# Preopenness Degree inRL-Fuzzy Bitopological Spaces

Authors: O. H. Khalil; K. E. El-Helow; A. Ghareeb
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

Based on the concept of pseudocomplement, we introduce a new representation of preopenness ofL-fuzzy sets in RL-fuzzy bitopological spaces. The concepts of pairwise RL-fuzzy precontinuous and pairwise RL-fuzzy preirresolute functions are extended and discussed based on the i,j-RL-preopen gradation. Further, we follow up with a study of pairwise RL-fuzzy precompactness in RL-fuzzy bitopological spaces of an L-fuzzy set. We find that our paper offers more general results since RL-fuzzy bitopology is a generalization of L-bitopology, RL-bitopology, and L-fuzzy topology.


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## Body

## 1. Introduction

The fuzzy set theory was introduced in the year 1965 by Zadeh [1]. The development of fuzzy set theory, since its introduction, has been dramatic and breathtaking! Thousands of research papers have appeared in various journals devoted entirely to theoretical and application aspects of fuzzy sets. Artificial intelligence, automata theory, computer science, control theory, decision-making, expert systems, medical diagnosis, neural networks, pattern recognition, robotics, and social sciences are a few fields where fuzzy sets find application. Within a short time since its introduction, the fuzzy sets have permeated almost every academic discipline and have made their way into consumer products! Apart from this, it is also used for the construction of machines by way of intelligent robotics (engines, cars, ships, turbines, etc.) and controls (Sendai subway train in Japan, etc.) as well as for military purposes.In 1991, Bin Shahna [2] introduced the concept of α-open and preopen sets in the context of fuzzy sets, and he introduced a preliminary study of fuzzy strong semicontinuity and fuzzy precontinuity as well. Later, the concepts of fuzzy α-open sets, fuzzy preopen, fuzzy α-continuous mappings, and fuzzy precontinuous mappings have been generalized to the setting of fuzzy bitopological spaces in [3], where some of their fundamental properties have been studied.Shi [4] presented the concept of an L-fuzzy preopen degree of L-fuzzy set in L-fuzzy topological spaces. In addition, he discussed the fundamental properties of L-fuzzy precontinuous and L-fuzzy preirresolute mappings. It has been found that Shi’s operator is incredibly useful in introducing other gradations as well as in analyzing many topological characteristics [5].The concept ofRL-topology has recently been introduced and studied by H. Li and Q. Li [6] as an extension of L-topology. A detailed discussion is also presented concerning RL-continuous mapping and RL-compactness by means of an inequality. As a generalization of RL-topology and L-fuzzy topology, H. Li and Q. Li [7] defined RL-fuzzy topology. Further investigations are conducted regarding the RL-fuzzy compactness in RL-fuzzy topological spaces. As a consequence of their work, Zhang et al. [8] presented the Lindelöf property degree as well as the countable RL-fuzzy compactness degree of an L-subset. It is clear that the gradation of fuzzy compactness and Lindelöf property in the sense of Kubiak and Šostak are special cases of the corresponding degrees in RL-fuzzy topology.In this paper, the pseudocomplement ofL-fuzzy sets is put forward as a basis for the definition of i,j-RL-preopen degree in RL-fuzzy bitopology. Additionally, we introduce and discuss pairwise RL-fuzzy precontinuous, pairwise RL-fuzzy preirresolute mappings, and pairwise RL-fuzzy precompactness.


## 2. Preliminaries

Throughout this paper,L,∨,∧,′ denotes a complete DeMorgan algebra [9, 10] and X is a nonempty crisp set. By LX, we refer to the family of all L-fuzzy sets presented on X. The greatest and the smallest elements in L and LX are 1L,0L and 1LX,0LX, respectively. For any α, β∈L, α≺β means that the element α is wedge below β in L [11]. JL (resp. PL) denotes the collection of nonzero coprime (resp. nonunit prime) members in L. The greatest minimal collection and the greatest maximal collection of α are denoted by αα and βα, respectively. Moreover, α∗α=αα∩JL and β∗α=βα∩PL. The valuable L-fuzzy set A is an L-fuzzy set that achieves the condition A≤A′. VXL refers to the family of valuable L-fuzzy sets on X, i.e, VXL=A∈LX:A≤A′. Moreover, ΨXLA=B∈LX:B≤A for every A∈VXL. Let ψ:X⟶Y be a mapping, A∈VXL, and B∈VYL, ψL⟶A:ΨXLA⟶LY with D∈ΨXLA↦ψL⟶D, is called an L-fuzzy function restriction (briefly, RL-fuzzy function), defined by ψL⟶,A:A⟶B with ψL⟶A≤B. Moreover, ψL⟵,AC=∨D∈ΨXLA:ψL⟶D≤C for each C∈ΨYLB. Clearly, ψL⟵,AC=A∧ψL⟵C. Let A∈VXL and B∈ΨXLA, then BLA=A∧B′ if B≠A and BLA=0LX if B=A. The operation BLA is called the pseudocomplement of B with respect to A [6, 7]. The following proposition lists some of its properties:Proposition 1 (see [6, 7]).
IfA∈VXL, H,G∈ΨXLA, andHii∈Ω⊆ΨXLA, then:
(1)
HLA=A⇔H≤A′(2)
H≤G⇒GLA≤HLA(3)
∧i∈ΩLAHi=∨i∈ΩHiLA(4)
∧i∈ΩLAHi≤∧i∈ΩHiLA, ∧i∈ΩLAHi=∧i∈ΩHiLA provided that ∨i∈ΩHi≠ALemma 2 (see [6]).
For anyA∈VXL, B∈VYL, ψL⟶,A:A⟶B, H∈ΨXLA, and U⊆ΨXLA, we have
(1)∨y∈YψL⟶,AHy∧∧G∈UGy=∨x∈XHx∧∧G∈UψL⟵,AGx.
Equivalently [8],
(2)∧y∈YψL⟶,ALAHy∨∨G∈PGy=∧x∈XHLAx∨∨G∈PψL⟵,AGx.A subcollectionT⊆LX is called an L-topology [9, 10, 12] (briefly, L-t) if T includes the smallest and the greatest L-fuzzy sets in LX. Moreover, the subcollection T is closed for every suprema and finite infima. If T is an L-topology on X, then the paire X,T is said to be an L-topological space on X. Furthermore, elements of T are said to be open, and their complements are said to be closed. For any mapping ψ:X,T1⟶Y,T2, ψ is said to be L-continuous iff ψL⟵C∈T1 for every C∈T2.The mappingT:LX⟶L is called an L-fuzzy topology on the set X [13–15] if it achieves the following statements:
(1)
T0LX=T1LX=1L(2)
TH1∧H2≥TH1∧TH2, for every H1,H2∈LX(3)
T∨i∈ΩHi≥∧i∈ΩTHi, for every Hii∈Ω⊆LXIfT is an L-topology on X, then the pair X,T is said to be an L-fuzzy topological space (briefly, L-fts). The degree of openness and the degree of closeness of G∈LX are represented by TG and T∗G=TG′, respectively. The mapping ψ:X,T1⟶Y,T2 is said to be an L-fuzzy continuous if and only if T1ψL⟵M≥T2M for every M∈LY.Definition 3 (see [6]).
LetA∈VXL and R⊆ΨXLA; then R is called a relative L-topology (briefly, RL-t) A if it achieves the next conditions:
(1)
A∈R and H∈R, for each H≤A′(2)
H1∧H2∈R, for any H1, H2∈R(3)
∨i∈ΩHi∈R, for any Hii∈Ω⊆RIfR is an RL-topology on A, then, the pair A,R is called an RL-topological space on A (briefly, RL-ts). The relative open L-fuzzy sets (briefly, RL-open fuzzy set) are the members of R while their pseudocomplements are said to be RL-closed fuzzy sets, i.e., BLA∈R. The family of all RL-closed fuzzy sets with respect to R is denoted by RLA. Let A∈VXL, B∈VYL and A,R1, B,R2 be two RL-ts. The RL-fuzzy mapping ψL⟶,A:A⟶B is called an RL-continuous if and only if ψL⟵,AC∈R1LA for each C∈R2LA. Equivalently, ψL⟶,A:A⟶B is called an RL-continuous if and only if ψL⟵,AH∈R1 for each H∈R2. If R1 and R2 are RL-topologies on A∈VXL, then A,R1,R2 is called an RL-bitopological space (briefly, RL-bts). Ri-RL-open (resp. closed) refers to the open (resp. closed) L-fuzzy set with respect to A,Ri, where i=1,2. In case of A=1X, we will get back to L-topology and L-bitopology.The concepts ofβα-cover, strong βα-cover, Qα-cover, α-shading, strong α-shading, α-remote collection, and strong α-remote family [16] are extended to RL-topological spaces in [17] as follows:Definition 4 (see [17]).
For eachA∈VXL, RL-topology R on A, H∈ΨXLA, and α∈L0, a set U⊆ΨXLA is said to be
(1)
βα-cover of H if α∈βHLAx∨∨G∈UGx for all x∈X and U is said to be strong βα-cover of B if a∈β∧x∈XHLAx∨∨G∈UGx(2)
Qα-cover of B if HLAx∨∨G∈UGx≥α for all x∈XDefinition 5 (see [17]).
For eachA∈VXL, RL-topology R on A, α∈L1, and H∈ΨXLA, a family A⊆ΨXLA is said to be
(1)
α-shading of H if HLAx∨∨G∈AGx≰α for all x∈X(2)
strongα-shading of H if ∧x∈XHLAx∨∨G∈AGx≰α(3)
α-remote family of H if Hx∧∧G∈AGx≱α for all x∈X(4)
strongα-remote family of H if ∨x∈XHx∧∧G∈AGx≱αTheorem 6 (see [6]).
For eachRL-ts A,R, the next statements are valid:
(1)
A∈RLA and H∈RLA for every H≤A′(2)
H1∨H2∈RLA for every H1, H2∈RLA(3)
∧i∈IHi∈RLA for every −Hi:i∈Ω⊆RLADefinition 7 (see [7]).
The mappingR:ΨXLA⟶L such that A∈VXL is called an RL-fuzzy topology on A if R achieves the next statements:
(1)
RA=1L, for every H≤A′, RH=1L(2)
RH1∧H2≥RH1∧RH2, for every H1, H2∈ΨXLA(3)
R∨i∈ΩHi≥∧i∈ΩRHi, for every Hii∈Ω⊆ΨXLAIfR is an RL-fuzzy topology on A, then the pair A,R is called an RL-fuzzy topological space (briefly, RL-fts). For every H∈ΨXLA, RH (resp. RHLA) refers to the degree of openness (resp. closeness) of H relative to R, respectively. Moreover, if RH=1L (resp. RHLA=1L), then the RL-openness (resp. RL-closeness) of an L-fuzzy set H is confirmed. Clearly, if A=1LX, then RL-fuzzy topology on A turn into Kubiak-Šostak’s L-fuzzy topology. Further, if R is an RL-topology on A and χR:ΨXLA⟶L is a mapping defined by χRH=1L if H∈R, and χRH=0L if H∉R, then χR introduces a special RL-ft on A.Theorem 8 (see [7]).
For everyA∈VXL and RL-ftR on A. The mapping RLA:ΨXLA⟶L defined byRLAH=RHLA for every H∈ΨXLA achieves the next statements:
(1)
RLAA=1L, for every H≤A′, RLAH=1L(2)
RLAH1∨H2≥RLAH1∧RLAH2, for every H1, H2∈ΨXLA(3)
RLA∧i∈ΩHi≥∧i∈IRLAHi, for every Hii∈Ω⊆ΨXLA
RLA is called an RL-fuzzy cotopology (briefly, RL-fct) on A.Definition 9 (see [7]).
LetA∈VXL,B∈VYL and A,R1,B,R2 be two RL-fts on A and B, respectively. The RL-fuzzy mapping ψL,A:A⟶B is called an RL-fuzzy continuous if and only if
(3)R1ψL⟵,AC≥R1C,that is,
(4)R1ψL⟵,ALAC≥R1CLB,for every C∈ΨYLB. Further, if A,R1LA and B,R2LB are RL-fcts with respect to A,R1 and B,R2 respectively, then ψL⟶,A is called an RL-fuzzy continuous if and only if
(5)R1LAψL⟵,AC≥R2LBC,for every C∈ΨXLB.Definition 10 (see [17, 18]).
LetA∈VXL, and A,R be an RL-fts on A. The mapping ClR:ΨXLA⟶LJΨXLA given by
(6)ClRHxλ=∧xλ≤G≥HRGLALAfor every xλ∈JΨXLA and H∈ΨXLA is said to be the induced RL-fuzzy closure operator by R.Definition 11 (see [7]).
For everyA∈VXL and RRL-ft on A, an L-fuzzy set B∈ΨXLA is called an RL-fuzzy compact (briefly, RL-fc) if for every P⊆ΨXLA; the next inequality is true:
(7)∨G∈PRGLA∨∨x∈XHx∧∧G∈PGx≥∧S∈2P∨x∈XHx∧∧G∈SGx.Theorem 12 (see [7]).
ForA=1LX, the next statements are true:
(1)
HLA=H′, H∈ΨXLA iff H∈LX(2)
RL-fc is turned into L-fc(3)
H is RL-fc iff H is L-fcTheorem 13 (see [7]).
For everyA∈VXL and RL-fts A,R, the next conclusions are true:
(1)
IfH1,H2∈ΨXLA and H1,H2 are RL-fc, then H1∨H2 is RL-fc(2)
IfH1,H2∈ΨXLA such that H1 is an RL-fc and H2 is an RL-closed, then H1∧H2 is an RL-fc


## 3. Thei,j-RL-Preopenness Degree of L-Fuzzy Set

IfR1 and R2 be two RL-fuzzy topologies on A∈VXL, the the triple A,R1,R2 is called an RL-fuzzy bitopological space (briefly, RL-fbts). Moreover, if S any topological property, then we refer to S with respect the RL-ft Ri by Ri-S. An L-fuzzy set H∈ΨXLA of an RL-bts A,Rl,R2 is said to be an i,j-RL-preopen if and only if H≤IntRiClRjH. In the remainder of this paper i, j=1,2 such that i≠j.Definition 14.
LetA∈VXL and A,R1,R2 be an RL-fbts on A. The i,j-RL-preopenness gradation of H∈ΨXLA with respect to Ri and Rj is the mapping i,j‐P:ΨXLA⟶L given by
(8)i,j‐PH=∧xλ≺H∨xλ≺MRiH∧∧yμ≺M∧yμ≰G≥HRjGLALA.The valuei,j‐PH introduces the gradation of i,j-RL-preopenness of H is and i,j‐P∗H=i,j‐PHLA introduces the gradation of i,j-RL-precloseness of H.The next corollary is a direct consequence of the above definition and Definition10:Corollary 15.
LetA,R1,R2 be an RL-fbts on A∈VXL. Then, for every H∈ΨXLA, we have
(9)i,j‐PH=∧xλ≺H∨xλ≺MRiM∧∧yμ≺MClRjHyμ.Theorem 16.
LetR1,R2:ΨXLA⟶0L,1L be RL-ts on A∈VXL, and i,j‐P:ΨXLA⟶0L,1L be the degree of i,j-RL-preopenness with respect to Ri and Rj with i≠j. Then i,j‐PH=1L if and only if H is an i,j-RL-preopen.Proof.
The next inequality provides the proof:(10)i,j‐PH=1L⇔∧xλ≺H∨xλ≺CRiC∧∧yμ≺CClRjHyμ=1L⇔∀xλ≺H,∃Csuchthatxλ≺C,RiC=1Land∧yμ≺CClRjHyμ=1L⇔∀xλ≺H,∃Csuchthatxλ≺C,RiC=1Land∀yμ≺C,ClRjHyμ=1L⇔∀xλ≺H,∃Csuchthatxλ≺C,RiC=1LandC≤ClRjH⇔H≤IntRiClRjH⇔Hisi,j‐RL‐preopen.Theorem 17.
LetA,R1,R2 be an RL-fbts on A∈VXL and i,j‐P be the degree of i,j-RL-preopenness with respect to Ri and Rj with i≠j. Then, for every H∈ΨXLA, we have RiH≤i,j‐PH.Proof.
The next inequality provides the proof:(11)i,j‐PH=∧xλ≺H∨xλ≺CRiC∧∧yμ≺CClRjHyμ≥∧xλ≺HRiH∧∧yμ≺HClRjHyμ=RiH∧1L=RiH.Corollary 18.
LetA,R1,R2 be an RL-fbts on A∈VXL and i,j‐P be the degree of i,j-RL-preopenness with respect to Ri and Rj with i≠j. Then, for every H∈ΨXLA, we have RiLAH≤i,j‐P∗H.Theorem 19.
IfA,R1,R2 be an RL-fbts on A∈VXL, and i,j‐P be the degree of i,j-RL-preopenness with respect to Ri and Rj with i≠j, then i,j‐P∨i∈ΩHi≥∧i∈Ωi,j‐PHifor every Hii∈Ω⊆ΨXLA.Proof.
Letα∈L and α≺∧i∈Ωi,j‐PHi. Then for any i∈Ω and for any xλ≺Hi, there exists Ci∈ΨXLA such that
(12)xλ≺Ci,a≺RiCi,α≺∧yμ≺Ci∧yμ≰D≥HiRjDLALA.
Hence,(13)α≤∧i∈ΩRCi≤Rj∨i∈ΩCi,α≤∧i∈Ω∧yμ≺Ci∧yμ≰D≥HiRjDLALA.
By(14)xλ:xλ≺∨i∈ICi=⋃i∈I−xλ:xλ≺Ci,we have
(15)i,j‐P∨i∈IHi=∧xλ≺∨i∈ΩHi∨xλ≺CRiC∧∧yμ≺C∧yμ≰D≥∨i∈ΩAiRjDLALA≥∧xλ≺∨i∈ΩHiRi∨i∈ΩCi∧∧xλ≺∨i∈ΩCi∧xλ≰D≥∨i∈ΩAiRjDLALA=∧xλ≺∨i∈ΩHiRi∨i∈ΩCi∧∧i∈Ω∧xλ≺Ci∧xλ≰D≥∨i∈ΩHiRjDLALA≥∧i∈Ω∧xλ≺HiRi∨i∈ΩCi∧∧i∈Ω∧xλ≺Ci∧xλ≰D≥HiRjDLALA≥α.
Then,i,j‐P∨i∈ΩHi≥∧i∈Ωi,j‐PHi.Corollary 20.
LetA,R1,R2 be an RL-fbts on A∈VXL, and i,j‐P be the degree of i,j-RL-preopenness with respect to Ri and Rj with i≠j. Then, i,j‐P∗∧i∈ΩHi≥∧i∈Ωi,j‐P∗Hi for any Hii∈Ω⊆ΨXLA.


## 4. A New Representation of Pairwise Fuzzy Precontinuous (Preirresolute) Functions

LetA,R1,R2,B,R1∗,R2∗ be RL-fbts on A∈VXL and B∈VYL, respectively. The RL-fuzzy mapping ψL,A:A⟶B is called pairwise RL-fuzzy continuous (briefly, PRL-fco) if and only if ψL,A:A,R1⟶B,R1∗ and ψL,A:A,R2⟶B,R2∗ are RL-fuzzy continuous. In a similar way, we define the concept pairwise RL-fuzzy open mapping (briefly, PRL-fo).Definition 21.
LetA,R1,R2 and B,R1∗,R2∗ be RL-fbts on A∈VXL and B∈VYL, respectively, and i,j‐P1,i,j‐P2 are the corresponding i,j-RL-preopenness degrees. An RL-fuzzy mapping ψL,A:A⟶B is said to be
(1)
pairwiseRL-fuzzy precontinuous (briefly, PRL-fpco) if and only if Ri∗M≤i,j‐P1ψL⟵,AM is true for every M∈ΨXLB(2)
pairwiseRL-fuzzy preirresolute (briefly, PRL-fpirr) if and only if i,j‐P2M≤i,j‐P1ψL⟵,AM is true for every M∈ΨXLBCorollary 22.
LetA,R1,R2 and B,R1∗,R2∗ be RL-fbts on A∈VXL and B∈VYL, respectively, and i,j‐P1,i,j‐P2 be the corresponding i,j-RL-preopenness degrees. Then,
(1)
ψL,A is PRL-fpco if and only if Ri∗LBM≤i,j‐P1∗ψL⟵,AM for every M∈ΨXLB(2)
ψL,A is PRL-fpirr if and only if i,j‐P2∗M≤i,j‐P1∗ψL⟵,AM for every M∈ΨXLBTheorem 23.
LetA,R1,R2 and B,R1∗,R2∗ be RL-fbts on A∈VXL and B∈VYL, respectively, and i,j‐P1,i,j‐P2 be the corresponding i,j-RL-preopenness degrees. Then,
(1)
ψL,A:A,R1,R2⟶B,R1∗,R2∗ is PRL-fpco if and only if ψL,A:A,R1α,R2α⟶B,R1∗α,R2∗α is PRL-pco for every α∈JL(2)
ψL,A:A,R1,R2⟶B,R1∗,R2∗ is PRL-fpirr if and only if ψL,A:A,R1α,R2α⟶B,R1∗α,R2∗α is PRL-pirr for every α∈JLProof.
(1)
IfM∈Ri∗α for every M∈ΨXLB and α∈JL, then Ri∗M≥α. Since ψL,A:A,R1,R2⟶B,R1∗,R2∗ is PRL-fpco, then i,j‐P1ψL⟵,AM≥Ri∗M≥α, i.e., i,j‐P1ψL⟵,AM≥α. Subsequently, ψL⟵,AM is i,j-RL-preopen L-fuzzy set in A,R1α,R2α. Thus, ψL,A:A,R1α,R2α⟶B,R1∗α,R2∗α is PRL-pco mapping
Now, ifRi∗M≥α for every M∈ΨXLB and α∈JL, then M∈Ri∗α. Since ψL,A:A,R1α,R2α⟶B,R1∗α,R2∗α is PRL-pco, we have ψL⟵,AM is i,j-RL-preopen in A,R1α,R2α. Then, i,j‐P1ψL⟵,AM≥α for every α∈JL∩JRi∗M, where JRi∗M=−α∈JL∣α≤Ri∗M. Hence i,j‐P1ψL⟵,AM≥∨JRi∗M=Ri∗M.
(2)
LetM be an i,j-RL-preopen L-fuzzy set with respect to B,R1∗α,R2∗α, then i,j‐P2M≥α. Since ψL,A:A,R1,R2⟶B,R1∗,R2∗ is PRL-fpirr, then i,j‐P1ψL⟵,AM≥i,j‐P2M≥α, so i,j‐P1ψL⟵,AM≥α, then ψL⟵,AM is i.j-RL-preopen L-fuzzy set with respect to A,R1α,R2α. Hence, ψL,A:A,R1α,R2α⟶B,R1∗α,R2∗α is PRL-pirr
Now, ifi,j‐P2M≥α for every α∈JL, then M is an α-i.j-RL-preopen with respect to B,R1∗α,R2∗α. Since ψL,A:A,R1α,R2α⟶B,R1∗α,R2∗α is PRL-pirr, fL⟵,AM is α-i,j-RL-preopen with respect to A,R1α,R2α. Therefore, i,j‐P1ψL⟵,AM≥α for all α∈JL∩Ji,j‐P2M, where Ji,j‐P2M=−α∈JL∣α≤i,j‐P2M. Hence, i,j‐P1ψL⟵,AM≥∨Ji,j‐P2M=i,j‐P2M.Theorem 24.
LetA,R1,R2,B,R1∗,R2∗ be RL-fbts on A∈VXL and B∈VYL, respectively. If ψL,A:A⟶B is PRL-fco, then, ψL,A is also PRL-fpco.Proof.
LetψL,A:A⟶B be PRL-fco; then Ri∗M≤RiψL⟵,AM for every M∈ΨXLB and i=1,2. Based on Theorem 17, we have
(16)Ri∗M≤RiψL⟵,AM≤i,j‐P1ψL⟵,AM,for every M∈ΨXLB. Hence, ψL,A is PRL-fpco.Theorem 25.
LetA,R1,R2, B,R1∗,R2∗ be two RL-fbts on A∈VXL and B∈VYL, respectively. If ψL,A:A,R1,R2⟶A,R1∗,R2∗ is PRL-fpirr, then ψL,A is PRL-fprco.Proof.
IfψL,A:A,R1,R2⟶B,R1∗,R2∗ be PRL-fpirr, then i,j‐P2M≤i,j‐P1ψL⟵,AM for every M∈ΨXLB. Based on Theorem 17, we have RiM≤i,j‐P2M≤i,j‐P1ψL⟵,AM. Hence, ψL,A is PRL-fpco.Theorem 26.
LetA,R1,R2, B,R1∗,R2∗, and C,R1∗∗,R2∗∗ be PRL-fbts on A∈VXL, B∈VYL, and C∈VZL, respectively. If ψL,A:A,R1,R2⟶B,R1∗,R2∗ is PRL-fpco and ϕL,B:B,R1∗,R2∗⟶C,R1∗∗,R2∗∗ is PRL-fco, then ϕ∘ψL,A:A,R1,R2⟶C,R1∗∗,R2∗∗ is PRL-fpco.Proof.
Straightforward.


## 5. A New Representation of Pairwise Fuzzy Precompactness

Definition 27.
For anyRL-fbt R1,R2 on A∈VXL, an L-fuzzy set H∈ΨXLA is called a pairwise RL-fuzzy precompact (briefly, PRL-fpcom) with respect to R1,R2 if for every R⊆ΨXLA, the next inequality is true:
(17)∧D∈Ri,j‐PD∧∧x∈XHLAx∨∨D∈RDx≤∨Q∈2R∧x∈XHLAx∨∨D∈QDx,where 2R denotes the family of all finite subfamilies of R.Theorem 28.
LetR1,R2 be RL-fbt on A∈VXL. An L-fuzzy set H∈ΨXLA is called a PRL-fpcom with respect to R1,R2 if for every W⊆ΨXLA, we have
(18)∨D∈Wi,j‐PDLA∨∨x∈XHx∧∧D∈WDx≥∧H∈2W∨x∈XHx∧∧D∈HDx.Proof.
Straightforward.Theorem 29.
IfR1,R2 is an RL-fbt on A∈VXL, and B∈ΨXLA, then the following conditions are equivalent:
(1)
H is a PRL-fpcom(2)
For everyα∈JL, each strong α-remote family R of H with ∧D∈Ri,j‐P∗D≰α′ has a finite subfamily H which is a (strong) α-remote family of H(3)
For everyα∈JL, in each strong α-remote family R of H with ∧D∈Ri,j‐P∗D≰α′, there exists a finite subfamily H of R and β∈β∗α with H is a (strong) β-remote family of H(4)
For everyα∈PL, each strong α-shading U of H with ∧D∈Ui,j‐PD≰α has a finite subfamily V which is a (strong) α-shading of H(5)
For everyα∈PL, in every strong α-shading U of H with ∧D∈Ui,j‐PD≰α, there exists a finite family V of U and β∈β∗α with V is a (strong) β-shading of H(6)
For everyα∈JL and β∈β∗α, every Qα-cover U of H with i,j‐PD≥α (for every D∈U) has a finite subfamily V which is a Qβ-cover of H(7)
For everyα∈JL and every β∈β∗α, Qα-cover U of H with i,j‐PD≥α (for every D∈U) has a finite subfamily V which is a (strong) βα-cover of HProof.
Straightforward.Theorem 30.
LetR1,R2 be an RL-fbt on A∈VXL, H∈ΨXLA and βα∧β=βα∧ββ for all α,β∈L; then, the following conditions are equivalent:
(1)
H is PRL-fpcom(2)
For everyα∈JL, every strong βα-cover U of B with α∈β∧D∈Ui,j‐PD has a finite subfamily V which is a (strong) βα-cover of H(3)
For everyα∈JL, in every strong βα-cover U of H with α∈β∧D∈Ui,j‐PD, there exists a finite subfamily V of U and β∈JL such that α∈β∗β with V is a (strongly) ββ-cover of HProof.
Straightforward.Definition 31.
LetA∈VXL, B∈ΨXLA, α∈JL, and A,R1,R2 be an RL-bts. An L-fuzzy set H is said to be an α-pairwise RL-fuzzy precompact (briefly, αPRL-fcom) if and only if for each β∈βα, Qα-i,j-RL-preopen cover U of H has a finite subfamily V which is a Qβ-i,j-RL-preopen cover of H.Theorem 32.
LetA∈VXL, and A,R1,R2 be an RL-bts. An L-fuzzy set H∈ΨXLA is PRL-fpcom if and only if H is αPRL-fpcom for every α∈JL.Proof.
IfH is aPRL-fpcom, then for every α∈L1, β∈βα, and U which can be any Qα-i,j-RL-preopen cover of H, we obtain
(19)∧x∈XHLAx∨∨D∈UDx≤∨V∈2U∧x∈XHLAx∨∨D∈VDx,and α≤∧x∈XHLAx∨∨D∈UDx, so that
(20)α≤∨V∈2U∧x∈XHLAx∨∨D∈VDx.
Byβ∈βα, we have
(21)β≤∨V∈2U∧x∈XHLAx∨∨D∈VDx.
Then, there isV∈2U such that β≤∧x∈XHLAx∨∨D∈VDx. It follows that V is Qβ-i,j-RL-preopen cover of H.
Now, suppose that everyQα-i,j-RL-preopen cover U of H has a finite subfamily V which is a Qα-i,j-RL-preopen cover of H for each β∈βα. Then, α≤∧x∈XHLAx∨∨D∈UDx implies that β≤∧x∈XHLAx∨∨D∈UDx. Hence, α≤∧x∈XHLAx∨∨D∈UDx implies that β≤∨V∈2U∧x∈XHLAx∨∨D∈UDx. So α≤∧x∈XHLAx∨∨D∈UDx implies that
(22)∨β∈βαβ≤∨V∈2U∧x∈XHLAx∨∨D∈UDx,that is,
(23)α≤∧x∈XHLAx∨∨D∈UDxyields
(24)α≤∨V∈2U∧x∈XHLAx∨∨D∈UDx.
Therefore,(25)∧x∈XHLAx∨∨D∈UDx≤∨V∈2U∧x∈XHLAx∨∨D∈VDx.Theorem 33.
LetA∈VXL and A,R1,R2 be an RL-fbts. An L-fuzzy set H∈ΨXLA is a PRL-fpcom in A,R1,R2 iff H is an αPRL-fpcom in A,R1α,R2α for each α∈JL.Proof.
IfH∈ΨXLA be a PRL-fpcom in A,R1,R2, then for every family U⊆ΨXLA, we have
(26)∧D∈Ui,j‐PD∧∧x∈XHLAx∨∨D∈UDx≤∨V∈2U∧x∈XHLAx∨∨D∈VDx.
Hence, for allα∈JL and U⊆i,j‐Pα, we have that
(27)α≤∧x∈XHLAx∨∨D∈UDx⇒α≤∨V∈2U∧x∈XHLAx∨∨D∈VDx.
Hence, for eachβ∈βα, there is V∈2U such that β≤∧x∈XHLAx∨∨D∈VDx, i.e., for each α∈JL and β∈βα, each Qα-i,j-RL-preopen cover U of H in A,R1α,R2α has a finite subfamily V which is a Qα-i.j-RL-preopen cover. Then, for each α∈JL, H is αPRL-fpcom in A,R1α,R2α.
Now, suppose that for eachα∈JL, H is αPRL-fpcom in A,R1α,R2α and let α≤∧D∈Ui,j‐PD∧∧x∈XHLAx∨∨D∈UDx for each U⊆ΨXLA, then α≤∧D∈Ui,j‐PD and α≤∧x∈XHLAx∨∨D∈UDx, i.e, U⊆i,j‐Pα and α≤∧x∈XHLAx∨∨D∈UDx. Then, for every β∈βα, there is V∈2U suth that
(28)β≤∧x∈XHLAx∨∨D∈VDx.
Then,(29)α≤∨V∈2U∧x∈XHLAx∨∨D∈VDx.
So thatH is a PRL-fpcom in A,R1,R2.Lemma 34.
LetA∈VXL and A,R1,R2 be an RL-bitopological space, α∈JL, and H,G∈ΨXLA. If H is αPRL-fpcom and G is i,j-RL-preclosed, then H∧G is αPRL-fpcom.As an immediate consequence from the above lemma, we have the following theorem:Theorem 35.
LetA∈VXL and A,R1,R2 be an RL-fbts, and H,G∈ΨXLA. If H is a PRL-fpcom and i,j‐P∗G=1L, then H∧G is a PRL-fpcom.Lemma 36.
LetA∈VXL and A,R1,R2 be an RL-bts, α∈JL, and H,G∈ΨXLA. If H,G are αPRL-fpcom, then H∨G is αPRL-fpcom.Theorem 37.
LetA∈VXL and A,R1,R2 be an RL-fbts, and H,G∈ΨXLA. If H,G are PRL-fpcom, then H∨G is PRL-fpcom.Proof.
Straightforward.Lemma 38.
LetA,R1,R2,B,R1∗,R2∗ be RL-bts’s on A∈VXL and B∈VYL, respectively, α∈JL, H∈ΨXLA, and ψL,A:A⟶B be a PRL-irresolute function. If H is αPRL-fpcom in A,R1,R2, then ψL⟶,AHis αPRL−fpcom in B,R1∗,R2∗.Theorem 39.
LetA,R1,R2and B,R1∗,R2∗ be two RL-fbts’s on A∈VXL and B∈VYL, respectively, H∈ΨXLA, and ψL,A:A⟶B be a PRL-fuzzy irresolute function. If H is a PRL-fpcom in A,R1,R2, then ψL⟶,AH is a PRL-fpcom in B,R1∗,R2∗.Proof.
LetH be a PRL-fpcom in A,R1,R2. By Theorem 33, we have H which is αPRL-fpcom in A,R1α,R2α for every α∈JL. By Theorem 33, ψL,A:A,R1α,R2α⟶B,R1∗α,R2∗α is PRL-irresolute. Therefore, based on Lemma 38, ψL⟶,AH is αPRL-fpcom in B,R1∗α,R2∗α. Thus, ψL⟶,AH is PRL-fpcom in B,R1∗,R2∗.


## 6. Conclusion

In this paper, we presented the gradation of preopenness ofL-fuzzy sets in RL-fuzzy bitopological spaces relied on pseudo-complement. The new gradation is used to extend and characterize pairwise RL-fuzzy precontinuous and pairwise RL-fuzzy preirresolute functions. Moreover, we discussed pairwise RL-fuzzy precompactness of an L-fuzzy set in RL-fuzzy bitopological spaces. We think that our findings present more general results and it will open the way for many other studies.



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