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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 399-406

Published online December 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.4.399

© The Korean Institute of Intelligent Systems

More on Pairwise Fuzzy Semi Volterra Spaces

V. Chandiran1 and G. Thangaraj2

1Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India
2Department of Mathematics, Thiruvalluvar University, Vellore, India

Correspondence to :
V. Chandiran (profvcmaths@gmail.com)

Received: November 3, 2023; Revised: October 13, 2024; Accepted: November 8, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper establishes that pairwise fuzzy semi σ-Baire spaces, pairwise fuzzy semi Baire with pairwise fuzzy semi P-spaces, pairwise fuzzy semi hyperconnected with pairwise fuzzy semi P-spaces are pairwise fuzzy semi Volterra spaces. Furthermore, it establishes that pairwise fuzzy semi σ-second category spaces, pairwise fuzzy semi strongly irresolvable, and pairwise fuzzy semi second category with pairwise fuzzy semi P-(resp. submaximal) spaces are pairwise fuzzy semi weakly Volterra spaces. The conditions for fuzzy bitopological spaces to become pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces, pairwise fuzzy semi Volterra spaces to become fuzzy pairwise semi Baire spaces, and vice versa are discussed. In addition, the conditions for the pairwise fuzzy semi second category using the pairwise fuzzy semi P-(resp. submaximal) spaces to become pairwise fuzzy semi weakly Volterra spaces, pairwise fuzzy semi almost P-spaces to become pairwise fuzzy semi weakly Volterra spaces are investigated.

Keywords: Pairwise fuzzy semi Baire space, Pairwise fuzzy semi P-space, Pairwise fuzzy semi almost P-space, Pairwise fuzzy semi almost GP-space, Pairwise fuzzy semi Volterra space, Pairwise fuzzy semi weakly Volterra space

Zadeh [1] introduced the fundamental concept of fuzzy sets in 1965, forming the backbone of fuzzy mathematics. In 1968, Chang [2] introduced fuzzy topological spaces. Fuzzy semi-open sets and fuzzy semi-continuous mapping was studied by Azad [3]. In 1989, Kandil and El-Shafee [4] introduced fuzzy bitopological spaces. The concept of pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces were introduced and studied in [5, 6]. Kareem and Kumara [7] introduced a CPU scheduling algorithm based on fuzzy logic, designed to facilitate more nuanced decisions in complex situations. The multi-objective optimization approach focused on by Alburaikan et al. [8] provided the best solution in cooperative continuous static games. Fuzzy-logic support systems were studied by Kamel and El-Mougi [9], which addresses uncertainty and imprecision in the medical field. The aim of this paper is to discuss the inter-relations between pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces and other fuzzy bitopological spaces, such as pairwise fuzzy semi hyperconnected spaces, pairwise fuzzy semi Baire spaces, pairwise fuzzy semi σ-Baire spaces, pairwise fuzzy semi P-spaces, pairwise fuzzy semi almost P-spaces, and pairwise fuzzy semi almost GP-spaces.

In this section, we review some basic notions and results that are used in the subsequent sections. In this paper, indices i and j take the values {1, 2} and ij.

Definition 2.1 [2]. A fuzzy topology is a family ‘T’ of fuzzy sets in X that satisfies the following conditions:

  • (1) 0X, 1XT,

  • (2) If a, bT, then abT,

  • (3) If aiT for each i, then ∪aiT.

T is called a fuzzy topology for X and the pair (X, T) is an fuzzy topology set (fts). Every member of T is called a fuzzy open (fo) set. A fuzzy set is a fuzzy closed (fc) set if and only if the complement is an fo set.

Lemma 2.1 [3]. For a family A = {λα} of the fuzzy sets in fuzzy space X, then, ∨(cl(λα)) ≤ cl(∨λα). If A is a finite set, ∨cl(λα) = cl(∨(λα)). In addition, ∨int(λα) ≤ int(∨λα).

Definition 2.2 [10]. A fuzzy set α in an fbts (X, T1, T2) is called a pariwise fuzzy semi open (pfso) set if αsclTisintTj (α), (ij and i, j = 1, 2).

Definition 2.3 [10]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi closed (pfsc) set if sintTisclTj (α) ≤ α, (ij and i, j = 1, 2).

Definition 2.4 [10]. Let (X, τ1, τ2) be an fbts. The (i, j)-semi closure (denoted by (i, j)-scl) and (i, j)-semi interior (denoted by (i, j)-sint) of a fuzzy set A are defined as follows:

  • (i, j)-scl(A) = inf{B : BA, B is (i, j)-fuzzy semi-closed},

  • (i, j)-sint(A) = sup{B : BA, B is (i, j)-fuzzy semi-open}.

Definition 2.5 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi (pfs) Gδ-set (pfsGδ-set) if α=k=1(αk), where (αk) are pfso sets.

Definition 2.6 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pfs Fσ-set (pfsFσ-set) if α=k=1(αk), where (αk) are pfsc sets.

Definition 2.7 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi dense (pfsd) set if sclTisclTj (α) = 1, (ij and i, j = 1, 2).

Definition 2.8 [11]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi nowhere dense (pfsnd) set if sintTisclTj (α) = 0, (ij and i, j = 1, 2).

Definition 2.9 [11]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi first category (pfsfc) set if α=k=1(αk), where (αk) are pfsnd sets. Any other fuzzy set is considered to be a pairwise fuzzy semi second category (pfssc) set.

Definition 2.10 [5]. If a fuzzy set α is a pfsfc set in an fbts (X, T1, T2), then the fuzzy set 1 – α is called a pairwise fuzzy semi residual (pfsr) set.

Definition 2.11 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi σ-nowhere dense (pfsσ-nd) set if α is a pfsFσ-set such that sintTisintTj (α) = 0, (ij and i, j = 1, 2).

Definition 2.12 [12]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi σ-first category (pfsσ-fc) set if α=k=1(αk), where (αk) are pfsσ-nd sets. Any other fuzzy set is said to be a pairwise fuzzy semi σ-second category (pfsσ-sc) set.

Definition 2.13 [5]. An fbts (X, T1, T2) is said to be a pairwise fuzzy semi Volterra space (pfsVs) if sclTi(k=1N(αk))=1, (i = 1, 2) where (αk) are pfsd and pfsGδ-sets.

Definition 2.14 [6]. An fbts (X, T1, T2) is said to be a pairwise fuzzy semi weakly Volterra space (pfswVs) if k=1N(αk)0, where (αk) are pfsd and pfsGδ-sets.

Definition 2.15 [6]. An fbts (X, T1, T2) is called a pairwise fuzzy semi hyperconnected space (pfshs) if the fuzzy set α is a pfso set, then sclTi (α) = 1, (i = 1, 2).

Definition 2.16 [6]. An fbts (X, T1, T2) is called a pairwise fuzzy semi P-space (pfsP-s) if every non-zero pfsGδ-set is a pfso set. That is, if (X, T1, T2) is a pfsP-s if αTi, (i = 1, 2) for α=k=1N(αk), where (αk) are pfso sets.

Definition 2.17 [12]. An fbts (X, T1, T2) is called a pairwise fuzzy semi strongly irresolvable space (pfssis) if sclTisintTj (α) = 1, (ij and i, j = 1, 2) for each pfsd set α.

Definition 2.18 [12]. An fbts (X, T1, T2) is called a pairwise fuzzy semi submaximal space (pfsss) if each pfsd set is a pfso set. That is, if α is a pfsd set in an fbts (X, T1, T2), then αTi, (i = 1, 2).

Definition 2.19 [11]. An fbts (X, T1, T2) is called a pairwise fuzzy semi first category space (pfsfcs) if the fuzzy set 1X is a pfsfc set. That is, 1X=k=1(λk), where (λk) are pfsnd sets. Otherwise (X, T1, T2) is called a pairwise fuzzy semi second category space (pfsscs).

Theorem 2.1 [11]. If α is a pfsnd set in an fbts (X, T1, T2), then 1 – α is a pfsd set.

Theorem 2.2 [5]. In an fbts (X, T1, T2), a fuzzy set α is a pfsσ-nd set if and only if 1 – α is a pfsd and pfsGδ-set.

Theorem 2.3 [11]. If the pfsfc sets (βk) are formed from the pfsGδ-sets (αk) such that sclTi (αk) = 1, (i = 1, 2) in a pfsVs (X, T1, T2), then sintTi(k=1N(βk))=0.

Theorem 2.4 [13]. If the pfsnd sets are pfsFσ-sets in a pairwise fuzzy semi Baire space (pfsBs) (X, T1, T2), then (X, T1, T2) is a pfsσ-Bs.

Theorem 2.5 [5]. If a pfsGδ-set α in an fbts (X, T1, T2) is such that sclTi (α) = 1, (i = 1, 2) then, 1 – α is a pfsfc set.

Theorem 2.6 [13]. If each pfsnd set α in an fbts (X, T1, T2) is a pfsFσ-set, then (X, T1, T2) is a pfsBs.

Theorem 2.7 [5]. If a pfso set α in an fbts (X, T1, T2) is such that sclTi (α) = 1, (i = 1, 2), then 1 – α is a pfsnd set.

Theorem 2.8 [5]. If a pfso set α in an fbts (X, T1, T2) is such that sclTi (α) = 1, (i = 1, 2), then 1 – α is a pfsnd set.

Theorem 2.9 [12]. If sclTisclTj (α) = 1, (i, j = 1, 2 and ij) for a fuzzy set α in a pfssis (X, T1, T2), then sclTi (α) = 1.

Theorem 2.10 [13]. Let (αk) (k = 1 to ∞) be the pfsd and pfsGδ-sets in a pfsaP-s (X, T1, T2). Then (X, T1, T2) is a pfsσ-scs.

Theorem 2.11 [13]. Let k=1(αk)0, where (αk) are the pfsd and pfsGδ-sets in an fbts (X, T1, T2). Then, (X, T1, T2) is a pfsσ-scs.

Theorem 2.12 [11]. Let (X, T1, T2) be an fbts. Then the following are equivalent:

  • (1) (X, T1, T2) is a pfsBs.

  • (2) sintTi (α) = 0, (i = 1, 2), for every pfsfc set α.

  • (3) sclTi (β) = 1, (i = 1, 2), for every pfsr set β.

Theorem 2.13 [11]. If the pfsfc set α is a pfsc set in a pfsBs (X, T1, T2), then α is a pfsnd set.

Definition 3.1 [11]. An fbts (X, T1, T2) is called a pfsBs if sintTi(k=1(αk))=0, (i = 1, 2) where (αk) are pfsnd sets.

Example 3.1. Let X = {a, b, c}. The fuzzy sets λ, μ and ν are defined on X as follows:

  • λ : X → [0, 1] is defined as λ(a) = 0.5, λ(b) = 0.7, λ(c) = 0.6;

  • μ : X → [0, 1] is defined as μ(a) = 0.4, μ(b) = 0.6, μ(c) = 0.5;

  • ν : X → [0, 1] is defined as ν(a) = 0.6, ν(b) = 0.5, ν(c) = 0.4.

Then T1 = {0, λ, μ, ν, λν, μν, λν, μν, λ∧(μν), λμν, 1} and T2 = {0, λ, μ, 1} are fuzzy topologies on X. From the computations, the fuzzy sets 1 – λ, 1 – μ, 1 – (λν), 1 – (μν) and 1 – [λ ∧ (μν)] are pfnd sets. Thus sintT1{(1–λ)∨(1–μ)∨[1–(λν)]∨[1–(μν)]∨ [1–(λ ∧ (μν))]} = 0 and sintT2{(1–λ) ∨ (1–μ) ∨ [1– (λν)] ∨ [1–(μν)] ∨ [1–(λ ∧ (μν))]} = 0. Therefore, (X, T1, T2) is a pfsBs.

Proposition 3.1. If each pfsnd set α in an fbts (X, T1, T2) is a pfsFσ-set, then (X, T1, T2) is a pfsVs.

Proof. Suppose that the pfsnd set α is a pfsFσ-set. Then, from Theorem 2.6, (X, T1, T2) is a pfsBs. Then, sintTi(k=1(αk))=0, (i = 1, 2) where (αk) are pfsnd sets. This implies that sclTi(k=1(1-αk))=1. Because (αk) are pfsnd sets and from Theorem 2.1, (1 – αk) are pfsd sets. In addition, because (αk) are pfsFσ-sets, and (1 – αk) are pfsGδ-sets. Then, sclTi(k=1(1-αk))sclTi(k=1N(1-αk)), implies that 1sclTi(k=1N(1-αk)). That is, sclTi(k=1N(1-αk))=1, where (1–αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 3.2. If the pfsfc sets βk(k = 1 to N) are formed by pfsGδ-sets αk(k = 1 to N) such that sclTi (αk) = 1, (i = 1, 2) in a pfsVs (X, T1, T2), then (X, T1, T2) is a pfsBs.

Proof. Now k=1N(sintTi(βk))sintTi(k=1N(βk)), (i = 1, 2). From Theorem 2.3, sintTi(k=1N(βk))=0. Then, k=1N(sintTi(βk))=0, implies that sintTi (βk) = 0, where (βk) are pfsfc sets. Thus, from Theorem 2.12, (X, T1, T2) is a pfsBs.

Definition 3.2. An fbts (X, T1, T2) is called a pairwise fuzzy semi σ-Baire (pfsσ-Bs) space, if sintTi(k=1(αk))=0, (i = 1, 2), where (αk) are pfsσ-nd sets.

Example 3.2. Let X = {a, b, c}. The fuzzy sets α, β, δ, and μ are defined on X as follows:

  • α : X → [0, 1] is defined as α(a) = 0.2, α(b) = 0.4, and α(c) = 0.7;

  • β : X → [0, 1] is defined as β(a) = 0.2, β(b) = 0.2, and β(c) = 0.6;

  • δ : X → [0, 1] is defined as δ(a) = 0.1, δ(b) = 0.3, and δ(c) = 0.5;

  • μ : X → [0, 1] is defined as μ(a) = 0.4, μ(b) = 0.3, and μ(c) = 0.5.

Then, T1 = {0, α, β, δ, αβ, βδ, αβ, αδ, βδ, α ∧ [βδ], 1} and T2 = {0, α, β, μ, αβ, αμ, βμ, αβ, αμ, βμ, β ∨ [αμ], α ∧ [βμ], μ ∧ [αβ], αβμ, 1} represent the fuzzy topologies of X. From the computations, the fuzzy sets 1–β and 1–(αβ) are pfsFσ-sets. In addition, sintT1sintT2 (1 – β) = sintT2sintT1 (1 – β) = 0 and sintT1sintT2 (1–(αβ)) = sintT2sintT1 (1–(αβ)) = 0. Hence, 1 – β and 1 – (αβ) are pfsσ-nd sets. Thus, sintTi [1 – β] ∨ [1 – (αβ)] = 0, (i = 1, 2). Therefore (X, T1, T2) is a pfsσ-Bs.

Proposition 3.3. If the fbts (X, T1, T2) is a pfsσ-Bs, then (X, T1, T2) is a pfsVs.

Proof. Because (X, T1, T2) is a pfsσ-Bs, sintTi(k=1(αk))=0, (i = 1, 2), where (αk) are pfsσ-nd sets. Then, sintTi(k=1N(αk))sintTi(k=1(αk))=0 implies that sintTi(k=1N(αk))0; hence, sintTi(k=1N(αk))=0. Then, 1-sintTi(k=1N(αk))=1. That is, sclTi(k=1N(1-αk))=1. Because (αk) are pfsσ-nd sets and from Theorem 2.2, (1 – αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 3.4. If the pfsnd sets are pfsFσ-sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Suppose that every pfsnd set is a pfsFσ-set in a pfsBs (X, T1, T2). Then, from Theorem 2.4, (X, T1, T2) is a pfsσ-Bs. In addition, from Proposition 3.3, (X, T1, T2) is a pfsVs.

Proposition 3.5. If the pfsfc sets are pfsc sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Let (αk) (k = 1 to ∞) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2). From Theorem 2.5, (1 – αk) are pfsfc sets. Then, from the hypothesis, (1 – αk) are pfsc sets and thus, (αk) are pfso sets. Now (αk) are pfso sets such that sclTi (αk) = 1. Hence, from Theorem 2.7, (1 – αk) are pfsnd sets. Because (X, T1, T2) is a pfsBs, sintTi(k=1(1-αk))=0. Then, sclTi(k=1(αk))=1. Now sclTi(k=1(αk))sclTi(k=1N(αk)). This implies that 1clTi(k=1N(αk)). That is, sclTi(k=1(αk))=1. Because sclTi (αk) = 1, sclTjsclTi (αk) = sclTj (1) = 1, (ij and i, j = 1, 2). Thus (αk) are pfsd sets. Therefore, sclTi(k=1N(αk))=1, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 3.6. If the pfsr sets are pfso sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Let the pfsr sets (αk) (k = 1 to ∞) be the pfso sets in a pfsBs (X, T1, T2). Then, (1 – αk) are pfsfc sets such that (1–αk) are pfsc sets. Hence, from Proposition 3.5, (X, T1, T2) is a pfsVs.

Proposition 3.7. If the pfsfc sets αk (k = 1 to N) are pfsc and pfsFσ-sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Let the pfsfc sets αk (k = 1 to N) be pfsc sets. From Theorem 2.13, (αk) are pfsnd sets. In addition, from Theorem 2.1, (1 – αk) are pfsd sets. As (αk) are pfsFσ-sets, (1–αk) are pfsGδ-sets. Thus, (1–αk) are pfsd and pfsGδ-sets. Now sclTi(k=1N(1-αk))=1-sintTi(k=1N(αk)), (i = 1, 2) –→ (1). If (βk) are pfsnd sets in which the first N pfsnd sets are αk then, k=1N(αk)k=1(βk). This implies that sintTi(k=1N(αk))sintTi(k=1(βk))(2). As (X, T1, T2) is a pfsBs, sintTi(k=1(βk))=0, where (βk) are pfsnd sets. Then, from (1) and (2), sclTi(k=1N(1-αk))=1-sintTi(k=1N(αk))=1-0=1. In other words, sclTi(k=1N(1-αk))=1, where (1 – αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Remark 3.1. The conditions under which pairwise fuzzy semi Baire spaces become pairwise fuzzy semi σ-Baire spaces and the relationship between pairwise fuzzy semi σ-Baire spaces and pairwise fuzzy semi Volterra spaces can be summarized as Figure 1.

Definition 4.1 [6]. An fbts (X, T1, T2) is called a pfsP-s (pfsP-s), if every non-zero pfsGδ-set is a pfso set. That is, if (X, T1, T2) is a pfsP-s if αTi, (i = 1, 2) for α=k=1(αk), where (αk) are pfso sets.

Example 4.1. Let X = {a, b, c}. Consider the fuzzy sets λ, μ, δ and β are defined on X as follows:

  • λ : X → [0, 1] is defined as λ(a) = 0.3, λ(b) = 0.7, λ(c) = 0.9;

  • μ : X → [0, 1] is defined as μ(a) = 0.7, μ(b) = 0.5, μ(c) = 0.3;

  • δ : X → [0, 1] is defined as δ(a) = 0.5, δ(b) = 0.9, δ(c) = 0.7;

  • β : X → [0, 1] is defined as β(a) = 0.8, β(b) = 0.5, β(c) = 0.3.

Then, T1 = {0, λ, μ, δ, λμ, λδ, μδ, λμ, λδ, μδ, λ ∨ (μδ), μ ∨ (λδ), δ ∧ (λμ), λμδ, 1} and T2 = {0, λ, δ, β, λδ, λβ, δβ, λδ, λβ, δβ, λ ∨ (δβ), β ∨ (λδ), δ ∧ (λβ), λδβ, 1} represent the fuzzy topologies of X. From the computations, the fuzzy sets α = λ ∧ (λμ) ∧ (δ ∧ [λμ]) ∧ (λ ∨ [δβ]) and γ = (λδ) ∧ (λ ∨ [δβ]) ∧ (δ ∧ [λβ]) are pfsGδ-sets and αTi (i = 1, 2) and γTi. Therefore, (X, T1, T2) is a pfsP-s.

Proposition 4.1. If an fbts (X, T1, T2) is a pfsBs and pfsP-s; then, (X, T1, T2) is a pfsVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets, such that sclTi (αk) = 1, (i = 1, 2). Because (X, T1, T2) is a pfsPs, the pfsGδ-sets (αk) are pfso sets. Then, (αk) are pfso sets such that sclTi (αk) = 1. From Theorem 2.7, (1–αk) are pfsnd sets. Because (X, T1, T2) is an pfsBs, sintTi(i=1(βi))=0, where (βi) are pfsnd sets. Let us consider the first N pfsnd sets in (βi), as (1 – αk). Then, sintTi(k=1N(1-αk))sintTi(k=1(βi)) implies that sintTi(k=1N(1-αk)=0. Hence, sclTi(k=1N(αk))=1. Because sclTi (αk) = 1, sclTisclTj (αk) = sclTi (1) = 1. This implies that (αk) are pfsd sets. Thus, sclTi(k=1N(αk))=1, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 4.2. If an fbts (X, T1, T2) is a pfshs and pfsP-s, then (X, T1, T2) is a pfsVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets. Because (X, T1, T2) is a pfsP-s, the pfsGδ-sets (αk) are the pfso sets. That is, αkTi, (i = 1, 2). Then, k=1N(αk)Ti. Thus, k=1N(αk) is a pfso set. In addition, because (X, T1, T2) is a pfshs, the pfso sets (αk) are pfsd sets. Hence, the fuzzy sets (αk) are pfsd and pfsGδ-sets. Because k=1N(αk) is a pfso set in a pfshs (X, T1, T2), sclTi(k=1N(αk))=1. Thus sclTi(k=1N(αk))=1, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Remark 4.1. The inter-relations between pairwise fuzzy semi Volterra space and other fuzzy bitopological spaces such as the pairwise fuzzy semi Baire space, pairwise fuzzy semi P-space, pairwise fuzzy semi hyperconnected space can be summarized as Figure 2.

Proposition 5.1. Let (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2) in the pfsscs and pfsP-s (X, T1, T2). Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2). Because sclTi (αk) = 1, sclT1clT2 (αk) = clT1 (1) = 1 and clT2clT1 (αk) = clT2 (1) = 1. Thus clTiclTj (αk) = 1, (ij and i, j = 1, 2) and hence, (αk) are pfsd sets. In addition, because (X, T1, T2) is a pfsP-s, the pfsGδ-sets (αk) are pfso sets. Because (αk) are pfso sets such that sclTi (αk) = 1 and from Theorem 2.8, (1 – αk) are pfsnd sets. Because (X, T1, T2) is a pfsscs, k=1(βk)1, where (βk) are pfsnd sets. Consider the first N pfsnd sets in (βk) as (1–αk). Now k=1N(1-αk)i=1(βk) and k=1(βk)1, implies that k=1N(1-αk)1. Hence, k=1N(αk)0, where (αk) denote pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfswVs.

Proposition 5.2. Let the fbts (X, T1, T2) be the pfssis, pfsscs and pfsP-s. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be pfsd and pfsGδ-sets. Because (αk) are pfsd sets, sclTisclTj (αk) = 1, (ij and i, j = 1, 2). In addition, because (X, T1, T2) is a pfssis and from Theorem 2.9, sclTi (αk) = 1. Because (αk) denote pfsGδ-sets such that sclTi (αk) = 1 and from Proposition 5.1, (X, T1, T2) is a pfswVs.

Proposition 5.3. Let the fuzzy sets (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2) in the pfsscs and pfsss. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2). Then, sclTisclTj (αk) = 1, (ij and i, j = 1, 2) and hence (αk) are pfsd sets. Because (X, T1, T2) is a pfsss, the pfsd sets (αk) are pfso sets. Because (αk) are pfso sets such that sclTi (α) = 1 and from Theorem 2.8, (1 – αk) are pfsnd sets. Because (X, T1, T2) is a pfsscs, k=1(βk)1, where (βk) are pfsnd sets. Consider the first N pfsnd sets in (βk) as (1–αk). Now k=1N(1-αk)k=1(βk) and k=1(βk)1, implies that k=1N(1-αk)1. Hence, k=1N(αk)0, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfswVs.

Proposition 5.4. Let the fbts (X, T1, T2) be the pfssis, pfsscs and pfsss. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsd and pfsGδ-set. Because (αk) are pfsd sets, sclTisclTj (αk) = 1, (ij and i, j = 1, 2). In addition, because (X, T1, T2) is a pfssis and from Theorem 2.9, sclTi (αk) = 1. Thus, from Proposition 5.3, (X, T1, T2) is a pfswVs.

Remark 5.1. The inter-relations between the pairwise fuzzy semi weakly Volterra spaces and other fuzzy bitopological spaces such as pairwise fuzzy semi strongly irresolvable spaces, pairwise fuzzy semi second category spaces, pairwise fuzzy semi submaximal spaces and pairwise fuzzy semi P-spaces can be summarized as Figure 3.

Definition 6.1. An fbts (X, T1, T2) is said to be a pairwise fuzzy semi almost P-space or pfsaP-s in short if for each nonzero pfsGδ-set α, sintTisintTj (α) ≠ 0, (ij and i, j = 1, 2). That is, (X, T1, T2) is a pfsaP-s if sintT1sintT2 (α) ≠ 0 and sintT2sintT1 (α) ≠ 0, for a pfsGδ-set α.

Example 6.1. Let X = {a, b, c}. Consider the fuzzy sets α, β, γ, β and α defined on X as follows:

  • α : X → [0, 1] is defined as α(a) = 0.6, α(b) = 0.4, and α(c) = 0.5;

  • β : X → [0, 1] is defined as β(a) = 0.4, β(b) = 0.7, and β(c) = 0.6;

  • γ : X → [0, 1] is defined as γ(a) = 0.5, γ(b) = 0.3, and γ(c) = 0.7;

  • δ : X → [0, 1] is defined as δ(a) = 0.5, δ(b) = 0.2, and δ(c) = 0.7;

  • η : X → [0, 1] is defined as η(a) = 0.5, η(b) = 0.4, and η(c) = 0.6.

Clearly, T1 = {0, α, β, γ, αβ, αγ, βγ, αβ, αγ, βγ, γ ∧ (αβ), β ∧ (αγ), α ∧ (βγ), γ ∨ (αβ), β ∨ (αγ), α ∨ (βγ), αβγ, αβγ, 1} and T2 = {0, α, β, δ, αβ, αδ, βδ, αβ, αδ, βδ, δ ∧ (αβ), β ∧ (αδ), α ∧ (βδ), δ ∨ (αβ), β ∨ (αδ), α ∨ (βδ), αβδ, αβδ, 1} represent the fuzzy topologies of X. From the computations, it can be observed that α, β, αβ, αγ, βγ, αβ, β ∧ (αγ), α ∧ (βγ), γ ∨ (αβ), β ∨ (αγ), α ∨ (βγ), αβγ, αδ, βδ, β ∧ (αδ), α ∧ (βδ), δ ∨ (αβ), β ∨ (αδ), α ∨ (βδ), and αβδ are pfso sets. In addition, from the computations, η = [α∨(βγ)]∧[β ∨(αγ)]∧[γ ∨(αβ)]∧[αβδ]∧ [αβ]∧[αδ] and αβ = [αβ]∧[αδ]∧[βγ]∧[α∧ (βγ)]∧[β ∧(αδ)]. Then, η and αβ are pfsGδ-sets. Now, sintT2sintT1 (η) = sintT2 ([α ∧ (βγ)] = α ∧ (βδ) ≠ 0; sintT1sintT2 (η) = sintT1 ([α ∧ (βδ)] = α ∧ (βγ) ≠ 0 and sintT2sintT1 (αβ) = sintT2 (αβ) = αβ ≠ 0 and sintT1sintT2 (αβ) = sintT1 (αβ) = αβ ≠ 0. Thus, for the pfsGδ-sets η and αβ, sintTisintTj (η) ≠ 0 and sintTisintTj (αβ) ≠ 0 (ij and i, j = 1, 2) implies that the fbts (X, T1, T2) is a pfsaP-s.

Definition 6.2. An fbts (X, T1, T2) is called a pairwise fuzzy semi σ-first category space (pfsσ-fcs) if the fuzzy set 1X is a pfsσ-fc set. That is, 1X=k=1(αk), where (αk) are pfsσ-nd sets. Otherwise, (X, T1, T2) is considered a pairwise fuzzy semi σ-second category space (pfsσ-scs).

Proposition 6.1. Let the fbts (X, T1, T2) be a pfsσ-scs. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsd and pfsGδ-sets. Then, from Theorem 2.2, (1–αk) are the pfsσ-nd sets. Consider (βl) (l = 1 to ∞) are the pfsσ-nd sets, in which we take the first N(βl) as (1–αk). In other words, βl = 1–αk. As (X, T1, T2) is a pfsσ-scs, l=1(βl)1. Then, 1-l=1(βl)0. This will imply that l=1(1-βl)0. However, l=1(1-βl)l=1N(1-βl). Then, l=1N(1-βl)0. That is, k=1N(αk)0 [because αk = 1 – βl]. Hence, l=1N(αk)0, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfswVs.

Proposition 6.2. Let (αk) (k = 1 to ∞) be the pfsd and pfsGδ-sets in a pfsaP-s (X, T1, T2). Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to ∞) be the pfsd and pfsGδ-sets. As (X, T1, T2) is a pfsaP-s and from Theorem 2.10, (X, T1, T2) is a pfsσ-scs. From Proposition 6.1, (X, T1, T2) is a pfswVs.

Remark 6.1. The inter-relations between the pairwise fuzzy semi weakly Volterra spaces and other fuzzy bitopological spaces such as pairwise fuzzy semi σ-second category spaces and pairwise fuzzy semi almost P-spaces can be summarized as Figure 4.

Definition 7.1. Let (X, T1, T2) be an fbts. Then, (X, T1, T2) is said to be a pairwise fuzzy semi almost GP-space (pfsaGP-s) if for each pfsGδ-set α such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk), sintT1sintT2 (α) ≠ 0 ≠ sintT2sintT1 (α).

Proposition 7.1. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk) in an fbts (X, T1, T2). Then, (X, T1, T2) is a pfsaGP-s.

Proof. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)). Then,sintT1sintT2k=1(αk)k=1sintT1sintT2(αk)). This will imply that k=1(sintT1sintT2(αk))0. That is, sintT1sintT2 (αk) ≠ 0. Similarly, sintT2sintT1 (αk) ≠ 0. Therefore, for the pfsGδ-sets (αk), such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk), (X, T1, T2) is a pfsaGP-s.

Proposition 7.2. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-set such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk) in a fbts (X, T1, T2). Then, (X, T1, T2) is a pfsaGP-s and pfsσ-scs.

Proof. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)). Subsequently, from Proposition 7.1, (X, T1, T2) is a pfsaGP-s. As sintT1sintT2(k=1(αk))k=1(αk) together with sintT2sintT1(k=1(αk))k=1(αk),k=1(αk))0. From Theorem 2.11, (X, T1, T2) is a pfsσ-scs. So, (X, T1, T2) is a pfsaGP-s and pfsσ-scs.

Proposition 7.3. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk) in a fbts (X, T1, T2). Then, (X, T1, T2) is a pfsaGP-s and pfswVs.

Proof. Let (αk) (k = 1 to ∞) be pfsd and pfsGδ-sets. That is, (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk). From the hypothesis, sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk). From Proposition 7.2, (X, T1, T2) is a pfsaGP-s and pfsσ-scs. However, from Proposition 6.1, the pfsσ-scs (X, T1, T2) is a pfswVs. Therefore, (X, T1, T2) is a pfsaGP-s and pfswVs.

This paper successfully explores the inter-relationships between fuzzy bitopological spaces and various types of fuzzy bitopological spaces such as pairwise fuzzy semi Baire space, pairwise fuzzy semi σ-Baire space, pairwise fuzzy semi P-space, pairwise fuzzy semi hyperconnected space, pairwise fuzzy semi almost P-spaces and pairwise fuzzy semi almost GP-spaces. This paper presents a thorough analysis of the necessary conditions for pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces.

Future research should aim to broaden the classification of fuzzy bitopological spaces and their inter-relations beyond those covered in this paper, particularly by incorporating additional space types and new properties. Investigating the practical applications of pairwise fuzzy semi Volterra spaces in different fields such as decision-making, data analysis, and optimization problems. Exploring the connections between the fuzzy bitopological spaces and other fields such as machine learning and network theory.

Fig. 1.

The relationship between pairwise fuzzy semi Baire spaces and pairwise fuzzy semi Volterra spaces.


Fig. 2.

The relationship between pairwise fuzzy semi Baire, pairwise fuzzy semi hyperconnected, pairwise fuzzy semi P-spaces and pairwise fuzzy semi Volterra spaces.


Fig. 3.

The relationship between pairwise fuzzy semi strongly irresolvable, pairwise fuzzy semi second category, pairwise fuzzy semi submaximal, pairwise fuzzy semi P-spaces and pairwise fuzzy semi weakly Volterra spaces.


Fig. 4.

The relationship between pairwise fuzzy semi σ-second category, pairwise fuzzy semi almost P-spaces and pairwise fuzzy semi weakly Volterra spaces.


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V. Chandiran received an M.Sc. degree from University of Madras, Tamil Nadu, India and a Ph.D. degree at Thiruvalluvar University, Tamil Nadu, India. Since 2022, he has been at Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Tamil Nadu, India. His research interests include topology and fuzzy topology. He has published more than 20 articles in publications in peer-reviewed journals, conferences, and books fuzzy topology.

G. Thangaraj received an M.Sc. degree from the Madurai Kamaraj University, Tamil Nadu, India and a Ph.D. degree from University of Madras, Tamil Nadu, India. He is currently a professor at Thiruvalluvar University, Tamil Nadu, India since 2011. He received “The Best Teacher Award” by the Thiruvalluvar University, August 15, 2023. His research interests include topology and fuzzy topology. He has published more than 200 publications in peer-reviewed journals or conferences and books on fuzzy topology.

E-mail: g.thangaraj@rediffmail.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 399-406

Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.399

Copyright © The Korean Institute of Intelligent Systems.

More on Pairwise Fuzzy Semi Volterra Spaces

V. Chandiran1 and G. Thangaraj2

1Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India
2Department of Mathematics, Thiruvalluvar University, Vellore, India

Correspondence to:V. Chandiran (profvcmaths@gmail.com)

Received: November 3, 2023; Revised: October 13, 2024; Accepted: November 8, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper establishes that pairwise fuzzy semi σ-Baire spaces, pairwise fuzzy semi Baire with pairwise fuzzy semi P-spaces, pairwise fuzzy semi hyperconnected with pairwise fuzzy semi P-spaces are pairwise fuzzy semi Volterra spaces. Furthermore, it establishes that pairwise fuzzy semi σ-second category spaces, pairwise fuzzy semi strongly irresolvable, and pairwise fuzzy semi second category with pairwise fuzzy semi P-(resp. submaximal) spaces are pairwise fuzzy semi weakly Volterra spaces. The conditions for fuzzy bitopological spaces to become pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces, pairwise fuzzy semi Volterra spaces to become fuzzy pairwise semi Baire spaces, and vice versa are discussed. In addition, the conditions for the pairwise fuzzy semi second category using the pairwise fuzzy semi P-(resp. submaximal) spaces to become pairwise fuzzy semi weakly Volterra spaces, pairwise fuzzy semi almost P-spaces to become pairwise fuzzy semi weakly Volterra spaces are investigated.

Keywords: Pairwise fuzzy semi Baire space, Pairwise fuzzy semi P-space, Pairwise fuzzy semi almost P-space, Pairwise fuzzy semi almost GP-space, Pairwise fuzzy semi Volterra space, Pairwise fuzzy semi weakly Volterra space

1. Introduction

Zadeh [1] introduced the fundamental concept of fuzzy sets in 1965, forming the backbone of fuzzy mathematics. In 1968, Chang [2] introduced fuzzy topological spaces. Fuzzy semi-open sets and fuzzy semi-continuous mapping was studied by Azad [3]. In 1989, Kandil and El-Shafee [4] introduced fuzzy bitopological spaces. The concept of pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces were introduced and studied in [5, 6]. Kareem and Kumara [7] introduced a CPU scheduling algorithm based on fuzzy logic, designed to facilitate more nuanced decisions in complex situations. The multi-objective optimization approach focused on by Alburaikan et al. [8] provided the best solution in cooperative continuous static games. Fuzzy-logic support systems were studied by Kamel and El-Mougi [9], which addresses uncertainty and imprecision in the medical field. The aim of this paper is to discuss the inter-relations between pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces and other fuzzy bitopological spaces, such as pairwise fuzzy semi hyperconnected spaces, pairwise fuzzy semi Baire spaces, pairwise fuzzy semi σ-Baire spaces, pairwise fuzzy semi P-spaces, pairwise fuzzy semi almost P-spaces, and pairwise fuzzy semi almost GP-spaces.

2. Preliminaries

In this section, we review some basic notions and results that are used in the subsequent sections. In this paper, indices i and j take the values {1, 2} and ij.

Definition 2.1 [2]. A fuzzy topology is a family ‘T’ of fuzzy sets in X that satisfies the following conditions:

  • (1) 0X, 1XT,

  • (2) If a, bT, then abT,

  • (3) If aiT for each i, then ∪aiT.

T is called a fuzzy topology for X and the pair (X, T) is an fuzzy topology set (fts). Every member of T is called a fuzzy open (fo) set. A fuzzy set is a fuzzy closed (fc) set if and only if the complement is an fo set.

Lemma 2.1 [3]. For a family A = {λα} of the fuzzy sets in fuzzy space X, then, ∨(cl(λα)) ≤ cl(∨λα). If A is a finite set, ∨cl(λα) = cl(∨(λα)). In addition, ∨int(λα) ≤ int(∨λα).

Definition 2.2 [10]. A fuzzy set α in an fbts (X, T1, T2) is called a pariwise fuzzy semi open (pfso) set if αsclTisintTj (α), (ij and i, j = 1, 2).

Definition 2.3 [10]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi closed (pfsc) set if sintTisclTj (α) ≤ α, (ij and i, j = 1, 2).

Definition 2.4 [10]. Let (X, τ1, τ2) be an fbts. The (i, j)-semi closure (denoted by (i, j)-scl) and (i, j)-semi interior (denoted by (i, j)-sint) of a fuzzy set A are defined as follows:

  • (i, j)-scl(A) = inf{B : BA, B is (i, j)-fuzzy semi-closed},

  • (i, j)-sint(A) = sup{B : BA, B is (i, j)-fuzzy semi-open}.

Definition 2.5 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi (pfs) Gδ-set (pfsGδ-set) if α=k=1(αk), where (αk) are pfso sets.

Definition 2.6 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pfs Fσ-set (pfsFσ-set) if α=k=1(αk), where (αk) are pfsc sets.

Definition 2.7 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi dense (pfsd) set if sclTisclTj (α) = 1, (ij and i, j = 1, 2).

Definition 2.8 [11]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi nowhere dense (pfsnd) set if sintTisclTj (α) = 0, (ij and i, j = 1, 2).

Definition 2.9 [11]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi first category (pfsfc) set if α=k=1(αk), where (αk) are pfsnd sets. Any other fuzzy set is considered to be a pairwise fuzzy semi second category (pfssc) set.

Definition 2.10 [5]. If a fuzzy set α is a pfsfc set in an fbts (X, T1, T2), then the fuzzy set 1 – α is called a pairwise fuzzy semi residual (pfsr) set.

Definition 2.11 [5]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi σ-nowhere dense (pfsσ-nd) set if α is a pfsFσ-set such that sintTisintTj (α) = 0, (ij and i, j = 1, 2).

Definition 2.12 [12]. A fuzzy set α in an fbts (X, T1, T2) is called a pairwise fuzzy semi σ-first category (pfsσ-fc) set if α=k=1(αk), where (αk) are pfsσ-nd sets. Any other fuzzy set is said to be a pairwise fuzzy semi σ-second category (pfsσ-sc) set.

Definition 2.13 [5]. An fbts (X, T1, T2) is said to be a pairwise fuzzy semi Volterra space (pfsVs) if sclTi(k=1N(αk))=1, (i = 1, 2) where (αk) are pfsd and pfsGδ-sets.

Definition 2.14 [6]. An fbts (X, T1, T2) is said to be a pairwise fuzzy semi weakly Volterra space (pfswVs) if k=1N(αk)0, where (αk) are pfsd and pfsGδ-sets.

Definition 2.15 [6]. An fbts (X, T1, T2) is called a pairwise fuzzy semi hyperconnected space (pfshs) if the fuzzy set α is a pfso set, then sclTi (α) = 1, (i = 1, 2).

Definition 2.16 [6]. An fbts (X, T1, T2) is called a pairwise fuzzy semi P-space (pfsP-s) if every non-zero pfsGδ-set is a pfso set. That is, if (X, T1, T2) is a pfsP-s if αTi, (i = 1, 2) for α=k=1N(αk), where (αk) are pfso sets.

Definition 2.17 [12]. An fbts (X, T1, T2) is called a pairwise fuzzy semi strongly irresolvable space (pfssis) if sclTisintTj (α) = 1, (ij and i, j = 1, 2) for each pfsd set α.

Definition 2.18 [12]. An fbts (X, T1, T2) is called a pairwise fuzzy semi submaximal space (pfsss) if each pfsd set is a pfso set. That is, if α is a pfsd set in an fbts (X, T1, T2), then αTi, (i = 1, 2).

Definition 2.19 [11]. An fbts (X, T1, T2) is called a pairwise fuzzy semi first category space (pfsfcs) if the fuzzy set 1X is a pfsfc set. That is, 1X=k=1(λk), where (λk) are pfsnd sets. Otherwise (X, T1, T2) is called a pairwise fuzzy semi second category space (pfsscs).

Theorem 2.1 [11]. If α is a pfsnd set in an fbts (X, T1, T2), then 1 – α is a pfsd set.

Theorem 2.2 [5]. In an fbts (X, T1, T2), a fuzzy set α is a pfsσ-nd set if and only if 1 – α is a pfsd and pfsGδ-set.

Theorem 2.3 [11]. If the pfsfc sets (βk) are formed from the pfsGδ-sets (αk) such that sclTi (αk) = 1, (i = 1, 2) in a pfsVs (X, T1, T2), then sintTi(k=1N(βk))=0.

Theorem 2.4 [13]. If the pfsnd sets are pfsFσ-sets in a pairwise fuzzy semi Baire space (pfsBs) (X, T1, T2), then (X, T1, T2) is a pfsσ-Bs.

Theorem 2.5 [5]. If a pfsGδ-set α in an fbts (X, T1, T2) is such that sclTi (α) = 1, (i = 1, 2) then, 1 – α is a pfsfc set.

Theorem 2.6 [13]. If each pfsnd set α in an fbts (X, T1, T2) is a pfsFσ-set, then (X, T1, T2) is a pfsBs.

Theorem 2.7 [5]. If a pfso set α in an fbts (X, T1, T2) is such that sclTi (α) = 1, (i = 1, 2), then 1 – α is a pfsnd set.

Theorem 2.8 [5]. If a pfso set α in an fbts (X, T1, T2) is such that sclTi (α) = 1, (i = 1, 2), then 1 – α is a pfsnd set.

Theorem 2.9 [12]. If sclTisclTj (α) = 1, (i, j = 1, 2 and ij) for a fuzzy set α in a pfssis (X, T1, T2), then sclTi (α) = 1.

Theorem 2.10 [13]. Let (αk) (k = 1 to ∞) be the pfsd and pfsGδ-sets in a pfsaP-s (X, T1, T2). Then (X, T1, T2) is a pfsσ-scs.

Theorem 2.11 [13]. Let k=1(αk)0, where (αk) are the pfsd and pfsGδ-sets in an fbts (X, T1, T2). Then, (X, T1, T2) is a pfsσ-scs.

Theorem 2.12 [11]. Let (X, T1, T2) be an fbts. Then the following are equivalent:

  • (1) (X, T1, T2) is a pfsBs.

  • (2) sintTi (α) = 0, (i = 1, 2), for every pfsfc set α.

  • (3) sclTi (β) = 1, (i = 1, 2), for every pfsr set β.

Theorem 2.13 [11]. If the pfsfc set α is a pfsc set in a pfsBs (X, T1, T2), then α is a pfsnd set.

3. Pairwise Fuzzy Semi Volterra Spaces and Pairwise Fuzzy Semi Baire Spaces

Definition 3.1 [11]. An fbts (X, T1, T2) is called a pfsBs if sintTi(k=1(αk))=0, (i = 1, 2) where (αk) are pfsnd sets.

Example 3.1. Let X = {a, b, c}. The fuzzy sets λ, μ and ν are defined on X as follows:

  • λ : X → [0, 1] is defined as λ(a) = 0.5, λ(b) = 0.7, λ(c) = 0.6;

  • μ : X → [0, 1] is defined as μ(a) = 0.4, μ(b) = 0.6, μ(c) = 0.5;

  • ν : X → [0, 1] is defined as ν(a) = 0.6, ν(b) = 0.5, ν(c) = 0.4.

Then T1 = {0, λ, μ, ν, λν, μν, λν, μν, λ∧(μν), λμν, 1} and T2 = {0, λ, μ, 1} are fuzzy topologies on X. From the computations, the fuzzy sets 1 – λ, 1 – μ, 1 – (λν), 1 – (μν) and 1 – [λ ∧ (μν)] are pfnd sets. Thus sintT1{(1–λ)∨(1–μ)∨[1–(λν)]∨[1–(μν)]∨ [1–(λ ∧ (μν))]} = 0 and sintT2{(1–λ) ∨ (1–μ) ∨ [1– (λν)] ∨ [1–(μν)] ∨ [1–(λ ∧ (μν))]} = 0. Therefore, (X, T1, T2) is a pfsBs.

Proposition 3.1. If each pfsnd set α in an fbts (X, T1, T2) is a pfsFσ-set, then (X, T1, T2) is a pfsVs.

Proof. Suppose that the pfsnd set α is a pfsFσ-set. Then, from Theorem 2.6, (X, T1, T2) is a pfsBs. Then, sintTi(k=1(αk))=0, (i = 1, 2) where (αk) are pfsnd sets. This implies that sclTi(k=1(1-αk))=1. Because (αk) are pfsnd sets and from Theorem 2.1, (1 – αk) are pfsd sets. In addition, because (αk) are pfsFσ-sets, and (1 – αk) are pfsGδ-sets. Then, sclTi(k=1(1-αk))sclTi(k=1N(1-αk)), implies that 1sclTi(k=1N(1-αk)). That is, sclTi(k=1N(1-αk))=1, where (1–αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 3.2. If the pfsfc sets βk(k = 1 to N) are formed by pfsGδ-sets αk(k = 1 to N) such that sclTi (αk) = 1, (i = 1, 2) in a pfsVs (X, T1, T2), then (X, T1, T2) is a pfsBs.

Proof. Now k=1N(sintTi(βk))sintTi(k=1N(βk)), (i = 1, 2). From Theorem 2.3, sintTi(k=1N(βk))=0. Then, k=1N(sintTi(βk))=0, implies that sintTi (βk) = 0, where (βk) are pfsfc sets. Thus, from Theorem 2.12, (X, T1, T2) is a pfsBs.

Definition 3.2. An fbts (X, T1, T2) is called a pairwise fuzzy semi σ-Baire (pfsσ-Bs) space, if sintTi(k=1(αk))=0, (i = 1, 2), where (αk) are pfsσ-nd sets.

Example 3.2. Let X = {a, b, c}. The fuzzy sets α, β, δ, and μ are defined on X as follows:

  • α : X → [0, 1] is defined as α(a) = 0.2, α(b) = 0.4, and α(c) = 0.7;

  • β : X → [0, 1] is defined as β(a) = 0.2, β(b) = 0.2, and β(c) = 0.6;

  • δ : X → [0, 1] is defined as δ(a) = 0.1, δ(b) = 0.3, and δ(c) = 0.5;

  • μ : X → [0, 1] is defined as μ(a) = 0.4, μ(b) = 0.3, and μ(c) = 0.5.

Then, T1 = {0, α, β, δ, αβ, βδ, αβ, αδ, βδ, α ∧ [βδ], 1} and T2 = {0, α, β, μ, αβ, αμ, βμ, αβ, αμ, βμ, β ∨ [αμ], α ∧ [βμ], μ ∧ [αβ], αβμ, 1} represent the fuzzy topologies of X. From the computations, the fuzzy sets 1–β and 1–(αβ) are pfsFσ-sets. In addition, sintT1sintT2 (1 – β) = sintT2sintT1 (1 – β) = 0 and sintT1sintT2 (1–(αβ)) = sintT2sintT1 (1–(αβ)) = 0. Hence, 1 – β and 1 – (αβ) are pfsσ-nd sets. Thus, sintTi [1 – β] ∨ [1 – (αβ)] = 0, (i = 1, 2). Therefore (X, T1, T2) is a pfsσ-Bs.

Proposition 3.3. If the fbts (X, T1, T2) is a pfsσ-Bs, then (X, T1, T2) is a pfsVs.

Proof. Because (X, T1, T2) is a pfsσ-Bs, sintTi(k=1(αk))=0, (i = 1, 2), where (αk) are pfsσ-nd sets. Then, sintTi(k=1N(αk))sintTi(k=1(αk))=0 implies that sintTi(k=1N(αk))0; hence, sintTi(k=1N(αk))=0. Then, 1-sintTi(k=1N(αk))=1. That is, sclTi(k=1N(1-αk))=1. Because (αk) are pfsσ-nd sets and from Theorem 2.2, (1 – αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 3.4. If the pfsnd sets are pfsFσ-sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Suppose that every pfsnd set is a pfsFσ-set in a pfsBs (X, T1, T2). Then, from Theorem 2.4, (X, T1, T2) is a pfsσ-Bs. In addition, from Proposition 3.3, (X, T1, T2) is a pfsVs.

Proposition 3.5. If the pfsfc sets are pfsc sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Let (αk) (k = 1 to ∞) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2). From Theorem 2.5, (1 – αk) are pfsfc sets. Then, from the hypothesis, (1 – αk) are pfsc sets and thus, (αk) are pfso sets. Now (αk) are pfso sets such that sclTi (αk) = 1. Hence, from Theorem 2.7, (1 – αk) are pfsnd sets. Because (X, T1, T2) is a pfsBs, sintTi(k=1(1-αk))=0. Then, sclTi(k=1(αk))=1. Now sclTi(k=1(αk))sclTi(k=1N(αk)). This implies that 1clTi(k=1N(αk)). That is, sclTi(k=1(αk))=1. Because sclTi (αk) = 1, sclTjsclTi (αk) = sclTj (1) = 1, (ij and i, j = 1, 2). Thus (αk) are pfsd sets. Therefore, sclTi(k=1N(αk))=1, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 3.6. If the pfsr sets are pfso sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Let the pfsr sets (αk) (k = 1 to ∞) be the pfso sets in a pfsBs (X, T1, T2). Then, (1 – αk) are pfsfc sets such that (1–αk) are pfsc sets. Hence, from Proposition 3.5, (X, T1, T2) is a pfsVs.

Proposition 3.7. If the pfsfc sets αk (k = 1 to N) are pfsc and pfsFσ-sets in a pfsBs (X, T1, T2), then (X, T1, T2) is a pfsVs.

Proof. Let the pfsfc sets αk (k = 1 to N) be pfsc sets. From Theorem 2.13, (αk) are pfsnd sets. In addition, from Theorem 2.1, (1 – αk) are pfsd sets. As (αk) are pfsFσ-sets, (1–αk) are pfsGδ-sets. Thus, (1–αk) are pfsd and pfsGδ-sets. Now sclTi(k=1N(1-αk))=1-sintTi(k=1N(αk)), (i = 1, 2) –→ (1). If (βk) are pfsnd sets in which the first N pfsnd sets are αk then, k=1N(αk)k=1(βk). This implies that sintTi(k=1N(αk))sintTi(k=1(βk))(2). As (X, T1, T2) is a pfsBs, sintTi(k=1(βk))=0, where (βk) are pfsnd sets. Then, from (1) and (2), sclTi(k=1N(1-αk))=1-sintTi(k=1N(αk))=1-0=1. In other words, sclTi(k=1N(1-αk))=1, where (1 – αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Remark 3.1. The conditions under which pairwise fuzzy semi Baire spaces become pairwise fuzzy semi σ-Baire spaces and the relationship between pairwise fuzzy semi σ-Baire spaces and pairwise fuzzy semi Volterra spaces can be summarized as Figure 1.

4. Pairwise Fuzzy Semi Volterra Spaces and Pairwise Fuzzy Semi P-Spaces

Definition 4.1 [6]. An fbts (X, T1, T2) is called a pfsP-s (pfsP-s), if every non-zero pfsGδ-set is a pfso set. That is, if (X, T1, T2) is a pfsP-s if αTi, (i = 1, 2) for α=k=1(αk), where (αk) are pfso sets.

Example 4.1. Let X = {a, b, c}. Consider the fuzzy sets λ, μ, δ and β are defined on X as follows:

  • λ : X → [0, 1] is defined as λ(a) = 0.3, λ(b) = 0.7, λ(c) = 0.9;

  • μ : X → [0, 1] is defined as μ(a) = 0.7, μ(b) = 0.5, μ(c) = 0.3;

  • δ : X → [0, 1] is defined as δ(a) = 0.5, δ(b) = 0.9, δ(c) = 0.7;

  • β : X → [0, 1] is defined as β(a) = 0.8, β(b) = 0.5, β(c) = 0.3.

Then, T1 = {0, λ, μ, δ, λμ, λδ, μδ, λμ, λδ, μδ, λ ∨ (μδ), μ ∨ (λδ), δ ∧ (λμ), λμδ, 1} and T2 = {0, λ, δ, β, λδ, λβ, δβ, λδ, λβ, δβ, λ ∨ (δβ), β ∨ (λδ), δ ∧ (λβ), λδβ, 1} represent the fuzzy topologies of X. From the computations, the fuzzy sets α = λ ∧ (λμ) ∧ (δ ∧ [λμ]) ∧ (λ ∨ [δβ]) and γ = (λδ) ∧ (λ ∨ [δβ]) ∧ (δ ∧ [λβ]) are pfsGδ-sets and αTi (i = 1, 2) and γTi. Therefore, (X, T1, T2) is a pfsP-s.

Proposition 4.1. If an fbts (X, T1, T2) is a pfsBs and pfsP-s; then, (X, T1, T2) is a pfsVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets, such that sclTi (αk) = 1, (i = 1, 2). Because (X, T1, T2) is a pfsPs, the pfsGδ-sets (αk) are pfso sets. Then, (αk) are pfso sets such that sclTi (αk) = 1. From Theorem 2.7, (1–αk) are pfsnd sets. Because (X, T1, T2) is an pfsBs, sintTi(i=1(βi))=0, where (βi) are pfsnd sets. Let us consider the first N pfsnd sets in (βi), as (1 – αk). Then, sintTi(k=1N(1-αk))sintTi(k=1(βi)) implies that sintTi(k=1N(1-αk)=0. Hence, sclTi(k=1N(αk))=1. Because sclTi (αk) = 1, sclTisclTj (αk) = sclTi (1) = 1. This implies that (αk) are pfsd sets. Thus, sclTi(k=1N(αk))=1, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Proposition 4.2. If an fbts (X, T1, T2) is a pfshs and pfsP-s, then (X, T1, T2) is a pfsVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets. Because (X, T1, T2) is a pfsP-s, the pfsGδ-sets (αk) are the pfso sets. That is, αkTi, (i = 1, 2). Then, k=1N(αk)Ti. Thus, k=1N(αk) is a pfso set. In addition, because (X, T1, T2) is a pfshs, the pfso sets (αk) are pfsd sets. Hence, the fuzzy sets (αk) are pfsd and pfsGδ-sets. Because k=1N(αk) is a pfso set in a pfshs (X, T1, T2), sclTi(k=1N(αk))=1. Thus sclTi(k=1N(αk))=1, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfsVs.

Remark 4.1. The inter-relations between pairwise fuzzy semi Volterra space and other fuzzy bitopological spaces such as the pairwise fuzzy semi Baire space, pairwise fuzzy semi P-space, pairwise fuzzy semi hyperconnected space can be summarized as Figure 2.

5. Pairwise Fuzzy Semi Weakly Volterra Spaces and Pairwise Fuzzy Semi P-Spaces

Proposition 5.1. Let (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2) in the pfsscs and pfsP-s (X, T1, T2). Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2). Because sclTi (αk) = 1, sclT1clT2 (αk) = clT1 (1) = 1 and clT2clT1 (αk) = clT2 (1) = 1. Thus clTiclTj (αk) = 1, (ij and i, j = 1, 2) and hence, (αk) are pfsd sets. In addition, because (X, T1, T2) is a pfsP-s, the pfsGδ-sets (αk) are pfso sets. Because (αk) are pfso sets such that sclTi (αk) = 1 and from Theorem 2.8, (1 – αk) are pfsnd sets. Because (X, T1, T2) is a pfsscs, k=1(βk)1, where (βk) are pfsnd sets. Consider the first N pfsnd sets in (βk) as (1–αk). Now k=1N(1-αk)i=1(βk) and k=1(βk)1, implies that k=1N(1-αk)1. Hence, k=1N(αk)0, where (αk) denote pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfswVs.

Proposition 5.2. Let the fbts (X, T1, T2) be the pfssis, pfsscs and pfsP-s. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be pfsd and pfsGδ-sets. Because (αk) are pfsd sets, sclTisclTj (αk) = 1, (ij and i, j = 1, 2). In addition, because (X, T1, T2) is a pfssis and from Theorem 2.9, sclTi (αk) = 1. Because (αk) denote pfsGδ-sets such that sclTi (αk) = 1 and from Proposition 5.1, (X, T1, T2) is a pfswVs.

Proposition 5.3. Let the fuzzy sets (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2) in the pfsscs and pfsss. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsGδ-sets such that sclTi (αk) = 1, (i = 1, 2). Then, sclTisclTj (αk) = 1, (ij and i, j = 1, 2) and hence (αk) are pfsd sets. Because (X, T1, T2) is a pfsss, the pfsd sets (αk) are pfso sets. Because (αk) are pfso sets such that sclTi (α) = 1 and from Theorem 2.8, (1 – αk) are pfsnd sets. Because (X, T1, T2) is a pfsscs, k=1(βk)1, where (βk) are pfsnd sets. Consider the first N pfsnd sets in (βk) as (1–αk). Now k=1N(1-αk)k=1(βk) and k=1(βk)1, implies that k=1N(1-αk)1. Hence, k=1N(αk)0, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfswVs.

Proposition 5.4. Let the fbts (X, T1, T2) be the pfssis, pfsscs and pfsss. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsd and pfsGδ-set. Because (αk) are pfsd sets, sclTisclTj (αk) = 1, (ij and i, j = 1, 2). In addition, because (X, T1, T2) is a pfssis and from Theorem 2.9, sclTi (αk) = 1. Thus, from Proposition 5.3, (X, T1, T2) is a pfswVs.

Remark 5.1. The inter-relations between the pairwise fuzzy semi weakly Volterra spaces and other fuzzy bitopological spaces such as pairwise fuzzy semi strongly irresolvable spaces, pairwise fuzzy semi second category spaces, pairwise fuzzy semi submaximal spaces and pairwise fuzzy semi P-spaces can be summarized as Figure 3.

6. Pairwise Fuzzy Semi Weakly Volterra Spaces and Pairwise Fuzzy Semi Almost P-Spaces

Definition 6.1. An fbts (X, T1, T2) is said to be a pairwise fuzzy semi almost P-space or pfsaP-s in short if for each nonzero pfsGδ-set α, sintTisintTj (α) ≠ 0, (ij and i, j = 1, 2). That is, (X, T1, T2) is a pfsaP-s if sintT1sintT2 (α) ≠ 0 and sintT2sintT1 (α) ≠ 0, for a pfsGδ-set α.

Example 6.1. Let X = {a, b, c}. Consider the fuzzy sets α, β, γ, β and α defined on X as follows:

  • α : X → [0, 1] is defined as α(a) = 0.6, α(b) = 0.4, and α(c) = 0.5;

  • β : X → [0, 1] is defined as β(a) = 0.4, β(b) = 0.7, and β(c) = 0.6;

  • γ : X → [0, 1] is defined as γ(a) = 0.5, γ(b) = 0.3, and γ(c) = 0.7;

  • δ : X → [0, 1] is defined as δ(a) = 0.5, δ(b) = 0.2, and δ(c) = 0.7;

  • η : X → [0, 1] is defined as η(a) = 0.5, η(b) = 0.4, and η(c) = 0.6.

Clearly, T1 = {0, α, β, γ, αβ, αγ, βγ, αβ, αγ, βγ, γ ∧ (αβ), β ∧ (αγ), α ∧ (βγ), γ ∨ (αβ), β ∨ (αγ), α ∨ (βγ), αβγ, αβγ, 1} and T2 = {0, α, β, δ, αβ, αδ, βδ, αβ, αδ, βδ, δ ∧ (αβ), β ∧ (αδ), α ∧ (βδ), δ ∨ (αβ), β ∨ (αδ), α ∨ (βδ), αβδ, αβδ, 1} represent the fuzzy topologies of X. From the computations, it can be observed that α, β, αβ, αγ, βγ, αβ, β ∧ (αγ), α ∧ (βγ), γ ∨ (αβ), β ∨ (αγ), α ∨ (βγ), αβγ, αδ, βδ, β ∧ (αδ), α ∧ (βδ), δ ∨ (αβ), β ∨ (αδ), α ∨ (βδ), and αβδ are pfso sets. In addition, from the computations, η = [α∨(βγ)]∧[β ∨(αγ)]∧[γ ∨(αβ)]∧[αβδ]∧ [αβ]∧[αδ] and αβ = [αβ]∧[αδ]∧[βγ]∧[α∧ (βγ)]∧[β ∧(αδ)]. Then, η and αβ are pfsGδ-sets. Now, sintT2sintT1 (η) = sintT2 ([α ∧ (βγ)] = α ∧ (βδ) ≠ 0; sintT1sintT2 (η) = sintT1 ([α ∧ (βδ)] = α ∧ (βγ) ≠ 0 and sintT2sintT1 (αβ) = sintT2 (αβ) = αβ ≠ 0 and sintT1sintT2 (αβ) = sintT1 (αβ) = αβ ≠ 0. Thus, for the pfsGδ-sets η and αβ, sintTisintTj (η) ≠ 0 and sintTisintTj (αβ) ≠ 0 (ij and i, j = 1, 2) implies that the fbts (X, T1, T2) is a pfsaP-s.

Definition 6.2. An fbts (X, T1, T2) is called a pairwise fuzzy semi σ-first category space (pfsσ-fcs) if the fuzzy set 1X is a pfsσ-fc set. That is, 1X=k=1(αk), where (αk) are pfsσ-nd sets. Otherwise, (X, T1, T2) is considered a pairwise fuzzy semi σ-second category space (pfsσ-scs).

Proposition 6.1. Let the fbts (X, T1, T2) be a pfsσ-scs. Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to N) be the pfsd and pfsGδ-sets. Then, from Theorem 2.2, (1–αk) are the pfsσ-nd sets. Consider (βl) (l = 1 to ∞) are the pfsσ-nd sets, in which we take the first N(βl) as (1–αk). In other words, βl = 1–αk. As (X, T1, T2) is a pfsσ-scs, l=1(βl)1. Then, 1-l=1(βl)0. This will imply that l=1(1-βl)0. However, l=1(1-βl)l=1N(1-βl). Then, l=1N(1-βl)0. That is, k=1N(αk)0 [because αk = 1 – βl]. Hence, l=1N(αk)0, where (αk) are pfsd and pfsGδ-sets. Therefore, (X, T1, T2) is a pfswVs.

Proposition 6.2. Let (αk) (k = 1 to ∞) be the pfsd and pfsGδ-sets in a pfsaP-s (X, T1, T2). Then, (X, T1, T2) is a pfswVs.

Proof. Let (αk) (k = 1 to ∞) be the pfsd and pfsGδ-sets. As (X, T1, T2) is a pfsaP-s and from Theorem 2.10, (X, T1, T2) is a pfsσ-scs. From Proposition 6.1, (X, T1, T2) is a pfswVs.

Remark 6.1. The inter-relations between the pairwise fuzzy semi weakly Volterra spaces and other fuzzy bitopological spaces such as pairwise fuzzy semi σ-second category spaces and pairwise fuzzy semi almost P-spaces can be summarized as Figure 4.

7. Pairwise Fuzzy Semi Weakly Volterra Spaces and Pairwise Fuzzy Semi Almost GP-Spaces

Definition 7.1. Let (X, T1, T2) be an fbts. Then, (X, T1, T2) is said to be a pairwise fuzzy semi almost GP-space (pfsaGP-s) if for each pfsGδ-set α such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk), sintT1sintT2 (α) ≠ 0 ≠ sintT2sintT1 (α).

Proposition 7.1. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk) in an fbts (X, T1, T2). Then, (X, T1, T2) is a pfsaGP-s.

Proof. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)). Then,sintT1sintT2k=1(αk)k=1sintT1sintT2(αk)). This will imply that k=1(sintT1sintT2(αk))0. That is, sintT1sintT2 (αk) ≠ 0. Similarly, sintT2sintT1 (αk) ≠ 0. Therefore, for the pfsGδ-sets (αk), such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk), (X, T1, T2) is a pfsaGP-s.

Proposition 7.2. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-set such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk) in a fbts (X, T1, T2). Then, (X, T1, T2) is a pfsaGP-s and pfsσ-scs.

Proof. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)). Subsequently, from Proposition 7.1, (X, T1, T2) is a pfsaGP-s. As sintT1sintT2(k=1(αk))k=1(αk) together with sintT2sintT1(k=1(αk))k=1(αk),k=1(αk))0. From Theorem 2.11, (X, T1, T2) is a pfsσ-scs. So, (X, T1, T2) is a pfsaGP-s and pfsσ-scs.

Proposition 7.3. Let sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk) in a fbts (X, T1, T2). Then, (X, T1, T2) is a pfsaGP-s and pfswVs.

Proof. Let (αk) (k = 1 to ∞) be pfsd and pfsGδ-sets. That is, (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk). From the hypothesis, sintT1sintT2(k=1(αk))0sintT2sintT1(k=1(αk)), where (αk) are pfsGδ-sets such that sclT1sclT2 (αk) = 1 = sclT2sclT1 (αk). From Proposition 7.2, (X, T1, T2) is a pfsaGP-s and pfsσ-scs. However, from Proposition 6.1, the pfsσ-scs (X, T1, T2) is a pfswVs. Therefore, (X, T1, T2) is a pfsaGP-s and pfswVs.

8. Conclusion and Future Works

This paper successfully explores the inter-relationships between fuzzy bitopological spaces and various types of fuzzy bitopological spaces such as pairwise fuzzy semi Baire space, pairwise fuzzy semi σ-Baire space, pairwise fuzzy semi P-space, pairwise fuzzy semi hyperconnected space, pairwise fuzzy semi almost P-spaces and pairwise fuzzy semi almost GP-spaces. This paper presents a thorough analysis of the necessary conditions for pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces.

Future research should aim to broaden the classification of fuzzy bitopological spaces and their inter-relations beyond those covered in this paper, particularly by incorporating additional space types and new properties. Investigating the practical applications of pairwise fuzzy semi Volterra spaces in different fields such as decision-making, data analysis, and optimization problems. Exploring the connections between the fuzzy bitopological spaces and other fields such as machine learning and network theory.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Fig 1.

Figure 1.

The relationship between pairwise fuzzy semi Baire spaces and pairwise fuzzy semi Volterra spaces.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 399-406https://doi.org/10.5391/IJFIS.2024.24.4.399

Fig 2.

Figure 2.

The relationship between pairwise fuzzy semi Baire, pairwise fuzzy semi hyperconnected, pairwise fuzzy semi P-spaces and pairwise fuzzy semi Volterra spaces.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 399-406https://doi.org/10.5391/IJFIS.2024.24.4.399

Fig 3.

Figure 3.

The relationship between pairwise fuzzy semi strongly irresolvable, pairwise fuzzy semi second category, pairwise fuzzy semi submaximal, pairwise fuzzy semi P-spaces and pairwise fuzzy semi weakly Volterra spaces.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 399-406https://doi.org/10.5391/IJFIS.2024.24.4.399

Fig 4.

Figure 4.

The relationship between pairwise fuzzy semi σ-second category, pairwise fuzzy semi almost P-spaces and pairwise fuzzy semi weakly Volterra spaces.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 399-406https://doi.org/10.5391/IJFIS.2024.24.4.399

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