{\land} {f^{\to} {(U)}^{'} (y) & \lor \underset{_{V \in P}}{\lor} V (y)} = \\ \underset{_{x \in X}}{\land} {U^{'} (x) \lor \underset{_{V \in U}}{\lor} f^{\leftarrow} (V) (x)} .} \end{array}$

Definition 2.30. Let $T : L^{X} ⟶ M$ be an (L, M)-fuzzy topology and L = M. For any U ∈ L^X, let $\begin{array}{l} s c d_{T} (U) \\ = \underset{U \subseteq \in L^{X}}{\land} {S (U) \mapsto ([U \tilde{\subset} \lor U] \mapsto \underset{_{V \in 2^{(U)}}}{\lor} [U \tilde{\subset} \lor v])} \\ \begin{array}{l} = \underset{U \subseteq \in L^{X}}{\land} {\underset{_{U_{1} \in U}}{\land} S (U_{1}) \mapsto {\underset{x \in X}{\lor} (U^{'} \lor \underset{U_{1} \in U}{\lor} U_{1}) (x) \\ \mapsto \underset{_{V \in 2^{(U)}}}{\lor} \underset{x \in X}{\land} (U^{'} \lor \underset{_{U_{1} \in v}}{} U_{1}) (x)}}, \end{array} \end{array}$ and $\begin{array}{l} p c d_{T} (U) = \\ \underset{U \subseteq \in L^{X}}{\land} {P (U) \mapsto ([U \tilde{\subset} \lor U] \mapsto \underset{v \in 2 (U)}{\lor} [U \tilde{\subset} \lor V])} \\ \underset{U \subseteq \in L^{X}}{\land} {\underset{U_{1} \in U}{\land} P (U_{1}) \mapsto {\underset{x \in X}{\land} (U^{'} \lor \underset{U_{1} \in U}{\lor} U_{1}) (x) \\ \mapsto \underset{v \in 2 (U)}{\lor} \underset{x \in X}{\land} (U^{'} \lor \underset{U_{1} \in v}{\lor} U_{1}) (x)}} \end{array}$

Then ${scd}_{T} (U)$ and ${pcd}_{T} (U)$ are called the semi-compactness and pre-compactness degrees of U with respect to $T$ , respectively.

Definition 2.31. Let $(X, T)$ be an (L, M)-fuzzy topological space, we define the degree $Semi - T_{1} (X, T)$ and $Pre - T_{1} (X, T)$ to which $(X, T)$ is Semi - T₁ and Pre - T₁ as follows: $S e m i - T_{1} (X, T) = \underset{_{u_{1} u_{2}}}{\land} \underset{_{u_{1} U \geq u_{2}}}{\lor} S (U^{'}),$ and $Pre - T_{1} (X, T) = \underset{u_{1} \leq u_{2}}{\land} \underset{_{u_{1} \leq U \leq u_{2}}}{\lor} P (U^{'}),$ respectively.

Definition 2.32. Let $(X, T)$ be an (L, M)-fuzzy topological space, we define the degree $Semi - T_{2} (X, T)$ and $Pre - T_{2} (X, T)$ to which $(X, T)$ is Semi - T₂ and Pre - T₂ as follows: $\begin{matrix} S e m i - T_{2} (X, T) = \underset{_{u_{1} u_{2}}}{\land} & \lor {S (U_{1}^{'}) \land S (U_{2}) ∣ \\ u_{1} \leq U_{1} \geq U_{2} \geq u_{2}}, \end{matrix}$ and $\begin{matrix} P r e - T_{2} (X, T) = \underset{_{u_{1} u_{2}}}{\land} & \lor {P (U_{1}^{'}) \land P (U_{2}) ∣ \\ u_{1} U_{1} \geq U_{2} \geq u_{2}}, \end{matrix}$ respectively.

3 Degree of semiopenness, semicontinuity, preopenness, precontinuity, irresolutness and preirresolutness for functions in (L, M)-fuzzy topological spaces

In this section, we present the degree of semiopenness, semicontinuity, preopenness, precontinuity, irresolutness and preirresolutness for functions in (L, M)-fuzzy topological spaces. Moreover, we show that this notions can be characterized by (L, M)-fuzzy semi quasi-neighborhood systems, (L, M)-fuzzy semi neighborhood systems, (L, M)-fuzzy semi-interior operators and (L, M)-fuzzy semiclosure operators.

Definition 3.1. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a function between two (L, M)-fuzzy topological spaces. Then:

The degree Sc (f) to which f is semicontinuous is defined by $S c (f) = \underset{_{V \in L^{Y}}}{\land} {T_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))} .$

The degree So (f) to which f semiopen is defined by $S o (f) = \underset{_{U \in L^{X}}}{\land} {S_{1} (U) \mapsto S_{2} (f^{\to} (U))} .$

The degree Irr (f) to which f is irresolute is defined by $I r r (f) = \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\to} (V))} .$

The degree Pc (f) to which f is precontinuous is defined by $P c (f) = \underset{_{V \in L^{Y}}}{\land} {T_{2} (V) \mapsto P_{1} (f^{\to} (V))} .$

The degree Po (f) to which f preopen is defined by $P o (f) = \underset{_{U \in L^{X}}}{\land} {P_{1} (U) \mapsto P_{2} (f^{\to} (U))} .$

The degree Pir (f) to which f is preirresolute is defined by $P i r (f) = \underset{_{V \in L^{Y}}}{\land} {P_{2} (V) \mapsto P_{1} (f^{\to} (V))} .$

Definition 3.2. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between two (L, M)-fuzzy topological spaces. Then:

The degree Semi - Hom (f) (resp. Pre - Hom (f)) to which f semi-homomorphism (resp. pre- homomorphism) is defined by Semi - Hom (f) = Irr (f) ∧ So (f) (resp. Pre - Hom (f) = Pir (f) ∧ Po (f)). This study focusing on this kind of homomorphism.

The degree S - Hom (f) (resp. P - Hom (f)) to which f S-homomorphism (resp. P- homomorphism) is defined by S - Hom (f) = Sc (f) ∧ So (f) (resp. P - Hom (f) = Pc (f) ∧ Po (f)).

Remark 3.3.

By Lemma 2.1 (2), if Sc (f) = ⊤ _M, we have $S_{1} (f^{\leftarrow} (V)) \geq T_{2} (V)$ for any V ∈ L^Y. This is just the definition of semicontinuous function between two (L, M)-fuzzy topological spaces. Analogously, if Pc (f) = ⊤ _M (resp. Ir (f) = ⊤ _M, Pir (f) = ⊤ _M), then (2) , (3) , (4) , (5) , and (6) in Definition 3.1 are precisely the definition of semiopen function (resp. irresolute function, precontinuous function, preopen function, and preirresolute function) between two (L, M)-fuzzy topological spaces in the sense of Shi [22].

If $id : (X, T_{1}) \to (X, T_{1})$ is the identity function, then Irr (id) = So (id) = Semi - Hom (id) = ⊤ _M and Pir (id) = Po (id) = Pre - Hom (id) = ⊤ _M.

Combining Definition 3.1 and Corollary 2.11, we can state the following corollary.

Corollay 3.4. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a function between two (L, M)-fuzzy topological spaces. Then:

The degree Sc (f) to which f is semicontinuous can be given by $S c (f) = \underset{_{V \in L^{Y}}}{\land} {T_{2}^{*} (V) \mapsto S_{1}^{*} (f^{\leftarrow} (V))} .$

The degree Irr (f) to which f is irresolute can be given by $I r r (f) = \underset{_{V \in L^{Y}}}{\land} {S_{2}^{*} (V) \mapsto S_{1}^{*} (f^{\to} (V))} .$

The degree Pc (f) to which f is precontinuous can be given by $P c (f) = \underset{_{V \in L^{Y}}}{\land} {T_{2}^{*} (V) \mapsto P_{1}^{*} (f^{\to} (V))} .$

The degree Pir (f) to which f is preirresolute can be given by $P i r (f) = \underset{_{V \in L^{Y}}}{\land} {P_{2}^{*} (V) \mapsto P_{1}^{*} (f^{\to} (V))} .$

Definition 3.5. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a function between two (L, M)-fuzzy topological spaces. Then:

The degree Scl (f) to which f semiclosed is defined by $S c l (f) = \underset{_{U \in L^{X}}}{\land} {S_{1}^{*} (U) \mapsto S_{2}^{*} (f^{\to} (U))} .$

The degree Pcl (f) to which f preclosed is defined by $P c l (f) = \underset{_{U \in L^{X}}}{\land} {P_{1}^{*} (U) \mapsto P_{2}^{*} (f^{\to} (U))} .$

Theorem 3.6. Let $(X, T_{1})$ , $(Y, T_{2})$ and $(Z, T_{3})$ be (L, M)-fuzzy topological spaces, f : X ⟶ Y and g : Y ⟶ Z be two functions. Then:

Irr (f) ∧ Irr (g) ≤ Irr (g ∘ f).

So (f) ∧ So (g) ≤ So (g ∘ f).

Scl (f) ∧ Scl (g) ≤ Scl (g ∘ f).

Proof. The proofs of (2) and (3) are straightforward. We will just prove (1). Based on Definition 3.1 and Lemma 2.1 (4), we get $\begin{array}{l} I r r (f) \land I r r (g) = \\ \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))} \\ \land \underset{_{W \in L^{Z}}}{\land} {S_{3} (W) \mapsto S_{2} (g^{\leftarrow} (W))} \\ \leq \underset{_{W \in L^{Z}}}{\land} {S_{2} (g^{\leftarrow} (W)) \mapsto S_{1} (f^{\leftarrow} (g^{\leftarrow} (W)))} \\ \land \underset{_{W \in L^{Z}}}{\land} {S_{3} (W) \mapsto S_{2} (g^{\leftarrow} (W))} \\ = \underset{_{W \in L^{Z}}}{\land} {(S_{2} (g^{\leftarrow} (W)) \mapsto S_{1} ({(g \circ f)}^{\leftarrow} (W))) \\ \land (S_{3} (W) \mapsto S_{2} (g^{\leftarrow} (W)))} \\ \leq \underset{_{W \in L^{Z}}}{\land} {S_{3} (g^{\leftarrow} (W)) \mapsto S_{1} ({(g \circ f)}^{\leftarrow} (W))} \\ = I r r (g \circ f) . \end{array}$

Theorem 3.7. Let $(X, T_{1})$ , $(Y, T_{2})$ and $(Z, T_{3})$ be (L, M)-fuzzy topological spaces, f : X ⟶ Y and g : Y ⟶ Z be two functions. Then:

Pir (f) ∧ Pir (g) ≤ Pir (g ∘ f).

Po (f) ∧ Po (g) ≤ Po (g ∘ f).

Pcl (f) ∧ Pcl (g) ≤ Pcl (g ∘ f).

Proof. Similar to the proof of Theorem 3.6. □

Combining Definition 3.2 with Theorems 3.6 and 3.7, we can state the following corollary.

Corollay 3.8. Let $(X, T_{1})$ , $(Y, T_{2})$ and $(Z, T_{3})$ be (L, M)-fuzzy topological spaces, f : X ⟶ Y and g : Y ⟶ Z be two bijective functions. Then:

Semi - Hom (f) ∧ Semi - Hom (g) ≤ Semi - Hom (f ∘ g).

Pre - Hom (f) ∧ Pre - Hom (g) ≤ Pre - Hom (f ∘ g).

Theorem 3.9. Let $(X, T_{1})$ , $(Y, T_{2})$ and $(Z, T_{3})$ be (L, M)-fuzzy topological spaces and g : Y ⟶ Z be a functions. If f is a surjective function, then:

So (g ∘ f) ∧ Irr (f) ≤ So (g).

Scl (g ∘ f) ∧ Irr (f) ≤ Scl (g).

Po (g ∘ f) ∧ Pir (f) ≤ Po (g).

Pcl (g ∘ f) ∧ Pir (f) ≤ Pcl (g).

Proof. (1) From the surjectivity of the function f, we have (g ∘ f) ^→ (f^← (V)) = g^→ (V), for any V ∈ L^Y. By Lemma 2.1 (4), we have $\begin{array}{l} S o (g \circ f) \land I r r (f) = \\ \underset{_{U \in L^{X}}}{\land} {S_{1} (U) \mapsto S_{3} {(g \circ f)}^{\to} (U))} \\ \land \underset{V \in L^{Y}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))} \\ \leq & \underset{_{V \in L^{Y}}}{\land} {S_{1} (f^{\leftarrow} (V)) \mapsto S_{3} {(g \circ f)}^{\to} (f^{\leftarrow} (V))} \\ \land \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))} \\ = & \underset{_{V \in L^{Y}}}{\land} {(S_{1} (f^{\leftarrow} (V)) \mapsto S_{3} (g^{\to} (V))) \\ \land (S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V)))} \\ \leq & \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{3} (g^{\to} (V))} \\ = & S o (g) . \end{array}$ Similarly, we can prove (2), (3), and (4).□

In a similar way, we state the following theorem.

Theorem 3.10. Let $(X, T_{1})$ , $(Y, T_{2})$ and $(Z, T_{3})$ be (L, M)-fuzzy topological spaces, f : X ⟶ Y and g : Y ⟶ Z be two functions. If f is injective, then

So (g ∘ f) ∧ Irr (g) ≤ So (f).

Scl (g ∘ f) ∧ Irr (g) ≤ Scl (f).

Po (g ∘ f) ∧ Pir (g) ≤ Po (f).

Pcl (g ∘ f) ∧ Pir (g) ≤ Pcl (f).

Theorem 3.11. Let $(X, T_{1})$ and $(Y, T_{2})$ be two (L, M)-fuzzy topological spaces. If f : X ⟶ Y is a bijective function, then

$Irr (f) = ⋀_{U \in L^{X}} {S_{2} (f^{\leftarrow} (U) \mapsto S_{1} (U))}$ .

$So (f) = ⋀_{V \in L^{Y}} {S_{1} (f^{\leftarrow} (V) \mapsto S_{2} (V))}$ .

Irr (f^-1) = So (f) = Scl (f).

Proof. Since the proof of (2) analogous to (1), we will just prove (1) and (3).

(1) From the bijectivity of the function f, we have f^← (f^→ (U)) = U for any U ∈ L^X, and f^→ (f^← (V)) = V for any V ∈ L^Y. This implies $\begin{array}{l} \underset{_{U \in L^{X}}}{\land} {S_{2} (f^{\to} (U)) \mapsto S_{1} (U)} = \\ \underset{_{U \in L^{X}}}{\land} {S_{2} (f^{\to} (U)) \mapsto S_{1} (f^{\leftarrow} (f^{\to} (U)))} \\ \geq & \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))} \\ = & \underset{_{V \in L^{Y}}}{\land} {S_{2} (f^{\leftarrow} (f^{\to} (V))) \mapsto S_{1} (f^{\leftarrow} (V))} \\ \geq & \underset{_{U \in L^{X}}}{\land} {S_{2} (f^{\to} (V)) \mapsto S_{1} (U)} . \end{array}$ Hence $\begin{array}{l} I r r (f) & = \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))} \\ = \underset{_{U \in L^{X}}}{\land} {S_{2} (f^{\leftarrow} (U)) \mapsto S_{1} (U)} . \end{array}$

(3) From the bijectivity of f, we have (f^-1) ^← (U) = f^→ (U) and f^→ (U′) = f^→ (U) ′ for any U ∈ L^X. Then $\begin{array}{l} I r r (f^{- 1}) & = \underset{_{U \in L^{X}}}{\land} {S_{1} (U) \mapsto S_{2} {(f^{- 1})}^{\leftarrow} (U)} \\ = \underset{_{U \in L^{X}}}{\land} {S_{1} (U) \mapsto S_{2} (f^{\to} (U))} \\ = S o (f), \end{array}$ and $\begin{array}{l} S o (f^{- 1}) & = \underset{_{U \in L^{X}}}{\land} {S_{1} (A) \mapsto S_{2} (f^{\to} (U))} \\ = \underset{_{U \in L^{X}}}{\land} {S_{1} (U^{'}) \mapsto S_{2} (f^{\to} (U^{'}))} \\ = \underset{_{U \in L^{X}}}{\land} {S_{1} (U^{'}) \mapsto S_{2} (f^{\to} {(U)}^{'})} \\ = S c l (f) . \end{array}$

The proof is completed. □

Similarly, we can prove the following theorem.

Theorem 3.12. Let $(X, T_{1})$ and $(Y, T_{2})$ be two (L, M)-fuzzy topological spaces. If f is a bijective function, then

$Pir (f) = ⋀_{U \in L^{X}} {P_{2} (f^{\leftarrow} (U) \mapsto P_{1} (U))}$ .

$Po (f) = ⋀_{V \in L^{Y}} {P_{1} (f^{\leftarrow} (V) \mapsto P_{2} (V))}$ .

Pir (f^-1) = Po (f) = Pcl (f).

Corollay 3.13. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ , then

Semi - Hom (f) = Irr (f) ∧ Irr (f^-1) = Irr (f) ∧ Scl (f).

$S e m i - H o m (f) = \underset{_{U \in L^{X}}}{\land} {S_{2} (f^{\to} (U)) \mapsto S_{1} (U))}$ .

$S e m i - H o m (f) = \underset{_{V \in L^{Y}}}{\land} {S_{1} (f^{\leftarrow} (V)) \mapsto S_{2} (V))}$ .

Corollay 3.14. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ , then

Pre - Hom (f) = Pir (f) ∧ Pir (f^-1) = Pir (f) ∧ Pcl (f).

$P r e - H o m (f) = \underset{_{U \in L^{X}}}{\land} {P_{2} (f^{\to} (U)) \mapsto P_{1} (U))}$ .

$P r e - H o m (f) = \underset{_{V \in L^{Y}}}{\land} {P_{1} (f^{\leftarrow} (V)) \mapsto P_{2} (V))}$ .

The following corollaries and theorems characterize degrees of irresolutness, preirresolutness, semiopenness and preopenness by (L, M)-fuzzy semi (pre) quasi-neighborhood systems, (L, M)-fuzzy semi (pre) closure operators, and (L, M)-fuzzy semi (pre) interior operators.

Corollay 3.15. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a function between two (L, M)-fuzzy topological spaces. Then

$I r r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{x_{λ} \in J (L^{X})}{\land} {S_{f {(x)}_{λ}}^{S_{2}} (V) \mapsto S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V))}$ ;

$I r r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {N_{f {(x)}_{λ}}^{S_{2}} (V) (f {(x)}_{λ}) \mapsto N_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V))}$ ;

$I r r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {s I n t^{S_{2}} (V) (f {(x)}_{λ}) \mapsto s I n t^{S_{1}} (f^{\leftarrow} (V)) (x_{λ})}$ ;

$I r r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {s C l^{S_{2}} (V^{'}) {(f {(x)}_{λ})}^{'} \mapsto s C l^{S_{1}} (f^{\leftarrow} {(V)}^{'}) {(x_{λ})}^{'}}$ ;

Proof. (1) Since $S_{x_{λ}}^{S_{1}} (U) = ⋁_{x_{λ} U_{1}^{'} > U^{'}} S_{1} (U_{1})$ for any U ∈ L^X, and for all x_λ ∈ J (L^X) and V₁, V₂ ∈ L^Y, we have $f (x)_{λ} V_{1}^{'} \geq V_{2}^{'} \Rightarrow x_{λ} f^{\leftarrow} (V_{1})^{'} \geq f^{\leftarrow} (V_{2})$ . Therefore, we have $\begin{array}{l} \underset{_{V_{2} \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {S_{f {(x)}_{λ}}^{S_{2}} (V_{2}) \\ \mapsto S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V_{2}))} \\ = & \underset{_{V_{2} \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {\underset{_{f {(x)}_{λ} V_{1}^{'} \geq V_{2}^{'}}}{\lor} S_{2} (V_{1}) \\ \mapsto \underset{_{x_{λ} U_{1}^{'} \geq f^{\leftarrow} {(V_{2})}^{'}}}{\lor} S_{1} (U_{1})} \\ \geq & \underset{_{V_{2} \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {\underset{_{f {(x)}_{λ} V_{1}^{'} \geq V_{2}^{'}}}{\lor} S_{2} (V_{1}) \\ \mapsto \underset{_{x_{λ} f^{\leftarrow} {(V_{3})}^{'} \geq f^{\leftarrow} {(V_{2})}^{'}}}{\lor} S_{1} (f^{\leftarrow} (V_{3}))} \\ \geq & \underset{_{V_{2} \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} \underset{f {(x)}_{λ} V_{1}^{'} \geq V_{2}^{'}}{\land} {S_{2} (V_{1}) \\ \mapsto S_{1} (f^{\leftarrow} (V_{1}))} \\ \geq & \underset{_{V_{2} \in L^{Y}}}{\land} {S_{2} (V_{2}) \mapsto S_{1} (f^{\leftarrow} (V_{2}))} = I r r (f) . \end{array}$ Since $S_{1} (U) = \underset{_{x_{λ} \leq U^{'}}}{\land} S_{x_{λ}}^{S_{1}} (U)$ for any U ∈ L^X and x_λ≰f^← (V) ′ ⇒ f (x) _λ≰V′ for any x_λ ∈ J (L^X) and V ∈ L^Y. Then $\begin{array}{l} I r r (f) = \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))} \\ = \underset{_{V \in L^{Y}}}{\land} {\underset{_{y_{μ} \leq V^{'}}}{\land} S_{y_{μ}}^{S_{2}} (V) \\ \mapsto \underset{x_{λ} \leq f^{\leftarrow} {(V)}^{'}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V))} \\ \geq \underset{_{V \in L^{Y}}}{\land} {\underset{f {(x)}_{λ} \leq V}{\land} S_{f {(x)}_{λ}}^{S_{2}} (V) \\ \mapsto \underset{x_{λ} \leq f^{\leftarrow} {(V)}^{'}}{\land} S_{x_{λ}}^{S 1} (f^{\leftarrow} (V))} \\ \geq \underset{V \in L^{Y}}{\land} \underset{x_{λ} \leq f^{\leftarrow} {(V)}^{'}}{\land} {S_{f {(x)}_{λ}}^{S_{2}} (V) \\ \mapsto S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V))} \\ \geq \underset{V \in L^{Y}}{\land} \underset{x_{λ} \in J (L^{X})}{\land} {S_{f {(x)}_{λ}}^{S_{2}} (V) \mapsto S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V))} . \end{array}$ Thus we complete the proof of (1). The proof of (2), (3) and (4) can obtained from Theorem 2.25 and Theorem 2.21. Thus the proof is completed.

Corollay 3.16. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a function between two (L, M)-fuzzy topological spaces. then

$P i r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {P_{f {(x)}_{λ}}^{P_{2}} (V) \mapsto P_{x_{λ}}^{P_{1}} (f^{\leftarrow} (V))}$ ;

$P i r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} { ℳ_{f {(x)}_{λ}}^{P_{2}} (V) (f {(x)}_{λ}) \mapsto ℳ_{x_{λ}}^{P_{1}} (f^{\leftarrow} (V))}$ ;

$P i r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {p I n t^{P_{2}} (V) (f {(x)}_{λ}) \mapsto p I n t^{P_{1}} (f^{\leftarrow} (V)) (x_{λ})}$ ;

$P i r (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} { p C l^{P_{2}} (V^{'}) {(f {(x)}_{λ})}^{'} \mapsto p C l^{P_{1}} (f^{\leftarrow} {(V)}^{'}) {(x_{λ})}^{'}}$ ;

Proof. Analogues to Corollary 3.16. □

Theorem 3.17. If $f : (X, T_{1}) ⟶ (Y, T_{2})$ is a function between two (L, M)-fuzzy topological spaces, then

$S o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {S_{x_{λ}}^{S_{1}} (U) \mapsto S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U))}$ .

$S o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {N_{x_{λ}}^{S_{1}} (U) \mapsto N_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U))}$ .

$S o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {s I n t^{S_{1}} (U) (x_{λ}) \mapsto s I n t^{S_{2}} (f^{\to} (U)) (f {(x)}_{λ})}$ .

$S o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {s C l^{S_{1}} (U^{'}) {(x_{λ})}^{'} \mapsto s C l^{S_{2}} (f^{\to} {(U)}^{'}) {(f {(x)}_{λ})}^{'}}$ .

$S o (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \mapsto S_{f {(x)}_{λ}}^{S_{2}} (V)}$ .

$S o (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {N_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \mapsto N_{f {(x)}_{λ}}^{S_{2}} (V)}$ .

$S o (f) = \underset{_{V \in L^{Y}}}{\land} \underset{x_{λ} \in J (L^{X})}{\land} { s I n t^{S_{1}} (f^{\leftarrow} (V)) (x_{λ}) \mapsto s I n t^{S_{2}} (V) (f {(x)}_{λ})}$ .

$S o (f) = \underset{_{V \in L^{V}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} { s C l^{S_{1}} (f^{\leftarrow} {(V)}^{'}) {(x_{λ})}^{'} \mapsto s C l^{S_{2}} (V^{'}) {(f {(x)}_{λ})}^{'}}$ .

Proof. We will suffice with the proof of (5). Firstly, $\begin{array}{l} \underset{V \in L^{Y}}{\land} \underset{x_{λ} \in (L^{X})}{\land} {S_{f {(x)}_{λ}}^{S_{2}} (V)} \\ = \underset{V \in L^{Y}}{\land} \underset{x_{λ} \in (L^{X})}{\land} {\underset{x_{λ} \leq {U^{'}}_{1} \geq f^{\leftarrow} {(V)}^{'}}{\lor} S_{1} (U) \\ \mapsto \underset{f {(x)}_{λ} \leq {V^{'}}_{1} \geq V^{'}}{\lor} S_{2} (V_{1})} \\ \geq \underset{V \in L^{Y}}{\land} \underset{x_{λ} \in J (L^{X})}{\land} {\underset{x_{λ} \leq {U^{'}}_{1} \geq f^{\leftarrow} {(V)}^{'}}{\lor} S_{1} (U_{1}) \\ \mapsto \underset{f {(x)}_{λ} \leq f^{\to} {(U_{2})}^{'} \geq V^{'}}{\lor} S_{2} (f^{\to} (U_{2})) \\ \geq \underset{V \in L^{Y}}{\land} \underset{x_{λ} \in J (L^{X})}{\land} {\underset{x_{λ} \leq {U^{'}}_{1} \geq f^{\to} {(V)}^{'}}{\lor} S_{1} (U_{1}) \\ \mapsto \underset{f {(x)}_{λ} \leq {U^{'}}_{1} \geq f^{\leftarrow} {(V)}^{'}}{\lor} S_{2} (f^{\to} (U_{2}))} \\ \underset{V \in L^{Y}}{\land} \underset{x_{λ} \in (L^{X})}{\land} \underset{x_{λ} \leq {U^{'}}_{1} \geq f^{\leftarrow} {(V)}^{'}}{\land} {S_{1} (U_{1}) \\ \mapsto \underset{f {(x)}_{λ} \leq {U^{'}}_{1} \geq f^{\leftarrow} {(V)}^{'}}{\lor} S_{2} (f^{\to} (U_{2}))} \\ \geq \underset{V \in L^{Y}}{\land} \underset{x_{λ} \in (L^{X})}{\land} \underset{x_{λ} \leq {U^{'}}_{1} \geq f^{\leftarrow} {(V)}^{'}}{\land} {S_{1} (U_{1}) \\ \mapsto S_{2} (f^{\to} (U_{2}))} \\ \geq \underset{U \in L^{X}}{\land} {S_{1} (U) \mapsto S_{2} (f^{\to} (U))} = S o (f) \end{array}$

Now we need to prove the following fact, $\underset{_{y_{μ} f^{\to} {(U)}^{'}}}{\land} S_{y_{μ}}^{S_{2}} f^{\to} (U) = \underset{_{f {(x)}_{λ} f^{\leftarrow} {(U)}^{'}}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)),$ for all U ∈ L^X. It is clear that $\underset{_{y_{μ} f^{\to} {(U)}^{'}}}{\land} S_{y_{μ}}^{S_{2}} f^{\to} (U) \leq_{\underset{f {(x)}_{λ} f^{\leftarrow} {(U)}^{'}}{\land}} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) .$ Now we prove $\underset{_{y_{μ} f^{\to} {(U)}^{'}}}{\land} S_{y_{μ}}^{S_{2}} f^{\to} (U) \geq_{\underset{f {(x)}_{λ} f^{\to} {(U)}^{'}}{\land}} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) .$

For each y_μ ∈ J (L^Y) with y_μ≱f^→ (U) ′, we obtain μ≱ (f^→ (U) (y)) ′ = ⋀ _f(x)=yU (x) ′. Then there exists x ∈ X such that f (x) = y and μ ≤ U (x) ′. Therefore μ≱ ⋀ _f(x)=f(z)U (z) ′ = f^→ (U) (f (x)) ′. Thus f (x) _μ ≤ f^→ (U) ′. From $\begin{matrix} \underset{_{f {(x)}_{λ} f^{\to} {(U)}^{'}}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) & \leq S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) \\ = S_{y_{μ}}^{S_{2}} (f^{\to} (U)), \end{matrix}$ we have $\underset{_{f {(x)}_{λ} f^{\to} {(U)}^{'}}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) \leq_{\underset{y_{μ} f^{\to} {(U)}^{'}}{\land}} S_{y_{μ}}^{S_{2}} (f^{\to} (U)) .$

Hence $_{\underset{y_{μ} f^{\to} {(U)}^{'}}{\land}} S_{y_{μ}}^{S_{2}} (f^{\to} (U)) =_{\underset{f {(x)}_{λ} f^{\to} {(U)}^{'}}{\land}} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) .$

To prove the following fact: $\begin{array}{l} \underset{U \in L^{X}}{\land} {\underset{x_{λ} \leq U^{'}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U))) \\ \mapsto \underset{f {(x)}_{λ} \leq f^{\to} {(U)}^{'}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U))} \\ \geq \underset{_{V \in L^{Y}}}{\land} \underset{x_{λ} \in J (L^{X})}{\land} {S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \mapsto S_{f {(x)}_{λ}}^{S_{2}} (V)} \end{array}$

Suppose that α ∈ M such that $α ≪ \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \mapsto S_{f {(x)}_{λ}}^{S_{2}} (V)} .$

Then $α \leq S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \mapsto S_{f (x)_{λ}}^{S_{2}} (V)$ for each V ∈ L^Y and x_λ ∈ J (L^X). By Lemma 2.1(1), we have $α \land S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \leq S_{f (x)_{λ}}^{S_{2}} (V)$ . For all U ∈ L^X and f (x) _λ ≤ f^→ (U) ′, we have λ ≤ f^→ (U) (f (x)) ′ = ⋀ _f(x)=f(z)U (z) ′. Then there exists z ∈ X such that f (z) = f (x) and λ ≰ U (z) ′. This implies z_λ ≰ U′. From $\begin{array}{l} α \land \underset{x_{λ} \leq U^{'}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U))) \\ \leq α \land S_{z_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U))) \\ \leq S_{f {(z)}_{λ}}^{S_{2}} (f^{\to} (U)) = S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)), \end{array}$ we have $\begin{matrix} α \land \underset{_{x_{λ} \leq U^{'}}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U))) \\ \leq \underset{_{f {(x)}_{λ} \leq f^{\to} {(U)}^{'}}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) . \end{matrix}$

By Lemma 2.1 (1), we have $\begin{array}{l} α \leq \underset{_{x_{λ} \leq U^{'}}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U))) \\ \mapsto \underset{f {(x)}_{λ} \leq f^{\to} {(U)}^{'}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)) . \end{array}$

Hence $\begin{array}{l} α \leq \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \leq U^{'}}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U))) \\ \mapsto \underset{f {(x)}_{λ} f^{\to} {(U)}^{'}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U))} . \end{array}$

Since α is arbitrary, we have $\begin{array}{l} \underset{_{U \in L^{X}}}{\land} {\underset{_{x_{λ} \leq U^{'}}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U))) \\ \mapsto \underset{_{f {(x)}_{λ} \leq f^{\to} {(U)}^{'}}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U))} \\ \geq & \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \mapsto S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (V)) } . \end{array}$

Since U ≤ f^← (f^→ (U)) for all U ∈ L^X, then $\begin{array}{l} S o (f) = \underset{_{U \in L^{X}}}{\land} {S_{1} (A) \mapsto S_{2} (f^{\to} (U))} \\ = & \underset{U \in L^{X}}{\land} {(\underset{_{x_{λ} \leq U^{'}}}{\land} S_{x_{λ}}^{S_{1}} (U)) \\ \mapsto (\underset{y_{μ} \leq f^{\to} {(U)}^{'}}{\land} S_{y_{μ}}^{S_{2}} (f^{\to} (U)))} \\ = & \underset{_{U \in L^{X}}}{\land} {(\underset{x_{λ \leq} U^{'}}{\land} S_{x_{λ}}^{S_{1}} (U)) \\ \mapsto (\underset{f {(x)}_{λ} \leq f^{\to} {(U)}^{'}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)))} \\ \geq & \underset{U \in L^{X}}{\land} {(\underset{x_{λ} \leq U^{'}}{\land} S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (f^{\to} (U)))) \\ \mapsto (\underset{_{f {(x)}_{λ} \leq f^{\to} {(U)}^{'}}}{\land} S_{f {(x)}_{λ}}^{S_{2}} (f^{\to} (U)))} \\ \geq & \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {S_{x_{λ}}^{S_{1}} (f^{\leftarrow} (V)) \mapsto S_{f {(x)}_{λ}}^{S_{2}} (V)}, \end{array}$ which completes the proof.□

Theorem 3.18. If $f : (X, T_{1}) ⟶ (Y, T_{2})$ is a function between two (L, M)-fuzzy topological spaces, then

$P o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {P_{x_{λ}}^{P_{1}} (U) \mapsto P_{f {(x)}_{λ}}^{P_{2}} (f^{\to} (U))}$ .

$P o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {ℳ_{x_{λ}}^{P_{1}} (U) \mapsto ℳ_{f {(x)}_{λ}}^{P_{2}} (f^{\to} (U))}$ .

$P o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {p I n t^{P_{1}} (U) (x_{λ}) \mapsto p I n t^{P_{2}} (f^{\to} (U)) (f {(x)}_{λ})}$ .

$P o (f) = \underset{_{U \in L^{X}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {p C l^{P_{1}} (U^{'}) {(x_{λ})}^{'} \mapsto p C l^{P_{2}} (f^{\to} {(U)}^{'}) {(f {(x)}_{λ})}^{'}}$ .

$P o (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {P_{x_{λ}}^{P_{1}} (f^{\leftarrow} (V)) \mapsto P_{f {(x)}_{λ}}^{P_{2}} (V)}$ .

$P o (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {ℳ_{x_{λ}}^{P_{1}} (f^{\leftarrow} (V)) \mapsto ℳ_{f {(x)}_{λ}}^{P_{2}} (V)}$ .

$P o (f) = \underset{_{V \in L^{Y}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {p I n t^{P_{1}} (f^{\leftarrow} (V)) (x_{λ}) \mapsto p I n t^{P_{2}} (V) (f {(x)}_{λ})}$ .

$P o (f) = \underset{_{V \in L^{V}}}{\land} \underset{_{x_{λ} \in J (L^{X})}}{\land} {p C l^{P_{1}} (f^{\leftarrow} {(V)}^{'}) {(x_{λ})}^{'} \mapsto p C l^{P_{2}} (V^{'}) {(f {(x)}_{λ})}^{'}}$ .

Proof. Analogues to the proof of Theorem 3.17. □

4 Interrelation with semi(pre)-compactness, semi(pre)-connectedness, Semi(pre)-T₁, and Semi(pre)-T₂ degree

In this section we verify the relationships with semi(pre)-compactness, semi(pre)-connectedness, Semi (pre)-T₁, and Semi (pre)-T₂ degree.

Theorem 4.1. If $f : (X, T_{1}) ⟶ (Y, T_{2})$ is a function between two (L, M)-fuzzy topological spaces, then ${scd}_{S_{1}} (U) \land Irr (f) \leq {scd}_{S_{2}} (f^{\to} (U))$ for any U ∈ L^X.

Proof. Suppose that α ∈ M such that $α ≪ {scd}_{S_{1}} (U) \land Irr (f)$ . Then $α ≪ I r r (f) = \underset{_{V \in L^{Y}}}{\land} {S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))},$ and $\begin{array}{l} α & ≪ & s c d_{S_{1}} (U) \\ = & \underset{_{U \in L^{X}}}{\land} {(\underset{_{U_{1} \in U}}{\land} S_{1} (U_{1}) \land \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{U_{1} \in U}}{\lor} U_{1}) (x)) \\ \mapsto & \underset{_{V \in 2^{(U)}}}{\land} \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{U_{1} \in V}}{\lor} U_{1}) (x)} \end{array}$

Then for each V ∈ L^Y and $U \subseteq L^{X}$ , we have $α \leq S_{2} (V) \mapsto S_{1} (f^{\leftarrow} (V))$ and $\begin{array}{l} α & \leq & (\underset{U_{1} \in U}{\land} S_{1} (U_{1}) \land \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{U_{1} \in U}}{\lor} U_{1}) (x))) \\ \mapsto \underset{_{V \in 2^{(U)}}}{\lor} \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{U_{1} \in V}}{\lor} U_{1}) (x) . \end{array}$

Based on Lemma 2.1(1), we have $α \land S_{2} (V) \leq S_{1} (f^{\leftarrow} (V))$ for any V ∈ L^Y, and $\begin{array}{l} α & \land \underset{W \in U}{\land} S_{1} (W) \land \underset{x \in X}{\land} (U^{'} \lor \underset{_{W \in U}}{\land} W) (x) \\ \leq \underset{V \in 2^{(U)}}{\lor} \underset{x \in X}{\land} (U^{'} \lor \underset{W \in U}{\lor} W) (x) \end{array}$ To prove $\begin{array}{l} α \leq & s c d_{S_{2}} (f^{\to} (U)) \\ = & \underset{_{W \subseteq ℒ^{Y}}}{\land} {(\underset{_{V_{1} \in W}}{\land} S_{2} (V_{1}) \land \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in W}}{\lor} V_{1}) (y)) \\ \mapsto & \underset{_{D \in 2^{(W)}}}{\land} \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in W}}{\lor} V_{1}) (y)}, \end{array}$ for all $W \subseteq L^{Y}$ , let $f^{\leftarrow} (W) = {f^{\leftarrow} (V_{1}) ∣ V_{1} \in W} \subseteq L^{X}$ . Then by Lemma 2.29, we have $\begin{array}{l} α \land & \underset{_{V_{1} \in W}}{\land} S_{2} (V_{1}) \land \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in W}}{\lor} V_{1}) (y) \\ \leq & α \land \underset{_{V_{1} \in W}}{\land} S_{1} (f^{\leftarrow} (V_{1})) \land \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in W}}{\lor} V_{1}) (y) \\ = & α \land \underset{_{V_{1} \in W}}{\land} S_{1} (f^{\leftarrow} (V_{1})) \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{V_{1} \in W}}{\lor} f^{\leftarrow} (V_{1})) (x) \\ = & α \land \underset{_{U_{1} \in f^{\leftarrow} (W)}}{\land} S_{1} (U_{1}) \land \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{U_{1} \in f^{\leftarrow} (W)}}{\lor} U_{1}) (x) \\ \leq & \underset{_{V \in 2^{(f^{\leftarrow} (W))}}}{\lor} \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{U_{1} \in V}}{\lor} U_{1}) (x) \\ = & \underset{_{D \in 2^{(W)}}}{\lor} \underset{_{x \in X}}{\land} (U^{'} \lor \underset{_{V_{1} \in D}}{\lor} f^{\leftarrow} (V_{1})) (x) \\ = & \underset{_{D \in 2^{(W)}}}{\lor} \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in D}}{\lor} V_{1}) (y) . \end{array}$

By using Lemma 2.1(1), we have $\begin{matrix} α \leq & ( \underset{_{V_{1} \in W}}{\land} S_{2} (V_{1}) \land \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in W}}{\lor} V_{1}) (y)) \\ \mapsto & \underset{_{D \in 2^{(W)}}}{\lor} \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in D}}{\lor} V_{1}) (y) . \end{matrix}$

Thus $\begin{array}{l} α \leq & \underset{_{W \subseteq ℒ^{Y}}}{\land} {(\underset{_{V_{1} \in W}}{\land} S_{2} (V_{1}) \\ \land \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in W}}{\lor} V_{1}) (y)) \\ \mapsto & \underset{_{D \in 2^{(W)}}}{\lor} \underset{_{y \in Y}}{\land} (f^{\to} {(U)}^{'} \lor \underset{_{V_{1} \in D}}{\lor} V_{1}) (y)} \\ = s c d_{S_{2}} (f^{\to} (U)) . \end{array}$ Since α is arbitrary, we have ${scd}_{S_{1}} (U) \land Irr t (f) \leq {scd}_{S_{2}} (f^{\to} (U))$ . The proof is completed.

Theorem 4.2. If $f : (X, T_{1}) ⟶ (Y, T_{2})$ is a function between two (L, M)-fuzzy topological spaces and L = M, then ${pcd}_{P_{1}} (U) \land Pir (f) \leq {pcd}_{P_{2}} (f^{\to} (U))$ for any U ∈ L^X.

Proof. Similar to the proof of Theorem 4.1. □

Based on Theorems 4.1 and 4.2, we have the following corollary.

Corollay 4.3. If $f : (X, T_{1}) ⟶ (Y, T_{2})$ is surjective function between the two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ and L = M, then:

${scd}_{S_{1}} ({\underline{⊤}}_{L^{X}}) \land Irr t (f) \leq {scd}_{S_{2}} ({\underline{⊤}}_{L^{Y}})$ .

${pcd}_{P_{1}} ({\underline{⊤}}_{L^{X}}) \land Pir t (f) \leq {pcd}_{P_{2}} ({\underline{⊤}}_{L^{Y}})$ .

It’s well known that f (U) is connected subset if U is connected and f is continuous function. Now we extend it to (L, M)-fuzzy topological space as follows

Theorem 4.4. If f : X ⟶ Y is a function between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, S_{2})$ , then

${sCon}_{S_{2}} (f^{\to} (U))^{'} \land Irr (f) \leq {sCon}_{S_{1}} (U)^{'}$ ,

${pCon}_{P_{2}} (f^{\to} (U))^{'} \land Pir (f) \leq {pCon}_{P_{1}} (U)^{'}$ ,

for any U ∈ L^X.

Proof. We will prove only (1) and the proof of (2) is similar. Suppose α ∈ M such that $α ≪ {sCon}_{S_{2}} (f^{\to} (U))^{'} \land Irr (f)$ . From Theorems 2.27 and 3.4 (2), we have $\begin{array}{l} α ≪ & s C o n_{S_{2}} {(f^{\to} (U))}^{'} \\ = \underset{\begin{matrix} f^{\to} (U) \land V_{1} \neq ⊥ L^{Y}, \\ f^{\to} (U) \land V_{2} \neq ⊥ L^{Y}, \\ f^{\to} (U) \land V_{1} \land V_{2} \neq ⊥ L^{Y}, \\ f^{\to} (U) \leq E \lor F \end{matrix}}{\lor} \end{array} {S_{2} ({V^{'}}_{1}) \land S_{2} ({V^{'}}_{2})}$ and $α ≪ I r r (f) = \underset{_{V_{3} \in L^{Y}}}{\land} {S_{2} (V_{3}^{'}) \mapsto S_{1} (f^{\to} {(V_{3})}^{'})} .$

Then there exist V₁, V₂ ∈ L^Y such that f^→ (U) ∧ V₁ ≠ ⊥_{L
^Y}, f^→ (U) ∧ V₂ ≠ ⊥ _{L
^Y}, f^→ (U) ∧ V₁ ∧V₂ = ⊥_{L
^Y}, f^→ (U) ≤ V₁ ∨ V₂ with α ≤ $S_{2}$ $(V_{1}^{'})$ $\land S_{2}$ $(V_{2}^{'})$ , and α ≤ $S_{2}$ $(V_{3}^{'})$ ↦ $S_{1} (f^{\leftarrow} {(V_{3})}^{'})$ for any V₃ ∈ L^Y. This implies that there exist V₁, V₂ ∈ L^Y such that U ∧ f^← (V₁) ¬ = ⊥ _{L
^X}, U ∧ f^← (V₂) ¬ = ⊥ _{L
^X}, U ∧ f^← (V₁) ∧ f^← (V₂) = ⊥ _{L
^X}, U ≤ f^← (V₁) ∨ f^← (V₂) with $α \leq S_{2} (V_{1}) \land S_{2} (V_{2}),$ (1) and $α \land S_{2} (V_{3}^{'}) \leq S_{1} (f^{\leftarrow} (V_{3})^{'}),$ (2) for any V₃ ∈ L^Y. From Equations 4.1 and 4.2, we have $\begin{array}{l} \begin{array}{l} α = & α \land S_{2} ({V^{'}}_{1}) \land S_{2} (V_{2}^{'}) \\ \leq S_{1} (f^{\leftarrow} {(V_{1})}^{'}) \land S_{1} (f^{\leftarrow} {(V_{1})}^{'}) \\ \leq \underset{\begin{array}{l} U \land U_{1} \neq ⊥_{L^{X}}, U \land U_{2} \neq ⊥_{L^{X}}, \\ U \land U_{2} \land U_{1} = ⊥_{L^{X}}, U \leq U_{1} \lor U_{2} \end{array}}{\land} {S_{1} ({U^{'}}_{1}) \land S_{1} ({V^{'}}_{2})} \end{array} \\ = s C o n_{S 1} {(U)}^{'} . \end{array}$ Since α is arbitrary, we have ${sCon}_{S_{2}} (f^{\to} (U))^{'} \land Irr (f) \leq {sCon}_{S_{1}} (U)^{'}$ .

It is easy to verify that semi - T₁ and semi - T₂ in our sense are preserved by semi-homeomorphisms. In the remainder of this paper, we extend it into the (L, M)-fuzzy topology setting.

Lemma 4.5. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ . Then

$semi - T_{1} (X, T_{1}) \land So (f) \leq semi - T_{1} (Y, T_{2})$ ;

$semi - T_{2} (X, T_{1}) \land So (f) \leq semi - T_{2} (Y, T_{2})$ .

Proof. We only prove (1) and the proof of (2) is similar. Suppose α ∈ M such that $\begin{array}{l} α & ≪ s e m i - T_{1} (X, T_{1}) \land S o (f) \\ = & \underset{_{u \leq v}}{\land} \underset{_{u \leq U \geq v}}{\lor} S_{1} (U^{'}) \\ \land \underset{_{U_{1} \in L^{X}}}{\land} {S_{1} (U_{1}) \mapsto S_{2} (f^{\to} (U_{1}))} . \end{array}$

Thus for each u₁, u₂ ∈ J (L^X) such that u₁≱u₂, there exists U ∈ L^X with u₁≱U ≥ u₂ and $α \leq S_{1} (U^{'})$ . For all U₁ ∈ L^X, $α \leq S_{1} (U_{1}) \to S_{2} (f^{\to} (U_{1}))$ . From Lemma 2.1 (1), we obtain $α \land S_{1} (U_{1}) \leq S_{2} (f^{\to} (U_{1}))$ . To verify $α \leq s e m i - T_{1} (Y, T_{2}) = \underset{_{v_{1} \leq v_{2}}}{\land} \underset{_{v_{1} \leq V \geq v_{2}}}{\lor} S_{2} (V^{'}),$ let v₁, v₂ ∈ J (L^Y) such that v₁≱v₂. From the bijectivity of the function f, there exist u₁, u₂ ∈ J (L^X) such that u₁≱u₂ with v₁ = f^→ (u₁) and v₂ = f^→ (u₂). Since u₁≱u₂, there exists U ∈ L^X such that u₁≱U ≥ u₂ with $α \leq S_{1} (U^{'})$ . Thus v₁ = f^→ (u₁) ≱f^→ (U) ≥ f^→ (u₂) = v₂. Since f is a bijective function, we know $α = α \land S_{1} (U^{'}) \leq S_{2} (f^{\to} (U^{'})) = S_{2} (f^{\to} (U)^{'}) .$ Therefore $α \leq \underset{_{v_{1} \leq v_{2}}}{\land} \underset{_{v_{1} \leq V \geq v_{2}}}{\lor} S_{2} (V^{'}) = s e m i - T_{1} (Y, T_{2}) .$ Since α is arbitrary, we proved that $semi - T_{1} (X, T_{1}) \land So (f) \leq semi - T_{1} (Y, T_{2})$ . This completes the proof.

Lemma 4.6. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ . Then

$Pre - T_{1} (X, T_{1}) \land Po (f) \leq Pre - T_{1} (Y, T_{2})$ .

$Pre - T_{2} (X, T_{1}) \land Po (f) \leq Pre - T_{2} (Y, T_{2})$ .

Proof. The proof is similar to Lemma 4.5. □

Similarly, we can stat the following lemma.

Lemma 4.7. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ . Then

$Semi - T_{1} (Y, T_{2}) \land Irr (f) \leq Semi - T_{1} (X, T_{1})$ .

$Semi - T_{2} (Y, T_{2}) \land Irr (f) \leq Semi - T_{2} (X, T_{1})$ .

$Pre - T_{1} (Y, T_{2}) \land Pir (f) \leq Pre - T_{1} (X, T_{1})$ .

$Pre - T_{2} (Y, T_{2}) \land Pir (f) \leq Pre - T_{2} (X, T_{1})$ .

By Combining Lemmas 4.5, 4.6, 4.7 and Definition 3.2 (1), we can stat the following theorem.

Theorem 4.8. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ . then

Semi - T₁ (X, $T_{1})$ ∧ Semi - Hom (f) ≤ Semi - T₁ (Y4, $T_{2})$ , Semi - T₁ (Y, $T_{2})$ ∧ Semi - Hom (f) ≤ Semi - T₁ (X, $T_{1})$ .

Semi - T₂ $(X, T_{1})$ ∧ Semi - Hom (f) ≤ Semi - T₂ (Y, $T_{2})$ , Semi - T₂ (Y, $T_{2})$ ∧ Semi - Hom (f) ≤ Semi - T₂ (X, $T_{1})$ .

Pre - T₁ (X, $T_{1})$ ∧ Pre - Hom (f) ≤ Pre - T₁ (Y, $T_{2})$ , Pre - T₁ (Y, $T_{2})$ ∧ Pre - Hom (f) ≤ Pre - T₁ (X, $T_{1})$ .

Pre - T₂ (X, $T_{1})$ ∧ Pre - Hom (f) ≤ Pre - T₂ (Y, $T_{2})$ , Pre - T₂ (Y, $T_{2})$ ∧ Pre - Hom (f) ≤ Pre - T₂ (X, $T_{1})$ .

Based on Theorem 4.8, we have the following corollary.

Corollay 4.9. Let $f : (X, T_{1}) ⟶ (Y, T_{2})$ be a bijective function between two (L, M)-fuzzy topological spaces $(X, T_{1})$ and $(Y, T_{2})$ . Then

Semi - T₁ (X, $T_{1})$ ∧ Semi - Hom (f) = Semi - T₁ (Y, $T_{2})$ ∧ Semi - Hom (f).

Semi - T₂ (X, $T_{1})$ ∧ Semi - Hom (f) = Semi - T₂ (Y, $T_{2})$ ∧ Semi - Hom (f).

Pre - T₁ (X, $T_{1})$ ∧ Pre - Hom (f) = Pre - T₁ (Y, $T_{2})$ ∧ Pre - Hom (f).

Pre - T₂ (X, $T_{1})$ ∧ Pre - Hom (f) = Pre - T₂ (Y, $T_{2})$ ∧ Pre - Hom (f).

5 Conclusion

Unlike classical sets theory, the theory of fuzzy sets, rough sets, and (fuzzy) soft sets deal competently with uncertainty. For instance, soft set theory has a great importance in decisions making [15, 39]. Alcantud and Zhan [37] reviewed some different algorithms of parameter reduction based on some types of (fuzzy) soft sets. Moreover, a comparison between these algorithms are given to emphasize their respective advantages and disadvantages with some examples illustrating the differences between them. In [38], they also presented a novel type of soft rough covering based on soft neighborhoods, and then they used it to improve decision making in a multicriteria group environment.

Fuzzy topology is a generalization of general topology in classical mathematics, but it also has its own marked characteristics. Some scholars used tools for examining fuzzy topological spaces and establishing new types from existing ones. Attention has been paid to define and characterize new weak forms of continuity.

In this paper, we provided a description of the (L, M)-fuzzy topological spaces by sub-varieties of new degrees of functions. We established new degrees of a weak forms of functions in (L, M)-fuzzy topological spaces by using the implication operation and Shi’s operators. We also investigated some properties of semiopenness, semicontinuity, preopenness, precontinuity, irresolutness and preirresolutness degree of functions in (L, M)-fuzzy topology. We found that the functions can be regard as semiopen, semicontinuous, preopen, precontinuous, irresolute and preirresolute to some degree. Furthermore, relations with semi (pre)-compactness, semi (pre)-connectedness, Semi-T₁, Pre-T₁, Semi-T₂, and Pre-T₂ have been constructed and analyzed.

Our findings may strengthen the theoretical foundation of fuzzy topology. This theoretical development can be applied in GIS, image processing, data mining, medical diagnosis etc.

Footnotes

Acknowledgments

The authors would like to thank the associate editor Prof. Xueling Ma and anonymous reviewers for their careful reading and constructive comments in improving this paper. Special thanks from the first author are due to Prof. Fu-Gui Shi for providing us with the necessary papers and answering our inquires.

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New degrees for functions in ( L,M )-fuzzy topological spaces based on ( L,M )-fuzzy semiopen and ( L,M )-fuzzy preopen operators

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Degree of semiopenness, semicontinuity, preopenness, precontinuity, irresolutness and preirresolutness for functions in (L, M)-fuzzy topological spaces

4 Interrelation with semi(pre)-compactness, semi(pre)-connectedness, Semi(pre)-T1, and Semi(pre)-T2 degree

Footnotes

Acknowledgments

References

4 Interrelation with semi(pre)-compactness, semi(pre)-connectedness, Semi(pre)-T₁, and Semi(pre)-T₂ degree