A New Representation of Semiopenness of L-fuzzy Sets in RL-fuzzy Bitopological Spaces

Abstract

In this paper, we introduce a new representation of semiopenness of L-fuzzy sets in $RL$-fuzzy bitopological spaces based on the concept of pseudo-complement. The concepts of pairwise $RL$-fuzzy semicontinuous and pairwise $RL$-fuzzy irresolute functions are extended and discussed based on the $(i,j)$-$RL$-semiopen gradation. Further, pairwise $RL$-fuzzy semi-compactness of an L-fuzzy set in $RL$-fuzzy bitopological spaces are given and characterized. As $RL$-fuzzy bitopology is a generalization of L-bitopology, $RL$-bitopology, L-fuzzy bitopology, and $RL$-fuzzy topology, the results of our paper are more general.

1. Introduction

In 1963, Levine [1] introduced the notion of semiopen set and its corresponding associated function in the realm of general topology. Afterwards, Azad [2] extended this notion and its related functions to the setting of L-topology. Thakur and Malviya [3] introduced and studied the concepts of $(i,j)$-semiopen and $(i,j)$-semiclosed L-fuzzy sets, pairwise fuzzy semicontinuous, and pairwise fuzzy semiopen functions in L-bitopology in the case of $L=[0,1]$. In [4], Shi introduced the notion of L-fuzzy semiopen and preopen gradations in L-fuzzy topological spaces. Furthermore, he introduced the notions of L-fuzzy semicontinuous functions, L-fuzzy precontinuous functions, L-fuzzy irresolute functions, and L-fuzzy pre-irresolute functions, and discussed some of their elementary properties. Shi’s operators have been found very useful in defining other gradations and also in studying many topological characteristics. In 2011, Ghareeb [5] used L-fuzzy preopen operator to introduce the degree of pre-separatedness and the degree of preconnectedness in L-fuzzy topological spaces. Many characterizations of the degree of preconnectedness are discussed in L-fuzzy topological spaces. Later, Ghareeb [6] introduced the concept of L-fuzzy semi-preopen operator in L-fuzzy topological spaces and studied some of its properties. The concepts of L-fuzzy $SP$-compactness and L-fuzzy $SP$-connectedness in L-fuzzy pretopological spaces are introduced and studied [7]. Further, a new operator in L-fuzzy topology introduced in [8] to measure the $\mathbf{F}$-openness of an L-fuzzy set in L-fuzzy topological spaces. Moreover, the new operator is used to introduce a new form of $\mathbf{F}$-compactness. Recently, we used the new operators to generalize several kinds of functions between L-fuzzy topological spaces [9,10,11,12].

Recently, Li and Li [13] defined and studied the concept of $RL$-topology as an extension of L-topology. Moreover, $RL$-compactness by means of an inequality and $RL$-continuous mapping are introduced and discussed in detail. In [14], they presented $RL$-fuzzy topology on an L-fuzzy set as a generalization of $RL$-topology and L-fuzzy topology. Some relevant properties of $RL$-fuzzy compactness in $RL$-fuzzy topological spaces are further investigated. Later on, Zhang et al. [15] defined the degree of Lindelöf property and the degree of countable $RL$-fuzzy compactness of an L-fuzzy set, where L is a complete DeMorgan algebra. Since L-fuzzy topology in the sense of Kubiak and Šostak is a special case of $RL$-fuzzy topology, the degree of $RL$-fuzzy compactness and the degree of Lindelöf property are extensions of the corresponding degrees in L-fuzzy topology.

The purpose of this paper is to introduce the $(i,j)$-$RL$-semiopen gradation in $RL$-fuzzy bitopological spaces based on the concept of pseudo-complement of L-fuzzy sets. We also define and characterize pairwise $RL$-fuzzy semicontinuous, pairwise $RL$-fuzzy irresolute functions, and pairwise $RL$-fuzzy semi-compactness. Our results are more general than those of the corresponding notions in L-bitopology, $RL$-bitopology, $RL$-fuzzy topology, L-fuzzy topology, and L-fuzzy bitopology.

2. Preliminaries

In this section, we give some basic preliminaries required for this paper. By $(L,\bigvee ,\bigwedge {,}^{\prime })$, we denote a complete DeMorgan algebra [16,17] (i.e., L is a completely distributive lattice with an order reversing involution ${}^{\prime }$, where ⋁ and ⋀ are join and meet operations, respectively), $X\ne \varnothing$ is a set, and $L^X$ is the family of each L-fuzzy sets defined on X. The largest and the smallest members in L and $L^X$ are denoted by ⊤, ⊥, and ${\top }_X$, ${\perp }_X$, respectively. For each any two L-fuzzy sets $B\in L^X$, $C\in L^Y$, and any mapping $f:X⟶Y$, we define $f_L^{\to }\left(B\right)\left(y\right)=\bigvee \{B\left(x\right):f\left(x\right)=y\}$ for all $y\in Y$ and $f_L^{\leftarrow }\left(C\right)\left(x\right)=\bigvee \{B\left(x\right):f_L^{\to }\left(B\right)\le C\}=C\left(f\left(x\right)\right)$ for all $x\in X$. For each $\alpha$, $\beta \in L$, $\alpha \prec \beta$ means that the element $\alpha$ is wedge below $\beta$ in L [18], i.e., $\alpha \prec \beta$ if for every arbitrary subset $\mathcal{D}\subseteq L$, $\bigvee \mathcal{D}\ge \beta$ implies $\alpha \le \gamma$ for some $\gamma \in \mathcal{D}$. An element $\alpha \in L$ is said to be co-prime if $\alpha \le \beta \vee \gamma$ implies that $\alpha \le \beta$ or $\alpha \le \gamma$ and $\alpha$ is said to be prime if and only if ${\alpha }^{\prime }$ is co-prime. The family of non-zero co-prime (resp. non-unit prime) members in L is denoted by $J\left(L\right)$ (resp. $P\left(L\right)$). By $\mathit{\alpha }\left(\beta \right)=\bigvee \{\alpha \in L:\alpha \prec \beta \}$ and $\mathit{\beta }\left(\beta \right)=\bigvee \{\alpha \in L:{\alpha }^{\prime }\prec {\beta }^{\prime }\}$, we denote the greatest minimal family and the greatest maximal family of $\beta$, respectively. ${\mathit{\alpha }}^{*}\left(\alpha \right)=\mathit{\alpha }\left(\alpha \right)\cap J\left(L\right)$ and ${\mathit{\beta }}^{*}\left(\alpha \right)=\mathit{\beta }\left(\alpha \right)\cap P\left(L\right)$ for all $\alpha \in L$.

An L-fuzzy set $A\in L^X$ is called valuable if $A\nleqq A^{\prime }$. The collection of valuable L-fuzzy sets on X is denoted by ${\mathcal{V}}_X^L$. In other words, ${\mathcal{V}}_X^L=\{A\in L^X:A\nleqq A^{\prime }\}$. For each $A\in {\mathcal{V}}_X^L$, we define the collection ${\mathcal{F}}_X^L\left(A\right)$ by ${\mathcal{F}}_X^L\left(A\right)=\{B\in L^X:B\le A\}$. In fact, ${\mathcal{F}}_X^L\left(A\right)$ introduces the powerset of L-fuzzy set $A\in L^X$. Let $A\in {\mathcal{V}}_X^L$ and $B\in {\mathcal{V}}_Y^L$, the restriction of $f_L^{\to }$ on A, i.e., $f_L^{\to }{|}_A:{\mathcal{F}}_X^L\left(A\right)⟶L^Y$ provided that $D\in {\mathcal{F}}_X^L\left(A\right)\mapsto f_L^{\to }\left(D\right)$, is said to be the restriction of L-fuzzy function ($RL$-fuzzy function, in short) from A to B, given by $f_{L,A}^{\to }:A⟶B$ if $f_L^{\to }\left(A\right)\le B$. The inverse of an L-fuzzy set $C\in {\mathcal{F}}_Y^L\left(B\right)$ under $f_{L,A}^{\to }$ is defined by $f_{L,A}^{\leftarrow }\left(C\right)=\bigvee \{D\in {\mathcal{F}}_X^L\left(A\right):f_L^{\to }\left(D\right)\le C\}$. It is clear that $f_{L,A}^{\leftarrow }\left(C\right)=A\wedge f_L^{\leftarrow }\left(C\right)$. The pseudo-complement of B relative to A [13,14], denoted by ${\langle }_L^{A}B$, is given by:

$${\langle }_L^{A}B=\left\{\begin{array}{cc}A\wedge B^{\prime },\hfill & \mathrm{if}B\ne A,\hfill \\ {\perp }_X,\hfill & \mathrm{if}B=A.\hfill \end{array}\right.$$

where $A\in {\mathcal{V}}_X^L$ and $B\in {\mathcal{F}}_X^L\left(A\right)$. Some properties of pseudo-complement operation ${\langle }_L^A$ are listed in the following proposition:

Proposition 1.

[13,14] If $A\in {\mathcal{V}}_X^L$, $B,C\in {\mathcal{F}}_X^L\left(A\right)$, and ${\left\{B_i\right\}}_{i\in I}\subseteq {\mathcal{F}}_X^L\left(A\right)$, then:

(1)

${\langle }_L^{A}B=A$ if and only if $B\le A^{\prime }$.

(2)

$B\le C$ implies ${\langle }_L^A{C\le \langle }_L^{A}B$.

(3)

${\langle }_L^A{\bigwedge }_{i\in I}{B}_i={\bigvee }_{i\in I}{\langle }_L^A{B}_i$.

(4)

${\langle }_L^A{\bigvee }_{i\in I}{B}_i\le {\bigwedge }_{i\in I}{\langle }_L^A{B}_i$ and ${\langle }_L^A{\bigvee }_{i\in I}{B}_i={\bigwedge }_{i\in I}{\langle }_L^A{B}_i$ if ${\bigvee }_{i\in I}{B}_i\ne A$.

Lemma 1.

[13] Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, $f_{L,A}^{\to }:A⟶B$ be $RL$-fuzzy function, and $D\in {\mathcal{F}}_X^L\left(A\right)$. Then for any $\mathcal{U}\subseteq {\mathcal{F}}_X^L\left(A\right)$, we have

$$\underset{y\in Y}{\bigvee }\left(f_{L,A}^{\to }\left(D\right)\left(y\right)\wedge \underset{E\in \mathcal{U}}{\bigwedge }E\left(y\right)\right)=\underset{x\in X}{\bigvee }\left(D\left(x\right)\wedge \underset{E\in \mathcal{U}}{\bigwedge }{f}_{L,A}^{\leftarrow }\left(E\right)\left(x\right)\right).$$

Equivalently [15],

$$\underset{y\in Y}{\bigwedge }\left({\langle }_L^A{f}_{L,A}^{\to }\left(D\right)\left(y\right)\vee \underset{E\in \mathcal{P}}{\bigvee }E\left(y\right)\right)=\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}D\left(x\right)\vee \underset{E\in \mathcal{P}}{\bigvee }{f}_{L,A}^{\leftarrow }\left(E\right)\left(x\right)\right).$$

An L-topology [16,17,19] (L-t, for short) $\tau$ is a subfamily of $L^X$ which contains ${\perp }_X$, ${\top }_X$ and is closed for any suprema and finite infima. Moreover, $(X,\tau )$ is called an L-topological space on X. Further, members of $\tau$ are called open L-fuzzy sets and their complements are called closed L-fuzzy sets. A mapping $f:(X,{\tau }_1)⟶(Y,{\tau }_2)$ is called L-continuous if and only if $f_L^{\leftarrow }\left(C\right)\in {\tau }_1$ for any $C\in {\tau }_2$. The notion of L-topology was generalized by Kubiak [20] and Šostak [21] independently as follows:

Definition 1.

[20,21,22] An L-fuzzy topology on the set X is the function $\tau :L^X⟶L$, which satisfies the following conditions:

($\mathcal{O}$1)

$\tau \left({\perp }_X\right)=\tau \left({\top }_X\right)=\top$.

($\mathcal{O}$2)

$\tau (A\wedge B)\ge \tau \left(A\right)\wedge \tau \left(B\right)$, for each $A,B\in L^X$.

($\mathcal{O}$3)

$\tau \left({\bigvee }_{i\in I}{A}_i\right)\ge {\bigwedge }_{i\in I}\tau \left(A_i\right)$, for each ${\left\{A_i\right\}}_{i\in I}\subseteq L^X$.

The pair $(X,\tau )$ is called an L-fuzzy topological space (L-fts, for short). The value $\tau \left(A\right)$ and ${\tau }^{*}\left(A\right)=\tau \left(A^{\prime }\right)$ represent the degree of openness and the degree of closeness of an L-fuzzy set A, respectively. A function $f:(X,{\tau }_1)⟶(Y,{\tau }_2)$ is called L-fuzzy continuous iff ${\tau }_1\left(f_L^{\leftarrow }\left(C\right)\right)\ge {\tau }_2\left(C\right)$ for any $C\in L^Y$.

One of the attempts to generalize L-topological spaces was the definition of $RL$-topology $\varkappa$ on an L-fuzzy set A by Li and Li [13] as follows:

Definition 2.

[13] Let $A\in {\mathcal{V}}_X^L$. A relative L-topology ($RL$-t, for short) ϰ on an L-fuzzy set A, is a subfamily of ${\mathcal{F}}_X^L\left(A\right)$, that satisfies the following statements:

(1)

$A\in \varkappa$ and $B\in \varkappa$, for each $B\le A^{\prime }$.

(2)

$B_1\wedge B_2\in \varkappa$, for any $B_1$, $B_2\in \varkappa$.

(3)

${\bigvee }_{i\in I}{B}_i\in \varkappa$, for any ${\left\{B_i\right\}}_{i\in I}\subseteq \varkappa$.

The pair $(A,\varkappa )$ is said to be a relative L-topological space on A ($RL$-ts, for short). The elements of ϰ are called relative open L-fuzzy sets ($RL$-open fuzzy set, for short) and an L-fuzzy set B is called relative L-closed fuzzy set ($RL$-closed fuzzy set, for short) if and only if ${\langle }_L^{A}B\in \varkappa$. The collection of all $RL$-closed fuzzy sets with respect to ϰ is denoted by ${\langle }_L^A\varkappa$, i.e., ${\langle }_L^A\varkappa =\{C:{\langle }_L^{A}C\in \varkappa \}$. Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, and $(A,{\varkappa }_1)$, $(B,{\varkappa }_2)$ be two $RL$-ts’s. The relative L-fuzzy function $f_{L,A}^{\to }:A⟶B$ is said to be an $RL$-continuous iff $f_{L,A}^{\leftarrow }{\left(C\right)\in \langle }_L^A{\varkappa }_1$ for any $C\in {\langle }_L^A{\varkappa }_2$. Equivalently, $f_{L,A}^{\to }:A⟶B$ is said to be an $RL$-continuous iff $f_{L,A}^{\leftarrow }\left(C\right)\in {\varkappa }_1$ for any $C\in {\varkappa }_2$. A triple $(A,{\varkappa }_1,{\varkappa }_2)$ consisting of an L-fuzzy set $A\in {\mathcal{V}}_X^L$ endowed with $RL$-topologies ${\varkappa }_1$ and ${\varkappa }_2$ on A is called an $RL$-bitopological space ($RL$-bts, for short). For any $B\in {\mathcal{F}}_X^L\left(A\right)$, ${\varkappa }_i$-$RL$-open (resp. closed) fuzzy set refers to the open (resp. closed) L-fuzzy set in $(A,{\varkappa }_i)$, for $i=1,2$. It is clear that we get L-topology and L-bitopology as a special case if $A={\top }_X$.

The following two definitions extend the notions of (strong) ${\mathit{\beta }}_{\alpha }$-cover, $Q_{\alpha }$-cover, (strong) $\alpha$-shading, (strong) $\alpha$-remote collection [23] to the setting of $RL$-topological spaces:

Definition 3.

For any $A\in {\mathcal{V}}_X^L$, $RL$-topology ϰ on A, $B\in {\mathcal{F}}_X^L\left(A\right)$, and $\alpha \in L_{\perp }$, a collection $\mathcal{U}\subseteq {\mathcal{F}}_X^L\left(A\right)$ is called:

(1)

${\mathit{\beta }}_{\alpha }$-cover of B if for any $x\in X$, it follows that $\alpha \in \mathit{\beta }({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{A\in \mathcal{U}}A\left(x\right))$ and $\mathcal{U}$ is called strong ${\mathit{\beta }}_{\alpha }$-cover of B if $a\in \mathit{\beta }({\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{A\in \mathcal{U}}A\left(x\right))\right)$.

(2)

$Q_{\alpha }$-cover of B if for any $x\in X$, it follows that ${\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{A\in \mathcal{U}}A\left(x\right)\ge \alpha$.

Definition 4.

For any $A\in {\mathcal{V}}_X^L$, $RL$-topology ϰ on A, $\alpha \in L_{\top }$ and $B\in {\mathcal{F}}_X^L\left(A\right)$, a collection $\mathcal{A}\subseteq {\mathcal{F}}_X^L\left(A\right)$ is called:

(1)

α-shading of B if for any $x\in X$, $({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{A\in \mathcal{A}}A\left(x\right))\nleqq \alpha$.

(2)

strong α-shading of B if ${\bigwedge }_{x\in X}({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{A\in \mathcal{A}}A\left(x\right))\nleqq \alpha$.

(3)

α-remote collection of B if for any $x\in X$, $(B\left(x\right)\wedge {\bigwedge }_{D\in \mathcal{A}}D\left(x\right))\ngeqq \alpha$.

(4)

strong α-remote collection of B if ${\bigvee }_{x\in X}(B\left(x\right)\wedge {\bigwedge }_{D\in \mathcal{A}}D\left(x\right))\ngeqq \alpha$.

Theorem 1.

[13] For any $RL$-ts $(A,\varkappa )$, the following statements are true:

(1)

$A\in {\langle }_L^A\varkappa$ and $B\in {\langle }_L^A\varkappa$ for all $B\le A^{\prime }$.

(2)

$B_1\vee B_2{\in \langle }_L^A\varkappa$ for each $B_1$, $B_2{\in \langle }_L^A\varkappa$,

(3)

${\bigwedge }_{i\in I}{B}_i{\in \langle }_L^A\varkappa$ for each $\{B_i:i\in I\}{\subseteq \langle }_L^A\varkappa$.

Definition 5.

[14] Let $A\in {\mathcal{V}}_X^L$. An $RL$-fuzzy topology on A is a function $\varkappa :{\mathcal{F}}_X^L\left(A\right)⟶L$ such that ϰ satisfying the following conditions:

($\mathcal{R}$1)

$\varkappa \left(A\right)=\top$, for each $B\le A^{\prime }$, $\varkappa \left(B\right)=\top$.

($\mathcal{R}$2)

$\varkappa (B_1\wedge B_2)\ge \varkappa \left(B_1\right)\wedge \varkappa \left(B_2\right)$, for each $B_1$, $B_2\in {\mathcal{F}}_X^L\left(A\right)$.

($\mathcal{R}$3)

$\varkappa \left({\bigvee }_{i\in I}{B}_i\right)\ge {\bigwedge }_{i\in I}\varkappa \left(B_i\right)$, for each ${\left\{B_i\right\}}_{i\in I}\subseteq {\mathcal{F}}_X^L\left(A\right)$.

The pair $(A,\varkappa )$ is said to be an $RL$-fuzzy topological space ($RL$-fts, for short) on A. For any $B\in {\mathcal{F}}_X^L\left(A\right)$, the gradation $\varkappa \left(B\right)$ (resp. $\varkappa ({\langle }_L^{A}B)$) can be viewed as the openness degree (resp. closeness degree) of B relative to ϰ, respectively. Further, $\varkappa \left(B\right)=\top$ (resp. $\varkappa ({\langle }_L^{A}B)=\top$) confirms the $RL$-openness (resp. $RL$-closeness) of an L-fuzzy set B. Obviously if $A={\top }_X$, then $RL$-fuzzy topology on A degenerates into Kubiak-Šostak’s L-fuzzy topology, that is, $RL$-fuzzy topology on A is a generalization of L-fuzzy topology. If $(A,\varkappa )$ is an $RL$-topological space and ${\chi }_{\varkappa }:{\mathcal{F}}_X^L\left(A\right)⟶L$ is a function given by ${\chi }_{\varkappa }\left(B\right)=\top$ if $B\in \varkappa$, and ${\chi }_{\varkappa }\left(B\right)=\perp$ if $B\notin \varkappa$, then $(A,{\chi }_{\varkappa })$ represents a special $RL$-fts, i.e., $(A,\varkappa )$ can also be seen as $RL$-fts.

Theorem 2.

[14] For each $A\in {\mathcal{V}}_X^L$ and $RL$-fts $(A,\varkappa )$ on A. The function ${\langle }_L^A\varkappa :{\mathcal{F}}_X^L\left(A\right)⟶L$ given by ${\langle }_L^A\varkappa \left(B\right)=\varkappa ({\langle }_L^{A}B)$ for any $B\in {\mathcal{F}}_X^L\left(A\right)$, satisfies the following conditions:

(1)

${\langle }_L^A\varkappa \left(A\right)=\top$, for each $B\le A^{\prime }$, ${\langle }_L^A\varkappa \left(B\right)=\top$.

(2)

${\langle }_L^A\varkappa (B_1\vee B_2)\ge {{\langle }_L^A\varkappa \left(B_1\right)\wedge \langle }_L^A\varkappa \left(B_2\right)$, for each $B_1$, $B_2\in {\mathcal{F}}_X^L\left(A\right)$.

(3)

${\langle }_L^A\varkappa \left({\bigwedge }_{i\in I}{B}_i\right)\ge {\bigwedge }_{i\in I}{\langle }_L^A\varkappa \left(B_i\right)$, for each ${\left\{B_i\right\}}_{i\in I}$$\subseteq {\mathcal{F}}_X^L\left(A\right)$.

${\langle }_L^A\varkappa$ is said to be an $RL$-fuzzy cotopology ($RL$-cft, for short) on A and the pair $(A,{\langle }_L^A\varkappa )$ is said to be an $RL$-fuzzy cotopological space ($RL$-cfts, for short).

Definition 6.

[14] Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, and $(A,{\varkappa }_1)$, $(B,{\varkappa }_2)$ be two $RL$-fuzzy topological spaces on A and B, respectively. The relative L-fuzzy function $f_{L,A}:A⟶B$ is said to be an $RL$-fuzzy continuous iff

$${\varkappa }_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge {\varkappa }_1\left(C\right),$$

equivalently,

$${\varkappa }_1({\langle }_L^A{f}_{L,A}^{\leftarrow }\left(C\right))\ge {\varkappa }_1({\langle }_L^{B}C),$$

for each $C\in {\mathcal{F}}_Y^L\left(B\right)$. If $(A,{\langle }_L^A{\varkappa }_1)$ and $(B,{\langle }_L^B{\varkappa }_2)$ are the associated $RL$-fuzzy cotopological spaces of $(A,{\varkappa }_1)$ and $(B,{\varkappa }_2)$ respectively, then $f_{L,A}^{\to }$ is said to be an $RL$-fuzzy continuous iff

$${\langle }_L^A{\varkappa }_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right){\ge \langle }_L^B{\varkappa }_2\left(C\right),$$

for each $C\in {\mathcal{F}}_X^L\left(B\right)$.

Shi [24] introduced L-fuzzy closure operators in L-fuzzy topological spaces. In the following definition, we introduce its equivalent form in $RL$-fuzzy topological spaces.

Definition 7.

Let $A\in {\mathcal{V}}_X^L$, and $(A,\varkappa )$ be an $RL$-fts on A. The function $C{l}^{\varkappa }:{\mathcal{F}}_X^L\left(A\right)\to L^{J\left({\mathcal{F}}_X^L\left(A\right)\right)}$ defined by

$$C{l}^{\varkappa }\left(B\right)\left(x_{\lambda }\right)=\underset{x_{\lambda }\nleqq D\ge B}{\bigwedge }{\langle }_L^A\left(\varkappa ({\langle }_L^{A}D)\right)$$

for each $x_{\lambda }\in J\left({\mathcal{F}}_X^L\left(A\right)\right)$ and $B\in {\mathcal{F}}_X^L\left(A\right)$ is called an $RL$-fuzzy closure operator induced by ϰ.

Definition 8.

[14] For any $A\in {\mathcal{V}}_X^L$ and an $RL$-fts $(A,\varkappa )$ on A, an L-fuzzy set $B\in {\mathcal{F}}_X^L\left(A\right)$ is called an $RL$-fuzzy compact with respect to ϰ if for any $\mathcal{P}\subseteq {\mathcal{F}}_X^L\left(A\right)$, the following inequality holds:

$$\underset{D\in \mathcal{P}}{\bigvee }\varkappa ({\langle }_L^{A}D)\vee \underset{x\in X}{\bigvee }\left(B\left(x\right)\wedge \underset{D\in \mathcal{P}}{\bigwedge }D\left(x\right)\right)\ge \underset{\mathcal{R}\in 2^{\mathcal{P}}}{\bigwedge }\underset{x\in X}{\bigvee }\left(B\left(x\right)\wedge \underset{D\in \mathcal{R}}{\bigwedge }D\left(x\right)\right).$$

Theorem 3.

[14] If $A={\top }_X$, then following statements hold:

(1)

${\langle }_L^{A}B=B^{\prime }$, $B\in {\mathcal{F}}_X^L\left(A\right)$⇔$B\in L^X$.

(2)

$RL$-fuzzy compactness is reduced to L-fuzzy compactness.

(3)

B is $RL$-fuzzy compact if and only if B is L-fuzzy compact.

Theorem 4.

[14] For any $A\in {\mathcal{V}}_X^L$ and an $RL$-ft ϰ on A, we have following conclusions:

(1)

If $B_1$, $B_2\in {\mathcal{F}}_X^L\left(A\right)$ and $B_1$, $B_2$ are $RL$-fuzzy compact, then $B_1\vee B_2$ is $RL$-fuzzy compact.

(2)

If $B_1$, $B_2\in {\mathcal{F}}_X^L\left(A\right)$ such that $B_1$ is an $RL$-fuzzy compact and $B_2$ is an $RL$-closed fuzzy set, then $B_1\wedge B_2$ is an $RL$-fuzzy compact.

3. The Gradation of Semiopenness in RL-fuzzy Bitopological Spaces

A system $(A,{\varkappa }_1,{\varkappa }_2)$ consisting of an L-fuzzy set $A\in {\mathcal{V}}_X^L$ with two $RL$-fuzzy topologies ${\varkappa }_1$ and ${\varkappa }_2$ on A is called an $RL$-fuzzy bitopological space. Throughout this paper i, $j=1$, 2 where $i\ne j$ and if P is any topological property then ${\varkappa }_i$-P refers to the property P with respect to the $RL$-fuzzy topology ${\varkappa }_i$. An L-fuzzy set $B\in {\mathcal{F}}_X^L\left(A\right)$ of an $RL$-bitopological space $(A,{\varkappa }_l,{\varkappa }_2)$ is called an $(i,j)$-$RL$-semiopen if there exists an L-fuzzy set $C\in {\varkappa }_i$ such that $C\le B\le C{l}^{{\varkappa }_j}\left(C\right)$.

Definition 9.

Let $A\in {\mathcal{V}}_X^L$ and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space on A. For any $B\in {\mathcal{F}}_X^L\left(A\right)$, define a function $(i,j)\text{-}\mathcal{S}:{\mathcal{F}}_X^L\left(A\right)\to L$ by

$$(i,j)\text{-}\mathcal{S}\left(B\right)=\underset{C\le B}{\bigvee }\left\{{\varkappa }_i\left(C\right)\wedge \underset{x_{\lambda }\prec B}{\bigwedge }\underset{x_{\lambda }\nleqq D\ge C}{\bigwedge }{\langle }_L^A\left({\varkappa }_j({\langle }_L^{A}D)\right)\right\}.$$

Then $(i,j)\text{-}\mathcal{S}\left(B\right)$ is called an $(i,j)$-$RL$-semiopenness gradation of B induced by ${\varkappa }_i$ and ${\varkappa }_j$ such that $i\ne j$, where $(i,j)\text{-}\mathcal{S}\left(B\right)$ represents the degree to which B is $(i,j)$-$RL$-semiopen and $(i,j)\text{-}{\mathcal{S}}^{*}\left(B\right)=(i,j)\text{-}\mathcal{S}({\langle }_L^{A}B)$ represents the degree to which B is $(i,j)$-$RL$-semiclosed.

Based on the above definition and Definition 7, we can state the following corollary:

Corollary 1.

Let $A\in {\mathcal{V}}_X^L$ and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space on A. Then for each $B\in {\mathcal{F}}_X^L\left(A\right)$, we have

$$(i,j)\text{-}\mathcal{S}\left(B\right)=\underset{C\le B}{\bigvee }\left\{{\varkappa }_i\left(C\right)\wedge \underset{x_{\lambda }\prec B}{\bigwedge }C{l}^{{\varkappa }_j}\left(C\right)\left(x_{\lambda }\right)\right\}.$$

Theorem 5.

Let $A\in {\mathcal{V}}_X^L$, ${\varkappa }_1$, ${\varkappa }_2:{\mathcal{F}}_X^L\left(A\right)\to \{\perp ,\top \}$ be $RL$-topologies on A, and $(i,j)\text{-}\mathcal{S}:{\mathcal{F}}_X^L\left(A\right)\to \{\perp ,\top \}$ be the gradation of $(i,j)$-$RL$-semiopenness induced by ${\varkappa }_i$ and ${\varkappa }_j$ such that $i\ne j$. Then $(i,j)\text{-}\mathcal{S}\left(B\right)=\top$ iff B is an $(i,j)$-$RL$-semiopen.

Proof. 

The proof can be obtained simply from the following inequality:

$$\begin{array}{cc}(i,j)\text{-}\mathcal{S}\left(B\right)=\top \hfill & \mathrm{iff}\underset{C\le B}{\bigvee }\left\{{\varkappa }_i\left(C\right)\wedge {\displaystyle \underset{x_{\lambda }\prec B}{\bigwedge }}C{l}^{{\varkappa }_j}\left(C\right)\left(x_{\lambda }\right)\right\}=\top \hfill \\ & \mathrm{iff}\exists C\le B\mathrm{such} \mathrm{that}{\varkappa }_i\left(C\right)=\top \mathrm{and}{\displaystyle \underset{x_{\lambda }\prec B}{\bigwedge }}C{l}^{{\varkappa }_j}\left(C\right)\left(x_{\lambda }\right)=\top \hfill \\ & \mathrm{iff}\exists C\le B\mathrm{such} \mathrm{that}{\varkappa }_i\left(C\right)=\top \mathrm{and} \mathrm{for} \mathrm{each}{x}_{\lambda }\prec B,C{l}^{{\varkappa }_j}\left(C\right)\left(x_{\lambda }\right)=\top \hfill \\ & \mathrm{iff}\exists C\in {\varkappa }_i\mathrm{such} \mathrm{that}C\le B\le C{l}^{{\varkappa }_j}\left(C\right)\hfill \\ & \mathrm{iff}Bis(i.j)-RL-semiopen.\hfill \end{array}$$

□

Theorem 6.

Let $A\in {\mathcal{V}}_X^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space on A, and $(i,j)\text{-}\mathcal{S}$ be the gradation of $(i,j)$-$RL$-semiopenness induced by ${\varkappa }_i$ and ${\varkappa }_j$ such that $i\ne j$. Then for each $B\in {\mathcal{F}}_X^L\left(A\right)$, we have ${\varkappa }_i\left(B\right)\le (i,j)\text{-}\mathcal{S}\left(B\right)$.

Proof. 

The proof can be obtained simply from the following inequality:

$$\begin{array}{cc}(i,j)\text{-}\mathcal{S}\left(B\right)\hfill & =\underset{C\le B}{\bigvee }\left\{{\varkappa }_i\left(C\right)\wedge {\displaystyle \underset{x_{\lambda }\prec B}{\bigwedge }}C{l}^{{\varkappa }_j}\left(C\right)\left(x_{\lambda }\right)\right\}\ge {\varkappa }_i\left(B\right)\wedge \underset{x_{\lambda }\prec B}{\bigwedge }C{l}^{{\varkappa }_j}\left(B\right)\left(x_{\lambda }\right)\hfill \\ & ={\varkappa }_i\left(B\right)\wedge \top ={\varkappa }_i\left(B\right).\hfill \end{array}$$

□

Corollary 2.

Let $A\in {\mathcal{V}}_X^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space on A, and $(i,j)\text{-}\mathcal{S}$ be the gradation of $(i,j)$-$RL$-semiopenness induced by ${\varkappa }_i$ and ${\varkappa }_j$ such that $i\ne j$. Then for each $B\in {\mathcal{F}}_X^L\left(A\right)$, we have ${\langle }_L^A{\varkappa }_i\left(B\right)\le (i,j)\text{-}{\mathcal{S}}^{*}\left(B\right)$.

Theorem 7.

If $A\in {\mathcal{V}}_X^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space on A, and $(i,j)\text{-}\mathcal{S}$ be the gradation of $(i,j)$-$RL$-semiopenness induced by ${\varkappa }_i$ and ${\varkappa }_j$ such that $i\ne j$, then $(i,j)\text{-}\mathcal{S}\left({\displaystyle \underset{i\in I}{\bigvee }}{B}_i\right)$$\ge {\displaystyle \underset{i\in I}{\bigwedge }}{(i,j)}_{\text{-}}\mathcal{S}\left(B_i\right)$ for each ${\left\{B_i\right\}}_{i\in I}\subseteq {\mathcal{F}}_X^L\left(A\right)$.

Proof. 

Let $\alpha \in L$ and $\alpha \prec {\displaystyle \underset{i\in I}{\bigwedge }}(i,j)\text{-}\mathcal{S}\left(B_i\right)$, then there exists $C_i\le B_i$ such that $\alpha \prec {\varkappa }_i\left(C_i\right)$ and $\alpha \prec {\displaystyle \underset{x_{\lambda }\prec B_i}{\bigwedge }}{\displaystyle \underset{x_{\lambda }\nleqq D\ge C_i}{\bigwedge }}{\langle }_L^A\left({\varkappa }_j({\langle }_L^{A}D)\right)$ for any $i\in I$. Hence $\alpha \le {\displaystyle \underset{i\in I}{\bigwedge }}{\varkappa }_i\left(C_i\right)\le {\varkappa }_i\left({\displaystyle \underset{i\in I}{\bigvee }}{C}_i\right)$ and $\alpha \le {\displaystyle \underset{i\in I}{\bigwedge }}{\displaystyle \underset{x_{\lambda }\prec B_i}{\bigwedge }}{\displaystyle \underset{x_{\lambda }\nleqq D\ge C_i}{\bigwedge }}{\langle }_L^A\left({\varkappa }_j({\langle }_L^{A}D)\right)$. Since $\{x_{\lambda }:x_{\lambda }\prec {\displaystyle \underset{i\in I}{\bigvee }}{B}_i\}={\displaystyle \bigcup _{i\in I}}\left\{x_{\lambda }:x_{\lambda }\prec B_i\right\},$ we have

$$\begin{array}{ccc}\hfill (i,j)\text{-}\mathcal{S}\left({\displaystyle \underset{i\in I}{\bigvee }}{B}_i\right)& =& \underset{C\le {\bigvee }_{i\in I}{B}_i}{\bigvee }\left\{{\varkappa }_i\left(C\right)\wedge \underset{x_{\lambda }\prec {\bigvee }_{i\in I}{B}_i}{\bigwedge }\underset{x_{\lambda }\nleqq D\ge C}{\bigwedge }{\langle }_L^A\left({\varkappa }_j({\langle }_L^{A}D)\right)\right\}\hfill \\ & \ge & {\varkappa }_i\left(\underset{i\in I}{\bigvee }{C}_i\right)\wedge \underset{i\in I}{\bigwedge }\underset{x_{\lambda }\prec B_i}{\bigwedge }\underset{x_{\lambda }\nleqq D\ge {\bigvee }_{i\in I}{C}_i}{\bigwedge }{\langle }_L^A\left({\varkappa }_j({\langle }_L^{A}D)\right)\hfill \\ & \ge & {\varkappa }_i\left(\underset{i\in I}{\bigvee }{C}_i\right)\wedge \underset{i\in I}{\bigwedge }\underset{x_{\lambda }\prec B_i}{\bigwedge }\underset{x_{\lambda }\nleqq D\ge C_i}{\bigwedge }{\langle }_L^A\left({\varkappa }_j({\langle }_L^{A}D)\right)\hfill \\ & \ge & \alpha .\hfill \end{array}$$

This shows that $(i,j)\text{-}\mathcal{S}\left(\underset{i\in I}{\bigvee }{B}_i\right)\ge \underset{i\in I}{\bigwedge }(i,j)\text{-}\mathcal{S}\left(B_i\right)$. □

Corollary 3.

Let $A\in {\mathcal{V}}_X^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space on A, and $(i,j)\text{-}\mathcal{S}$ be the gradation of $(i,j)$-$RL$-semiopenness induced by ${\varkappa }_i$ and ${\varkappa }_j$ such that $i\ne j$. Then $(i,j)\text{-}{\mathcal{S}}^{*}\left({\displaystyle \underset{i\in I}{\bigwedge }}{B}_i\right)\ge {\displaystyle \underset{i\in I}{\bigwedge }}(i,j)\text{-}{\mathcal{S}}^{*}\left(B_i\right)$ for any ${\left\{B_i\right\}}_{i\in I}\subseteq {\mathcal{F}}_X^L\left(A\right)$.

4. Pairwise Fuzzy Semicontinuous Functions Between RL-fuzzy Bitopological Spaces

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$, $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be $RL$-fbts’s on A and B, respectively. An $RL$-fuzzy function $f_{L,A}:A⟶B$ is said to be pairwise $RL$-fuzzy continuous (resp. open) iff $f_{L,A}:(A,{\varkappa }_1)⟶(B,{\varkappa }_1^{*})$ and $f_{L,A}:(A,{\varkappa }_2)⟶(B,{\varkappa }_2^{*})$ are $RL$-fuzzy continuous (resp. open).

Definition 10.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ and $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be $RL$-fbts’s on A and B, respectively, and $(i,j)\text{-}{\mathcal{S}}_1$, $(i,j)\text{-}{\mathcal{S}}_2$ their corresponding gradations of $(i,j)$-$RL$-semiopenness. An $RL$-fuzzy function $f_{L,A}:A⟶B$ is called:

(1)

pairwise $RL$-fuzzy semicontinuous iff ${\varkappa }_i^{*}\left(C\right)\le (i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$ holds for each $C\in {\mathcal{F}}_X^L\left(B\right)$.

(2)

pairwise $RL$-fuzzy irresolute iff $(i,j)\text{-}{\mathcal{S}}_2\left(C\right)\le (i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$ holds for each $C\in {\mathcal{F}}_X^L\left(B\right)$.

Corollary 4.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ and $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be $RL$-fbts’s on A and B, respectively, and $(i,j)\text{-}{\mathcal{S}}_1$, $(i,j)\text{-}{\mathcal{S}}_2$ their corresponding gradations of $(i,j)$-$RL$-semiopenness. Then:

(1)

$f_{L,A}$ is pairwise $RL$-fuzzy semicontinuous iff ${\langle }_L^B{\varkappa }_i^{*}\left(C\right)\le (i,j)\text{-}{\mathcal{S}}_1^{*}\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$ for each $C\in {\mathcal{F}}_X^L\left(B\right)$.

(2)

$f_{L,A}$ is pairwise $RL$-fuzzy irresolute iff $(i,j)\text{-}{\mathcal{S}}_2^{*}\left(C\right)\le (i,j)\text{-}{\mathcal{S}}_1^{*}\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$ for each $C\in {\mathcal{F}}_X^L\left(B\right)$.

Theorem 8.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ and $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be $RL$-fbts’s on A and B, respectively, and $(i,j)\text{-}{\mathcal{S}}_1$, $(i,j)\text{-}{\mathcal{S}}_2$ their corresponding gradations of $(i,j)$-$RL$-semiopenness. Then:

(1)

$f_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)\to (B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ is pairwise $RL$-fuzzy semicontinuous iff $f_{L,A}:(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})\to (B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$ is pairwise $RL$-semicontinuous for each $\alpha \in J\left(L\right)$.

(2)

$f_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)\to (B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ is pairwise $RL$-fuzzy irresolute iff $f_{L,A}$:$(A,{{\varkappa }_1}_{\left[\alpha \right]},$${{\varkappa }_2}_{\left[\alpha \right]})$$\to (B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$ is pairwise $RL$-irresolute for each $\alpha \in J\left(L\right)$.

Proof. 

(1)

Let $C\in {{\varkappa }_i^{*}}_{\left[\alpha \right]}$ for each $C\in {\mathcal{F}}_X^L\left(B\right)$ and $\alpha \in J\left(L\right)$, then ${\varkappa }_i^{*}\left(C\right)\ge \alpha$. Since $f_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)\to (B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ is pairwise $RL$-fuzzy semicontinuous, then $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge {\varkappa }_i^{*}\left(C\right)\ge \alpha$, i.e., $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge \alpha$. Therefore $f_{L,A}^{\leftarrow }\left(C\right)$ is $(i,j)$-$RL$-semiopen L-fuzzy set in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$. Hence $f_{L,A}:(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})\to (B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$ is pairwise $RL$-semicontinuous function.

Conversely, let ${\varkappa }_i^{*}\left(C\right)\ge \alpha$ for each $C\in {\mathcal{F}}_X^L\left(B\right)$ and $\alpha \in J\left(L\right)$, then $C\in {{\varkappa }_i^{*}}_{\left[\alpha \right]}$. By the pairwise semicontinuity of $f_{L,A}:(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})\to (B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$, we have $f_{L,A}^{\leftarrow }\left(C\right)$ is $(i,j)$-$RL$-semiopen with respect to $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$. Accordingly, $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge \alpha$ for each $\alpha \in J\left(L\right)\cap J\left({\varkappa }_i^{*}\left(C\right)\right)$, where $J\left({\varkappa }_i^{*}\left(C\right)\right)=\{\alpha \in J\left(L\right)|\alpha \le {\varkappa }_i^{*}\left(C\right)\}$. It follows that $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge \bigvee J\left({\varkappa }_i^{*}\left(C\right)\right)={\varkappa }_i^{*}\left(C\right)$.

(2)

Suppose that C is $(i,j)$-$RL$-semiopen L-fuzzy set in $(B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$, then $(i,j)\text{-}{\mathcal{S}}_2\left(C\right)$$\ge \alpha$. Since $f_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)\to (B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ is pairwise $RL$-fuzzy irresolute, then $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$$\ge (i,j)\text{-}{\mathcal{S}}_2\left(C\right)\ge \alpha$, so $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge \alpha$, therefore $f_{L,A}^{\leftarrow }\left(C\right)$ is $(i.j)$-$RL$-semiopen L-fuzzy set in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$. So that $f_{L,A}:(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})\to (B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$ is pairwise $RL$-irresolute.

Conversely, let $(i,j)\text{-}{\mathcal{S}}_2\left(C\right)\ge \alpha$ for each $\alpha \in J\left(L\right)$, then C is an $(i.j)$-$RL$-semiopen in $(B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$. Since $f_{L,A}:(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})\to (B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$ is pairwise $RL$-irresolute, then $f_{L,A}^{\leftarrow }\left(C\right)$ is $(i,j)$-$RL$-semiopen in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$. Accordingly, $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge \alpha$ for any $\alpha \in J\left(L\right)\cap J\left((i,j)\text{-}{\mathcal{S}}_2\left(C\right)\right)$, where $J\left((i,j)\text{-}{\mathcal{S}}_2\left(C\right)\right)=\{\alpha \in J\left(L\right)|\alpha \le (i,j)\text{-}{\mathcal{S}}_2\left(C\right)\}$. It follows that $(i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\ge \bigvee J\left((i,j)\text{-}{\mathcal{S}}_2\left(C\right)\right)=(i,j)\text{-}{\mathcal{S}}_2\left(C\right)$.

□

Theorem 9.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$, $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be $RL$-fbts’s on A and B, respectively. If an $RL$-fuzzy function $f_{L,A}:A⟶B$ is pairwise $RL$-fuzzy continuous, then $f_{L,A}$ is also pairwise $RL$-fuzzy semicontinuous.

Proof. 

Let $f_{L,A}:A⟶B$ be pairwise $RL$-fuzzy continuous, then ${\varkappa }_i^{*}\left(C\right)\le {\varkappa }_i\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$ for each $C\in {\mathcal{F}}_X^L\left(B\right)$ and $i=1,2$. By Theorem 6, we have

$${\varkappa }_i^{*}\left(C\right)\le {\varkappa }_i\left(f_{L,A}^{\leftarrow }\left(C\right)\right)\le (i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right),$$

for each $C\in {\mathcal{F}}_X^L\left(B\right)$. Therefore $f_{L,A}$ is pairwise $RL$-fuzzy semicontinuous. □

Theorem 10.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$, $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be two $RL$-fbts’s on A and B, respectively. If $f_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)⟶(A,{\varkappa }_1^{*},{\varkappa }_2^{*})$ is pairwise $RL$-fuzzy irresolute, then $f_{L,A}$ is pairwise $RL$-fuzzy semicontinuous.

Proof. 

Let $f_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)⟶(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be pairwise $RL$-fuzzy irresolute, then $(i,j)\text{-}{\mathcal{S}}_2\left(C\right)\le (i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$ for each $C\in {\mathcal{F}}_X^L\left(B\right)$. By Theorem 6, we have ${\varkappa }_i\left(C\right)\le (i,j)\text{-}{\mathcal{S}}_2\left(C\right)\le (i,j)\text{-}{\mathcal{S}}_1\left(f_{L,A}^{\leftarrow }\left(C\right)\right)$. Therefore $f_{L,A}$ is pairwise $RL$-fuzzy semicontinuous. □

Theorem 11.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, $C\in {\mathcal{V}}_Z^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$, $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$, $(C,{\varkappa }_1^{**},{\varkappa }_2^{**})$ be $RL$-fbts’s on A, B, and C, respectively. If $f_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)⟶(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ is pairwise $RL$-fuzzy semicontinuous and $g_{L,B}:(B,{\varkappa }_1^{*},{\varkappa }_2^{*})⟶(C,{\varkappa }_1^{**},{\varkappa }_2^{**})$ is pairwise $RL$-fuzzy continuous, then ${(g\circ f)}_{L,A}:(A,{\varkappa }_1,{\varkappa }_2)⟶(C,{\varkappa }_1^{**},{\varkappa }_2^{**})$ is pairwise $RL$-fuzzy semicontinuous.

Proof. 

Straightforward. □

5. Pairwise Fuzzy Semi-Compactness in RL-fuzzy Bitopological Spaces

Definition 11.

For any $A\in {\mathcal{V}}_X^L$ and $RL$-fbt $({\varkappa }_1,{\varkappa }_2)$ on A, an L-fuzzy set $B\in {\mathcal{F}}_X^L\left(A\right)$ is said to be a pairwise $RL$-fuzzy semi-compact with respect to $({\varkappa }_1,{\varkappa }_2)$ if for each $\mathcal{R}\subseteq {\mathcal{F}}_X^L\left(A\right)$, the following inequality holds:

$$\underset{D\in \mathcal{R}}{\bigwedge }(i,j)\text{-}\mathcal{S}\left(D\right)\wedge \underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{R}}{\bigvee }D\left(x\right)\right)\le \underset{\mathcal{Q}\in 2^{\left(\mathcal{R}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{Q}}{\bigvee }D\left(x\right)\right),$$

where $2^{\left(\mathcal{R}\right)}$ refers to the collection of all finite subcollection of $\mathcal{R}$.

Theorem 12.

Let $A\in {\mathcal{V}}_X^L$ and $RL$-fbt $({\varkappa }_1,{\varkappa }_2)$ on A. An L-fuzzy set $B\in {\mathcal{F}}_X^L\left(A\right)$ is said to be a pairwise $RL$-fuzzy semi-compact with respect to $({\varkappa }_1,{\varkappa }_2)$ if for each $\mathcal{W}\subseteq {\mathcal{F}}_X^L\left(A\right)$, it follows that

$$\underset{D\in \mathcal{W}}{\bigvee }(i,j)\text{-}\mathcal{S}({\langle }_L^{A}D)\vee \underset{x\in X}{\bigvee }\left(B\left(x\right)\wedge \underset{D\in \mathcal{W}}{\bigwedge }D\left(x\right)\right)\ge \underset{\mathcal{H}\in 2^{\left(\mathcal{W}\right)}}{\bigwedge }\underset{x\in X}{\bigvee }\left(B\left(x\right)\wedge \underset{D\in \mathcal{H}}{\bigwedge }D\left(x\right)\right).$$

Proof. 

Straightforward. □

Theorem 13.

If $A\in {\mathcal{V}}_X^L$, $({\varkappa }_1,{\varkappa }_2)$ be an $RL$-fbt on A, and $B\in {\mathcal{F}}_X^L\left(A\right)$, then the next statements are equivalent:

(1)

B is a pairwise $RL$-fuzzy semi-compact.

(2)

For all $\alpha \in J\left(L\right)$, every strong α-remote collection $\mathcal{R}$ of B such that ${\bigwedge }_{D\in \mathcal{R}}(i,j)\text{-}{\mathcal{S}}^{*}\left(D\right)\nleqq {\alpha }^{\prime }$ has a finite subcollection $\mathcal{H}$ which is a (strong) α-remote collection of B.

(3)

For all $\alpha \in J\left(L\right)$, every strong α-remote collection $\mathcal{R}$ of B such that ${\bigwedge }_{D\in \mathcal{R}}(i,j)\text{-}{\mathcal{S}}^{*}\left(D\right)\nleqq {\alpha }^{\prime }$, there exists a finite subcollection $\mathcal{H}$ of $\mathcal{R}$ and $\beta \in {\mathit{\beta }}^{*}\left(\alpha \right)$ such that $\mathcal{H}$ is a (strong) β-remote collection of B.

(4)

For all $\alpha \in P\left(L\right)$, every strong α-shading $\mathcal{U}$ of B such that ${\bigwedge }_{D\in \mathcal{U}}(i,j)\text{-}\mathcal{S}\left(D\right)\nleqq \alpha$ has a finite subcollection $\mathcal{V}$ which is a (strong) α-shading of B.

(5)

For all $\alpha \in P\left(L\right)$, each strong α-shading $\mathcal{U}$ of B such that ${\bigwedge }_{D\in \mathcal{U}}(i,j)\text{-}\mathcal{S}\left(D\right)\nleqq \alpha$, there exists a finite collection $\mathcal{V}$ of $\mathcal{U}$ and $\beta \in {\mathit{\beta }}^{*}\left(\alpha \right)$ such that $\mathcal{V}$ is a (strong) β-shading of B.

(6)

For all $\alpha \in J\left(L\right)$ and $\beta \in {\mathit{\beta }}^{*}\left(\alpha \right)$, each $Q_{\alpha }$-cover $\mathcal{U}$ of B such that $(i,j)\text{-}\mathcal{S}\left(D\right)\ge \alpha$ ( for each $D\in \mathcal{U}$) has a finite subcollection $\mathcal{V}$ which is a $Q_{\beta }$-cover of B.

(7)

For all $\alpha \in J\left(L\right)$ and any $\beta \in {\mathit{\beta }}^{*}\left(\alpha \right)$, $Q_{\alpha }$-cover $\mathcal{U}$ of B such that $(i,j)\text{-}\mathcal{S}\left(D\right)\ge \alpha$ ( for each $D\in \mathcal{U}$) has a finite subcollection $\mathcal{V}$ which is a (strong) ${\beta }_{\alpha }$-cover of B.

Proof. 

Straightforward. □

Theorem 14.

Let $A\in {\mathcal{V}}_X^L$, $({\varkappa }_1,{\varkappa }_2)$ be an $RL$-fbt on A, $B\in {\mathcal{F}}_X^L\left(A\right)$, and $\mathit{\beta }(\alpha \wedge \beta )=\mathit{\beta }\left(\alpha \right)\wedge \mathit{\beta }\left(\beta \right)$ for all α, $\beta \in L$, then the next statements are equivalent:

(1)

B is pairwise $RL$-fuzzy semi-compact.

(2)

For all $\alpha \in J\left(L\right)$, every strong ${\mathit{\beta }}_{\alpha }$-cover $\mathcal{U}$ of B such that $\alpha \in \mathit{\beta }\left({\bigwedge }_{D\in \mathcal{U}}(i,j)\text{-}\mathcal{S}\left(D\right)\right)$ has a finite subcollection $\mathcal{V}$ which is a (strong) ${\mathit{\beta }}_{\alpha }$-cover of B.

(3)

For all $\alpha \in J\left(L\right)$, every strong ${\mathit{\beta }}_{\alpha }$-cover $\mathcal{U}$ of B such that $\alpha \in \mathit{\beta }\left({\bigwedge }_{D\in \mathcal{U}}(i,j)\text{-}\mathcal{S}\left(D\right)\right)$, there exists a finite subcollection $\mathcal{V}$ of $\mathcal{U}$ and $\beta \in J\left(L\right)$ with $\alpha \in {\mathit{\beta }}^{*}\left(\beta \right)$ such that $\mathcal{V}$ is a (strongly) ${\mathit{\beta }}_{\beta }$-cover of B.

Proof. 

Straightforward. □

Definition 12.

Let $A\in {\mathcal{V}}_X^L$, $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-bitopological space, $\alpha \in J\left(L\right)$, and $B\in {\mathcal{F}}_X^L\left(A\right)$. An L-fuzzy set B is called an α-pairwise $RL$-fuzzy semi-compact iff for any $\beta \in \mathit{\beta }\left(\alpha \right)$, $Q_{\alpha }$-$(i,j)$-$RL$-semiopen cover $\mathcal{U}$ of B has a finite subcollection $\mathcal{V}$ which is a $Q_{\beta }$-$(i,j)$-$RL$-semiopen cover of B.

Theorem 15.

Let $A\in {\mathcal{V}}_X^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-bitopological space. An L-fuzzy set $B\in {\mathcal{F}}_X^L\left(A\right)$ is pairwise $RL$-fuzzy semi-compact iff B is α-pairwise fuzzy semi-compact for any $\alpha \in J\left(L\right)$.

Proof. 

Let B be a pairwise $RL$-fuzzy semi-compact, then for any $\alpha \in L_{\top }$, $\beta \in \mathit{\beta }\left(\alpha \right)$ and $\mathcal{U}$ be any $Q_{\alpha }$-$(i,j)$-$RL$-semiopen cover of B, we have

$$\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{U}}{\bigvee }D\left(x\right)\right)\le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right),$$

and $\alpha \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$, so that

$$\alpha \le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right).$$

By $\beta \in \mathit{\beta }\left(\alpha \right)$, we have

$$\beta \le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right).$$

Then there is $\mathcal{V}\in 2^{\left(\mathcal{U}\right)}$ with $\beta \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{V}}D\left(x\right)\right)$. This proves that $\mathcal{V}$ is $Q_{\beta }$-$(i,j)$-$RL$-semiopen cover of B.

Conversely, suppose that each $Q_{\alpha }$-$(i,j)$-$RL$-semiopen cover $\mathcal{U}$ of B has a finite subcollction $\mathcal{V}$ which is a $Q_{\beta }$-$(i,j)$-$RL$-semiopen cover of B for all $\beta \in \mathit{\beta }\left(\alpha \right)$. Hence, $\alpha \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$ yields to $\beta \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$. Therefore $\alpha \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$ implies that $\beta \le {\bigvee }_{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$. So $\alpha \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$ implies that

$$\underset{\beta \in \mathit{\beta }\left(\alpha \right)}{\bigvee }\beta \le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{U}}{\bigvee }D\left(x\right)\right),$$

i.e,

$$\alpha \le \underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{U}}{\bigvee }D\left(x\right)\right),$$

implies that

$$\alpha \le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{U}}{\bigvee }D\left(x\right)\right).$$

Hence

$$\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{U}}{\bigvee }D\left(x\right)\right)\le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right).$$

□

Theorem 16.

Let $A\in {\mathcal{V}}_X^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space. An L-fuzzy set $B\in {\mathcal{F}}_X^L\left(A\right)$ is a pairwise $RL$-fuzzy semi-compact in $(A,{\varkappa }_1,{\varkappa }_2)$ if and only if B is an α-pairwise $RL$-fuzzy semi-compact in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$ for all $\alpha \in J\left(L\right)$.

Proof. 

Let $B\in {\mathcal{F}}_X^L\left(A\right)$ be a pairwise $RL$-fuzzy semi-compact in $(A,{\varkappa }_1,{\varkappa }_2)$, then for each collection $\mathcal{U}\subseteq {\mathcal{F}}_X^L\left(A\right)$, we have

$$\underset{D\in \mathcal{U}}{\bigwedge }(i,j)\text{-}\mathcal{S}\left(D\right)\wedge \underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{U}}{\bigvee }D\left(x\right)\right)\le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right).$$

Then for all $\alpha \in J\left(L\right)$ and $\mathcal{U}\subseteq {\left((i,j)\text{-}\mathcal{S}\right)}_{\left[\alpha \right]}$, we have that

$$\alpha \le \underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{U}}{\bigvee }D\left(x\right)\right)\Rightarrow \alpha \le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right).$$

Hence, for every $\beta \in \mathit{\beta }\left(\alpha \right)$, there is $\mathcal{V}\in 2^{\left(\mathcal{U}\right)}$ with $\beta \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{V}}D\left(x\right)\right)$. i.e., for all $\alpha \in J\left(L\right)$ and $\beta \in \mathit{\beta }\left(\alpha \right)$, every $Q_{\alpha }$-$(i,j)$-$RL$-semiopen cover $\mathcal{U}$ of B in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$ has a finite subcollection $\mathcal{V}$ which is a $Q_{\alpha }$-$(i.j)$-$RL$-semiopen cover. Then for every $\alpha \in J\left(L\right)$, B is $\alpha$-pairwise $RL$-fuzzy semi-compact in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$.

Conversely, suppose that for every $\alpha \in J\left(L\right)$, B is $\alpha$-pairwise $RL$-fuzzy semi-compact in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$ and let $\alpha \le {\bigwedge }_{D\in \mathcal{U}}(i,j)\text{-}\mathcal{S}\left(D\right)\wedge {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$ for every $\mathcal{U}\subseteq {\mathcal{F}}_X^L\left(A\right)$, then $\alpha \le {\bigwedge }_{D\in \mathcal{U}}(i,j)\text{-}\mathcal{S}\left(D\right)$ and $\alpha \le {\bigwedge }_{x\in X}\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$, i.e, $\mathcal{U}\subseteq {\left((i,j)\text{-}\mathcal{S}\right)}_{\left[\alpha \right]}$ and $\alpha$≤${\bigwedge }_{x\in X}$$\left({\langle }_L^{A}B\left(x\right)\vee {\bigvee }_{D\in \mathcal{U}}D\left(x\right)\right)$. Hence for all $\beta \in \beta \left(\alpha \right)$, there is $\mathcal{V}\in 2^{\left(\mathcal{U}\right)}$ with

$$\beta \le \underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right).$$

So that

$$\alpha \le \underset{\mathcal{V}\in 2^{\left(\mathcal{U}\right)}}{\bigvee }\underset{x\in X}{\bigwedge }\left({\langle }_L^{A}B\left(x\right)\vee \underset{D\in \mathcal{V}}{\bigvee }D\left(x\right)\right).$$

Then B is a pairwise $RL$-fuzzy semi-compact in $(A,{\varkappa }_1,{\varkappa }_2)$. □

Lemma 2.

Let $A\in {\mathcal{V}}_X^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-bitopological space, $\alpha \in J\left(L\right)$, and $B,C\in {\mathcal{F}}_X^L\left(A\right)$. If B is α-pairwise $RL$-fuzzy semi-compact and C is $(i,j)$-$RL$-semiclosed, then $B\wedge C$ is α-pairwise $RL$-fuzzy semi-compact.

The next theorem is an immediate consequence from Lemma 2:

Theorem 17.

Let $A\in {\mathcal{V}}_X^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space, and $B,C\in {\mathcal{F}}_X^L\left(A\right)$. If B is a pairwise $RL$-fuzzy semi-compact and $(i,j)\text{-}{\mathcal{S}}^{*}\left(C\right)=\top$, then $B\wedge C$ is a pairwise $RL$-fuzzy semi-compact.

Lemma 3.

Let $A\in {\mathcal{V}}_X^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-bitopological space, $\alpha \in J\left(L\right)$, and $B,C\in {\mathcal{F}}_X^L\left(A\right)$. If B, C are α-pairwise $RL$-fuzzy semi-compact, then $B\vee C$ is α-pairwise $RL$-fuzzy semi-compact.

Theorem 18.

Let $A\in {\mathcal{V}}_X^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$ be an $RL$-fuzzy bitopological space, and $B,C\in {\mathcal{F}}_X^L\left(A\right)$. If B, C are pairwise $RL$-fuzzy semi-compact, then $B\vee C$ is pairwise $RL$-fuzzy semi-compact.

Proof. 

Straightforward. □

Lemma 4.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$, $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be $RL$-bts’s on A and B, respectively, $\alpha \in J\left(L\right)$, $D\in {\mathcal{F}}_X^L\left(A\right)$, and $f_{L,A}:A⟶B$ be a pairwise $RL$-irresolute mapping. If D is α-pairwise fuzzy semi-compact in $(A,{\varkappa }_1,{\varkappa }_2)$, then $f_{L,A}^{\to }\left(D\right)$ is α-pairwise fuzzy semi-compact in $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$.

Theorem 19.

Let $A\in {\mathcal{V}}_X^L$, $B\in {\mathcal{V}}_Y^L$, and $(A,{\varkappa }_1,{\varkappa }_2)$, $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$ be two $RL$-fbts’s on A and B, respectively, $D\in {\mathcal{F}}_X^L\left(A\right)$, and $f_{L,A}:A⟶B$ be a pairwise $RL$-fuzzy irresolute mapping. If D is a pairwise $RL$-fuzzy semi-compact in $(A,{\varkappa }_1,{\varkappa }_2)$, then $f_{L,A}^{\to }\left(D\right)$ is a pairwise $RL$-fuzzy semi-compact in $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$.

Proof. 

Let D be a pairwise $RL$-fuzzy semi -compact in $(A,{\varkappa }_1,{\varkappa }_2)$. Based on Theorem 16, we have D is $\alpha$-pairwise fuzzy semi-compact in $(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})$ for all $\alpha \in J\left(L\right)$. By Theorem 16, $f_{L,A}:(A,{{\varkappa }_1}_{\left[\alpha \right]},{{\varkappa }_2}_{\left[\alpha \right]})\to (B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$ is pairwise $RL$-irresolute. Therefore by using Lemma 4, $f_{L,A}^{\to }\left(D\right)$ is $\alpha$-pairwise $RL$-fuzzy semi-compact in $(B,{{\varkappa }_1^{*}}_{\left[\alpha \right]},{{\varkappa }_2^{*}}_{\left[\alpha \right]})$. Then $f_{L,A}^{\to }\left(D\right)$ is pairwise $RL$-fuzzy semi-compact in $(B,{\varkappa }_1^{*},{\varkappa }_2^{*})$. □

6. Conclusions

The idea of $RL$-fuzzy bitopological spaces extends the idea of $RL$-fuzzy topological spaces and as well as the idea of L-fuzzy topological spaces in Kubiak-Šostak’s sense. If we restrict the newly defined concepts by assuming that A equal to ${\top }_X$, we get L-fuzzy bitopological spaces. On the other hand, if we consider the case of $i=j$, we get L-fuzzy topological spaces in Kubiak-Šostak’s sense [20,21].

In this paper, we initiated the idea of $(i,j)$-$RL$-semiopen gradation of L-fuzzy sets in $RL$-fuzzy bitopological spaces based on the concept of pseudo-complement. We studied different properties regarding the degree of $(i,j)$-$RL$-semiopenness of L-fuzzy set. Moreover, we elaborated pairwise $RL$-fuzzy semicontinuous and pairwise $RL$-fuzzy irresolute functions and discussed some of their elementary properties based on the $(i,j)$-$RL$-semiopen gradation. Further, the pairwise $RL$-fuzzy semi-compactness of an L-fuzzy set in $RL$-fuzzy bitopological spaces is defined and explained.

In the future, we are focusing on representing several kinds of openness as gradation in $RL$-fuzzy bitopology and use it to extend the corresponding kinds of continuity, separation, connectedness, and compactness.