Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 150 https://internationalpubls.com Characterization of Pythagorean Fuzzy Bi-Interior Ideal and Bi-Quasi- Ideal in πšͺ-Semirings T. Anitha1, Y. Lavanya2 1Department of Mathematics, Annamalai University, Annamalainagar, 608002. anitha81t@gmail.com 2Department of Mathematics, Annamalai University, Annamalainagar, 608002. lavanyaannamalaiuniversity@gmail.com Article History: Received: 19-09-2024 Revised: 24-11-2024 Accepted: 01-12-2024 Abstract: In this paper, we introduce the Pythagorean fuzzy bi-interior-ideals and Pythagorean fuzzy bi-quasi-ideals in Ξ“ - semiring. More over we prove the every Pythagorean fuzzy left and right ideals are Pythagorean fuzzy bi-interior -ideal in Ξ“ - semiring. Also we study the notion of Pythagorean fuzzy bi-quasi-ideal in Ξ“ - semiring and characterize Pythagorean fuzzy bi-quasi-ideal in Ξ“ - semiring. Keywords: Fuzzy set, Bi-ideal, Semiring 1. Introduction As a generalization of ring, the notion of a Ξ“ - ring was introduced by Nobusawa [21] in 1964 and Iseki [7, 8, 9] studied the ideal theory in semiring. In 1995, Murali Krishna Rao [23, 24] introduced the notion of a Ξ“ - semiring as a generalization of Ξ“-ring, ring, ternary semiring and semiring. Ahsan et.al [1] introduced the concept of fuzzy semirings. The concept of soft set was established by Molodtsov [20], which deals with parametrized values of the alternative. Maji et al. [17, 18, 19] investigated the soft set views on decision-making issues and defined some important concepts for soft set with their properties. Maji et al.[18] offered the notion of the fuzzy soft set by merging two existing notions fuzzy sets and soft sets. Peng et al. [22] protracted the idea of intuitionistic fuzzy soft set to Pythagorean fuzzy soft set by upgrading the conditions. The fuzzy set was studied by Zadeh’s[36] in his seminal paper. Pythagorean fuzzy sets [34][35] characterized by the condition that the sum of the squares of membership and non-membership degrees is less than or equal to one, have been extensively investigated. Numerous authors have explored the algebraic properties of Pythagorean fuzzy ideals. This paper is structured into three sections. The initial two sections provide an introduction and lay down the preliminary concepts. The third section deals with Pythagorean fuzzy bi-interior ideal and its properties in Ξ“ - semiring. Also characterize Pythagorean fuzzy soft bi-interior ideal. Fourth section deals with the Pythagorean fuzzy bi-quasi-ideal and prove some important properties. 2. Preliminaries This section deals with the basic definitions. A semiring is a set 𝑆 with two binary operations + and . on 𝑆 called addition and multiplications such that, (i) (𝑆, +) is a semigroup, (ii)(𝑆, . ) is a semigroup and (iii) π‘Ž(𝑏 + 𝑐) = π‘Žπ‘ + π‘Žπ‘ and (π‘Ž + 𝑏)𝑐 = π‘Žπ‘ + 𝑏𝑐 for all π‘Ž, 𝑏, 𝑐 ∈ 𝑆. mailto:anitha81t@gmail.com mailto:lavanyaannamalaiuniversity@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 151 https://internationalpubls.com A nonempty subset 𝐴 of a semiring 𝑆 is called a left (right) ideal of 𝑆 if 𝐴 is closed under addition and 𝑆𝐴 βŠ† 𝐴(𝐴𝑆 βŠ† 𝐴). 𝐴 is an ideal of 𝑆 if it is both a left and a right ideal of the semiring 𝑆. Definition 2.1 [3] If (𝑆, +) and (𝛀, +) be two commutative semigroups then S is called a 𝛀 - semiring if there exists a structure 𝑆 Γ— 𝛀 Γ— 𝑆 denoted by 𝛼𝛾𝛽 for all 𝛼, 𝛽 ∈ 𝑆 and 𝛾 ∈ 𝛀 satisfying the following properties, 1. 𝛼𝛾(𝛽 + 𝜈) = 𝛼𝛾𝛽 + π›Όπ›Ύπœˆ, 2. (𝛽 + 𝜈)𝛾𝛼 = 𝛽𝛾𝛼 + πœˆπ›Ύπ›Ό, 3. 𝛼(𝛾 + 𝛾1)𝜈 = π›Όπ›Ύπœˆ + 𝛼𝛾1𝜈, 4. 𝛼𝛾(𝛽𝛾1𝜈) = (𝛼𝛾𝛽)𝛾1𝜈 for all 𝛼, 𝛽, 𝜈 ∈ 𝑆 and 𝛾, 𝛾1 ∈ Ξ“. Definition 2.2 [3] Define addition in the following way 𝐴, 𝐡 ∈ 𝑆,𝛾 ∈ 𝛀 , let 𝐴𝛾𝐡 denote the ideal generated by {𝛼𝛾𝛽/𝛼, 𝛽 ∈ 𝑆}. Then 𝑆 is a 𝛀 - semiring. Definition 2.3 [3] A 𝛀 - semiring S is said to be commutative if 𝛼𝛾𝛽 = 𝛽𝛾𝛼, for all 𝛼, 𝛽 ∈ 𝑆 and 𝛾 ∈ 𝛀. Definition 2.4 [3] A 𝛀-semiring S is said to have a zero element if 0𝛽𝛼 = 0 = 𝛼𝛽0 and 𝛼 + 0 = 𝛼 = 0 + 𝛼, for all 𝛼 ∈ 𝑆 and 𝛾 ∈ 𝛀. Definition 2.5 [3] S is said to have a identity element if there exists 𝛾 ∈ 𝛀 such that 1𝛾𝛼 = 𝛼 = 𝛼𝛾1, for all 𝛼 ∈ 𝑆. Definition 2.6 [3] S is said to have a strong identity element if for all 𝛼 ∈ 𝑆, 1𝛾𝛼 = 𝛼 = 𝛼𝛾1 for all 𝛾 ∈ 𝛀. Definition 2.7 [3] A nonempty subset R of a 𝛀 - semiring S is said to be a sub𝛀 semiring of S if (𝑅, +) is a sub semigroup of (𝑆, +) and 𝛼𝛾𝛽 ∈ 𝑆 for all 𝛼, 𝛽 ∈ 𝑆 and 𝛾 ∈ 𝛀. Definition 2.8 [3] A nonempty subset R of a 𝛀 - semiring S is called an ideal if 𝛼, 𝛽 ∈ 𝑅 implies 𝛼 + 𝛽 ∈ 𝑅 and π‘Ž ∈ 𝑅, 𝛼 ∈ 𝑆 and 𝛾 ∈ 𝛀 implies π›Όπ›Ύπ‘Ž ∈ 𝑅 and π‘Žπ›Όπ›Ύ ∈ 𝑅. Definition 2.9 [11] Let π‘ˆ be the universe and 𝐸 be the set of parameters. Let 𝑃(π‘ˆ) denote the power set of π‘ˆ and 𝐴 βŠ‚ 𝐸. A pair (𝐹, 𝐴) is called a soft set over π‘ˆ, where 𝐹 is a mapping given by 𝐹: 𝐴 β†’ 𝑃(π‘ˆ). Definition 2.10 [23] A nonempty set 𝐴 of 𝑆 is called a 𝛀 - subsemiring of 𝑆 if (𝐴, +) is a subsemigroup of (𝐴, +) and 𝐴𝛀𝐴 βŠ† 𝐴. Definition 2.11 [30] A is called a quasi-ideal of 𝑆 if 𝐴 is a 𝛀 - subsemiring of 𝑆 and 𝐴𝛀𝑆 ∩ 𝑆𝛀𝐴 βŠ† 𝐴. Definition 2.12 [30] A is called a bi-ideal of 𝑆 if 𝐴 is a 𝛀 - subsemiring of 𝑆 and 𝐴𝛀𝑆𝛀𝐴 βŠ† 𝐴. Definition 2.13 [30] A is called an interior-ideal of 𝑆 if 𝐡 is a 𝛀 - subsemiring of 𝑆 and 𝑆𝛀𝐴𝛀𝑆 βŠ† 𝐴. Definition 2.14 [30] A is called a left(right) ideal of 𝑆 if 𝐴 is a 𝛀 - subsemiring of 𝑆 and 𝑆𝛀𝐴 βŠ† 𝐴(𝐴𝛀𝑆 βŠ† 𝐴). Definition 2.15 [30] A is called an ideal of 𝑆 if 𝐴 is a 𝛀 - subsemiring of 𝑆 and 𝑆𝛀𝐴 βŠ† 𝐴, 𝐴𝛀𝑆 βŠ† 𝐴. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 152 https://internationalpubls.com Definition 2.16 [30] A is called a left(right) bi-quasi-ideal of 𝑆 if 𝐴 is a 𝛀-subsemiring of 𝑆 and 𝐴𝛀𝑆 ∩ 𝑆𝛀𝑆𝛀𝐴 βŠ† 𝐴(𝐴𝛀𝑆 ∩ 𝑆𝛀𝐴𝛀𝑆 βŠ† 𝐴). Definition 2.17 [9] Let π‘ˆ be the initial universe. 𝐸 be the set of parameters and 𝐹𝑆(π‘ˆ) denote the fuzzy power set of π‘ˆ and 𝐴 βŠ‚ 𝐸. A pair (𝐹, 𝐴) is called a fuzzy soft set over π‘ˆ, where 𝐹 is a mapping given by 𝐹: 𝐴 β†’ 𝐹𝑆(π‘ˆ). A fuzzy soft set is a parameterized family of fuzzy subsets of π‘ˆ. Definition 2.18 [16] Let 𝑋 be a non empty set. A Pythagorean Fuzzy Set 𝔄 in 𝑋 is given by 𝔄 = {𝛼, 𝔄π‘₯(𝛼), 𝔄𝑦(𝛼)/𝛼 ∈ 𝑋} where 𝔄π‘₯: 𝑋 β†’ [0,1] and 𝔄𝑦: 𝑋 β†’ [0,1] represent the degree of membership and degree of non membership of 𝔄 respectively. Also, 𝔄π‘₯ and 𝔄𝑦 satisfies the condition (𝔄π‘₯)2 + (𝔄𝑦)2 ≀ 1 for all 𝛼 ∈ 𝑋. Definition 2.19 [13] Let π‘ˆ be the initial universe. 𝐸 be the set of parameters and 𝑃𝐹𝑆(π‘ˆ) denote the Pythagorean fuzzy power set of π‘ˆ and 𝐴 βŠ‚ 𝐸. A pair (𝐹, 𝐴) is called a Pythagorean fuzzy soft set over π‘ˆ, where 𝐹 is a mapping given by 𝐹: 𝐴 β†’ 𝑃𝐹𝑆(π‘ˆ). A Pythagorean fuzzy soft set is a parameterized family of fuzzy subsets of π‘ˆ. Definition 2.20 [13] Let (𝐹, 𝐴) and (𝐺, 𝐡) be two Pythagorean fuzzy soft sets over π‘ˆ. Then the union of (𝐹, 𝐴) is called a Pythagorean fuzzy soft subset of (𝐺, 𝐡) if 1. 𝐴 βŠ‚ 𝐡 2. 𝐹(𝛼) is a Pythagorean fuzzy subset of 𝐺(𝛼), for all 𝛼 ∈ 𝐴. Definition 2.21 [13] Let (𝐹, 𝐴) and (𝐺, 𝐡) be two Pythagorean fuzzy soft sets over π‘ˆ. (𝐹, 𝐴)𝐴𝑁𝐷(𝐺, 𝐡) denoted by (𝐹, 𝐴) ∧ (𝐺, 𝐡), is defined by (𝐹, 𝐴) ∧ (𝐺, 𝐡) = (𝐻, 𝐴 Γ— 𝐡), where 𝐻(𝛼, 𝛽) = 𝐹(𝛼) ∩ 𝐺(𝛽), for all (𝛼, 𝛽) ∈ 𝐴 Γ— 𝐡. Definition 2.22 [13] Let (𝐹, 𝐴) and (𝐺, 𝐡) be two Pythagorean fuzzy soft sets over π‘ˆ. (𝐹, 𝐴)𝑂𝑅(𝐺, 𝐡) denoted by (𝐹, 𝐴) ∨ (𝐺, 𝐡), is defined by (𝐹, 𝐴) ∨ (𝐺, 𝐡) = (𝐻, 𝐴 Γ— 𝐡), where 𝐻(𝛼, 𝛽) = 𝐹(𝛼) βˆͺ 𝐺(𝛽), for all (𝛼, 𝛽) ∈ 𝐴 Γ— 𝐡. Definition 2.23 [13] The intersection of two Pythagorean fuzzy soft sets (𝐹, 𝐴) and (𝐺, 𝐡) over a universe π‘ˆ is a Pythagorean fuzzy soft set denoted by (𝐻, 𝐢), where 𝐢 = 𝐴 ∩ 𝐡 and 𝐻(𝛼) = { 𝐹(𝛼) if𝛼 ∈ 𝐴 βˆ’ 𝐡 𝐺(𝛼) if𝛼 ∈ 𝐡 βˆ’ 𝐴 π‘šπ‘–π‘›{𝐹(𝛼), 𝐺(𝛼)} if𝛼 ∈ 𝐴 ∩ 𝐡 For all 𝛼 ∈ 𝐢. It is denoted by (𝐻, 𝐢) = (𝐹, 𝐴) ∩ (𝐹, 𝐡). Definition 2.24 [13] The union of two Pythagorean fuzzy soft sets (𝐹, 𝐴) and (𝐺, 𝐡) over a universe π‘ˆ is a Pythagorean fuzzy soft set denoted by (𝐻, 𝐢), where 𝐢 = 𝐴 βˆͺ 𝐡 and 𝐻(𝛼) = { 𝐹(𝛼) if𝛼 ∈ 𝐴 βˆ’ 𝐡 𝐺(𝛼) if𝛼 ∈ 𝐡 βˆ’ 𝐴 π‘šπ‘–π‘›{𝐹(𝛼), 𝐺(𝛼)} if𝛼 ∈ 𝐴 βˆͺ 𝐡 For all 𝛼 ∈ 𝐢. It is denoted by (𝐻, 𝐢) = (𝐹, 𝐴) βˆͺ (𝐹, 𝐡). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 153 https://internationalpubls.com Definition 2.25 [13] Let (𝐹, 𝐴) and (𝐺, 𝐡) be two Pythagorean fuzzy soft sets over π‘ˆ such that 𝐴 βˆͺ 𝐡 β‰  βˆ…. The bi-union of (𝐹, 𝐴) and (𝐺, 𝐡) is defined to be the Pythagorean fuzzy soft set (𝐻, 𝐢), where 𝐢 = 𝐴 βˆͺ 𝐡 and 𝐻(𝛼) = 𝐹(𝛼) βˆͺ 𝐺(𝛼) for all 𝛼 ∈ 𝐢. It is denoted by (𝐻, 𝐢) = (𝐹, 𝐴) βŠ” (𝐺, 𝐡). Definition 2.26 [13] Let (𝐹, 𝐴) and (𝐺, 𝐡) be two Pythagorean fuzzy soft sets over π‘ˆ such that 𝐴 ∩ 𝐡 β‰  βˆ…. The bi-union of (𝐹, 𝐴) and (𝐺, 𝐡) is defined to be the Pythagorean fuzzy soft set (𝐻, 𝐢), where 𝐢 = 𝐴 ∩ 𝐡 and 𝐻(𝛼) = 𝐹(𝛼) ∩ 𝐺(𝛼) for all 𝛼 ∈ 𝐢. It is denoted by (𝐻, 𝐢) = (𝐹, 𝐴) βŠ“ (𝐺, 𝐡). Definition 2.27 [13] Let (𝐹, 𝐴) and (𝐺, 𝐡) two Pythagorean fuzzy soft sets over a universe π‘ˆ. The product of (𝐹, 𝐴) and (𝐺, 𝐡) is defined to be the Pythagorean fuzzy soft set denoted by (𝐹 ∘ 𝐺, 𝐢), where 𝐢 = 𝐴 βˆͺ 𝐡 and 𝐴π‘₯(𝐹∘𝐺)(𝛼)(𝑖) = { 𝐴π‘₯(𝐹)(𝛼)(𝑖) if𝛼 ∈ 𝐴 βˆ’ 𝐡 𝐴π‘₯(𝐺)(𝛼)(𝑖) if𝛼 ∈ 𝐡 βˆ’ 𝐴 sup 𝑖=π‘Žπ‘ π‘šπ‘–π‘›{𝐴π‘₯(𝐹)(𝛼)(𝑖), 𝐴π‘₯(𝐺)(𝛼)(𝑖)} if𝛼 ∈ 𝐴 ∩ 𝐡 and 𝐴𝑦(𝐹∘𝐺)(𝛼)(𝑖) = { 𝐴𝑦(𝐹)(𝛼)(𝑖) if𝛼 ∈ 𝐴 βˆ’ 𝐡 𝐴𝑦(𝐺)(𝛼)(𝑖) if𝛼 ∈ 𝐡 βˆ’ 𝐴 inf 𝑖=π‘Žπ‘ π‘šπ‘Žπ‘₯{𝐴𝑦(𝐹)(𝛼)(𝑖), 𝐴𝑦(𝐺)(𝛼)(𝑖)} if𝛼 ∈ 𝐴 ∩ 𝐡 For all 𝛼 ∈ 𝐢 and 𝑖 ∈ π‘ˆ . It is denoted by (𝐹 ∘ 𝐺, 𝐢) = (𝐹, 𝐴) ∘ (𝐺, 𝐡). Definition 2.28 [28] A fuzzy subset A of S is called a fuzzy bi-interior-ideal if 𝑆𝐴𝑆 ∩ 𝐴𝑆𝐴 βŠ† 𝐴. 3. Pythagorean fuzzy Bi-interior-ideals in πšͺ-semiring This section deals with the Pythagorean fuzzy bi-interior-ideals in Ξ“-semiring 𝑆. Definition 3.1 A PFS 𝐴 = (π΄πœ‡, 𝐴𝜈) of 𝑆 is said to be a 𝑃𝐹𝐡𝐼𝐼 of 𝑆 if the following conditions are holds: 1. π΄πœ‡(π‘₯ + 𝑦) β‰₯ π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΄πœ‡(𝑦)} 𝐴𝜈(π‘₯ + 𝑦) ≀ π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐴𝜈(𝑦)} 2. πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡ πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘† ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 βŠ† 𝐴𝜈. Theorem 3.2 Every PF left ideal of 𝑆 is a PFBII of 𝑆. Proof. Let 𝐴 be a 𝑃𝐹 left ideal of 𝑆 and π‘₯ ∈ 𝑆, 𝛼, 𝛽 ∈ Ξ“. πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{1, π΄πœ‡(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(𝑏)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 154 https://internationalpubls.com β‰₯ sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘Žπ‘)} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯) π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑒), πœ’π‘† ∘ π΄πœ‡(𝑣𝛽𝑠)}} β‰₯ sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑒), π΄πœ‡(𝑣𝛽𝑠)}} = π΄πœ‡(π‘₯) Now πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = π‘šπ‘–π‘›{πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘†, π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡} β‰₯ π‘šπ‘–π‘›{πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘†, π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯) Hence πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡. Next we have to prove for non membership function πœ’π‘† ∘ 𝐴𝜈(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{0, 𝐴𝜈(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(𝑏)} ≀ inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘Žπ‘)} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘₯)} = 𝐴𝜈(π‘₯) 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯) = inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈(𝑒), πœ’π‘† ∘ 𝐴𝜈(𝑣𝛽𝑠)}} ≀ inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈(𝑒), 𝐴𝜈(𝑣𝛽𝑠)}} = 𝐴𝜈(π‘₯) Now πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘† ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯) = π‘šπ‘Žπ‘₯{πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘†, 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈} ≀ π‘šπ‘Žπ‘₯{πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘†, 𝐴𝜈(π‘₯)} = 𝐴𝜈(π‘₯) Hence πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘† ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 βŠ† 𝐴𝜈. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 155 https://internationalpubls.com Theorem 3.3 Every PF right ideal of S is a PFBII of S. Proof. Let A be a PF right ideal of 𝑆 and π‘₯ ∈ 𝑆,𝛼, 𝛽 ∈ Ξ“. π΄πœ‡(π‘₯) ∘ πœ’π‘† = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{π΄πœ‡(π‘Ž), πœ’π‘†(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{π΄πœ‡(π‘Ž),1}} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘Ž)} ≀ sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘Žπ‘)} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯) π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡ ∘ πœ’π‘†(𝑒𝛼𝑣), π΄πœ‡(𝑠)}} ≀ sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑒𝛼𝑣), π΄πœ‡(𝑠)}} = π΄πœ‡(π‘₯) Now πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = π‘šπ‘–π‘›{πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘†, π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡} β‰₯ π‘šπ‘–π‘›{πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘†, π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯) Hence πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡. Next we have to prove for non membership function 𝐴𝜈(π‘₯) ∘ πœ’π‘† = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{𝐴𝜈(π‘Ž), πœ’π‘†(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{𝐴𝜈(π‘Ž),0}} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘Ž)} ≀ inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘Žπ‘)} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘₯)} = 𝐴𝜈(π‘₯) 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯) = inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈 ∘ πœ’π‘†(𝑒𝛼𝑣), 𝐴𝜈(𝑠)}} ≀ inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈(𝑒𝛼𝑣), 𝐴𝜈(𝑠)}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 156 https://internationalpubls.com = 𝐴𝜈(π‘₯) Now πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘† ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯) = π‘šπ‘Žπ‘₯{πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘†, 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈} ≀ π‘šπ‘Žπ‘₯{πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘†, 𝐴𝜈(π‘₯)} = 𝐴𝜈(π‘₯) Hence πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘† ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 βŠ† 𝐴𝜈. Corollary 3.4 Every PFI is PFBII of S. Proof. By Theorem 3.2 and 3.4 proof is obvious. Theorem 3.5 Let 𝐡 be a nonempty subset of 𝑆. Then 𝐡 is a bi-interior-ideal of 𝑆 ⟺ πœ’π΅ is an PFBII of S. Proof. Assume that 𝐡 is a bi-interior-ideal of 𝑆. Then πœ’π΅ is an 𝑃𝐹 sub-Ξ“ semiring of 𝑆. By hypothesis we’ve 𝑆Γ𝐡Γ𝑆 ∩ 𝐡Γ𝑆Γ𝐡 βŠ† 𝐡. Then, πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ ∩ πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ = πœ’π‘†Ξ“π΅Ξ“π‘† ∩ πœ’π΅Ξ“π‘†Ξ“π΅ = πœ’π‘†Ξ“π΅Ξ“π‘†βˆ©π΅Ξ“π‘†Ξ“π΅ βŠ† πœ’π΅ Hence πœ’π΅ is a 𝑃𝐹𝐡𝐼𝐼 of 𝑆. Conversely, let us assume that πœ’π΅ is a 𝑃𝐹𝐡𝐼𝐼 of 𝑆. Then 𝐡 is a sub-Ξ“ semiring of 𝑆. We have πœ’π‘† ∘ πœ’π΅ ∘ πœ’π‘† ∩ πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ βŠ† πœ’π΅ πœ’π‘†Ξ“π΅Ξ“π‘† ∩ πœ’π΅Ξ“π‘†Ξ“π΅ βŠ† πœ’π΅ πœ’π‘†π΅π‘†βˆ©π΅π‘†π΅ βŠ† πœ’π΅ 𝑆𝐡𝑆 ∩ 𝐡𝑆𝐡 βŠ† 𝐡 Hence 𝐡 is a bi-interior-ideal of 𝑆. Theorem 3.6 Let B be a nonempty subset of 𝑆. Then 𝐡 is a PFBII of 𝑆 ⟺ the nonempty level subset of 𝐡 is a bi-interior-ideal of 𝑆 for every 𝑑 ∈ [0,1] . Proof. Assume that 𝐡 is a 𝑃𝐹𝐡𝐼𝐼 of 𝑆. π΅πœ‡π‘‘ β‰  πœ™,𝑑 ∈ [0,1] and π‘Ž, 𝑏 ∈ π΅πœ‡π‘‘ Then, π΅πœ‡π‘‘ (π‘Ž) β‰₯ 𝑑, π΅πœ‡π‘‘ (𝑏) β‰₯ 𝑑 π΅πœ‡π‘‘ (π‘Ž + 𝑏) β‰₯ π‘šπ‘–π‘›{π΅πœ‡π‘‘ (π‘Ž), π΅πœ‡π‘‘ (𝑏)} β‰₯ 𝑑 π‘Ž + 𝑏 ∈ π΅πœ‡π‘‘ . Let π‘₯ ∈ π‘†Ξ“π΅πœ‡π‘‘ Γ𝑆 ∩ π΅πœ‡π‘‘ Ξ“π‘†Ξ“π΅πœ‡π‘‘ . Then π‘₯ = π‘π›Όπ‘Žπ›½π‘’ = 𝑐𝛾𝑑𝛿𝑒, 𝑏, 𝑒, 𝑑 ∈ 𝑆 and π‘Ž, 𝑐, 𝑒 ∈ π΅πœ‡π‘‘ , 𝛼, 𝛽, 𝛾, 𝛿 ∈ Ξ“ πœ’π‘† ∘ π΅πœ‡π‘‘ ∘ πœ’π‘† β‰₯ 𝑑 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 157 https://internationalpubls.com and π΅πœ‡π‘‘ ∘ πœ’π‘† ∘ π΅πœ‡π‘‘ β‰₯ 𝑑. π΅πœ‡π‘‘ (π‘₯) β‰₯ 𝑑 Hence π‘₯ ∈ π΅πœ‡π‘‘ Conversely, suppose that π΅πœ‡π‘‘ is a bi-interior-ideal of 𝑆 for all 𝑑 ∈ [0,1] Let π‘Ž, 𝑏 ∈ 𝑆, 𝛼 ∈ Ξ“, π΅πœ‡π‘‘ (π‘Ž) = 𝑑1, π΅πœ‡π‘‘ (𝑏) = 𝑑2 and 𝑑1 β‰₯ 𝑑2. Then π‘Ž, 𝑏 ∈ π΅πœ‡π‘‘ Then, π‘Ž + 𝑏 ∈ π΅πœ‡π‘‘ . Therefore π΅πœ‡π‘‘ =β‰₯ 𝑑2 = π‘šπ‘–π‘›{π΅πœ‡π‘‘ (π‘Ž), π΅πœ‡π‘‘ (𝑏)}. Hence we have, π‘†Ξ“π΅πœ‡π‘‘ Γ𝑆 ∩ π΅πœ‡π‘‘ Ξ“π‘†Ξ“π΅πœ‡π‘‘ βŠ† π΅πœ‡π‘‘ . Similarly we prove the non membership function. Theorem 3.7 If 𝐴 and B are PFBII of 𝑆 then 𝐴 ∩ 𝐡 is PFBII of 𝑆. Proof. Let 𝐴 and 𝐡 are 𝑃𝐹𝐡𝐼𝐼 of 𝑆 and π‘₯, 𝑦 ∈ 𝑆 and 𝛼, 𝛽 ∈ Ξ“. (π΄πœ‡ ∩ π΅πœ‡)(π‘₯ + 𝑦) = π‘šπ‘–π‘›{π΄πœ‡(π‘₯ + 𝑦), π΅πœ‡(π‘₯ + 𝑦)} β‰₯ π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΄πœ‡(𝑦)}, π‘šπ‘–π‘›{π΅πœ‡(π‘₯), π΅πœ‡(𝑦)}} = π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΅πœ‡(π‘₯)}, π‘šπ‘–π‘›{π΄πœ‡(𝑦), π΅πœ‡(𝑦)}} = π‘šπ‘–π‘›{(π΄πœ‡ ∩ π΅πœ‡)(π‘₯), (π΄πœ‡ ∩ π΅πœ‡)(𝑦)} πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡)(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡ ∩ π΅πœ‡(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘› {πœ’π‘†(π‘Ž), π‘šπ‘–π‘›{π΄πœ‡(𝑏), π΅πœ‡(𝑏)}}} = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘› {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}, π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΅πœ‡(𝑏)}}} = π‘šπ‘–π‘› { sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}} , sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΅πœ‡(𝑏)}}} = π‘šπ‘–π‘›{πœ’π‘† ∘ π΄πœ‡(π‘₯), πœ’π‘† ∘ π΅πœ‡(π‘₯)} = (πœ’π‘† ∘ π΄πœ‡) ∩ (πœ’π‘† ∘ π΅πœ‡)(π‘₯) (π΄πœ‡ ∩ π΅πœ‡) ∘ πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡)(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘›{π΄πœ‡ ∩ π΅πœ‡(π‘Ž), πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡(𝑏𝛽𝑐)}} = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘›{(π΄πœ‡ ∩ π΅πœ‡)(π‘Ž), πœ’π‘† ∘ π΄πœ‡ ∩ πœ’π‘† ∘ π΅πœ‡(𝑏𝛽𝑐)}} = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘Ž), π΅πœ‡(π‘Ž)}, π‘šπ‘–π‘›{(πœ’π‘† ∘ π΄πœ‡)(𝑏𝛽𝑐), (πœ’π‘† ∘ π΅πœ‡)(𝑏𝛽𝑐)}}} = π‘šπ‘–π‘›{π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯), π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡(π‘₯)} Therefore (π΄πœ‡ ∩ π΅πœ‡) ∘ πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡) = π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 158 https://internationalpubls.com Similarly we prove πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† = π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡ Hence πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ = (πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘†) ∩ (π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡) ∩ (πœ’π‘† ∘ π΅πœ‡ ∘ πœ’π‘†) ∩ (π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡) βŠ‡ π΄πœ‡ ∩ π΅πœ‡ Similarly we can prove for non membership (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯ + 𝑦) = π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯ + 𝑦), 𝐡𝜈(π‘₯ + 𝑦)} ≀ π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐴𝜈(𝑦)}, π‘šπ‘Žπ‘₯{𝐡𝜈(π‘₯), 𝐡𝜈(𝑦)}} = π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐡𝜈(π‘₯)}, π‘šπ‘Žπ‘₯{𝐴𝜈(𝑦), 𝐡𝜈(𝑦)}} = π‘šπ‘Žπ‘₯{(𝐴𝜈 ∩ 𝐡𝜈)(π‘₯), (𝐴𝜈 ∩ 𝐡𝜈)(𝑦)} πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈 ∩ π΅πœ‡(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯ {πœ’π‘†(π‘Ž), π‘šπ‘Žπ‘₯{𝐴𝜈(𝑏), π΅πœ‡(𝑏)}}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}, π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐡𝜈(𝑏)}}} = π‘šπ‘Žπ‘₯ { inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}}, inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐡𝜈(𝑏)}}} = π‘šπ‘Žπ‘₯{πœ’π‘† ∘ 𝐴𝜈(π‘₯), πœ’π‘† ∘ 𝐡𝜈(π‘₯)} = (πœ’π‘† ∘ 𝐴𝜈) ∩ (πœ’π‘† ∘ 𝐡𝜈)(π‘₯) (𝐴𝜈 ∩ 𝐡𝜈) ∘ πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{𝐴𝜈 ∩ 𝐡𝜈(π‘Ž), πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈(𝑏𝛽𝑐)}} = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{(𝐴𝜈 ∩ 𝐡𝜈)(π‘Ž), πœ’π‘† ∘ 𝐴𝜈 ∩ πœ’π‘† ∘ 𝐡𝜈(𝑏𝛽𝑐)}} = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘Ž), 𝐡𝜈(π‘Ž)}, π‘šπ‘–π‘›{(πœ’π‘† ∘ 𝐴𝜈)(𝑏𝛽𝑐), (πœ’π‘† ∘ 𝐡𝜈)(𝑏𝛽𝑐)}}} = π‘šπ‘Žπ‘₯{𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯), 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈(π‘₯)} Therefore (𝐴𝜈 ∩ 𝐡𝜈) ∘ πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈) = 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈 Similarly we prove πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† = 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈 Hence πœ’π‘† ∘ 𝐴𝜈 ∩ π΅πœ‡ ∘ πœ’π‘† ∩ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 ∩ π΅πœ‡ = (πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘†) ∩ (𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈) ∩ (πœ’π‘† ∘ 𝐡𝜈 ∘ πœ’π‘†) ∩ (𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈) βŠ† 𝐴𝜈 ∩ 𝐡𝜈 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 159 https://internationalpubls.com 4 Pythagorean fuzzy Bi-quasi-ideals in πšͺ-semiring This section deals with the Pythagorean fuzzy bi-interior-ideals in Ξ“-semiring 𝑆. Definition 4.1 A PFS 𝐴 = (π΄πœ‡, 𝐴𝜈) of 𝑆 is said to be a PFLBQI of 𝑆 if the following conditions are holds: 1. π΄πœ‡(π‘₯ + 𝑦) β‰₯ π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΄πœ‡(𝑦)} 𝐴𝜈(π‘₯ + 𝑦) ≀ π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐴𝜈(𝑦)} 2. πœ’π‘† ∘ π΄πœ‡ ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡ πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 βŠ† 𝐴𝜈 Definition 4.2 A PFS 𝐴 = (π΄πœ‡, 𝐴𝜈) of 𝑆 is said to be a PFRBQI of 𝑆 if the following conditions are holds: 1. π΄πœ‡(π‘₯ + 𝑦) β‰₯ π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΄πœ‡(𝑦)} 𝐴𝜈(π‘₯ + 𝑦) ≀ π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐴𝜈(𝑦)} 2. π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡ 𝐴𝜈 ∘ πœ’π‘† ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 βŠ† 𝐴𝜈. Definition 4.3 A PFS A = (π΄πœ‡, 𝐴𝜈) of 𝑆 is said to be a PFBQI of 𝑆 if it is both Pythagorean fuzzy left bi-quasi-ideal and right bi-quasi-ideal of 𝑆. Theorem 4.4 Every PF left ideal of 𝑆 is a PFLBQI of 𝑆. Proof. Let 𝐴 be a 𝑃𝐹 left ideal of 𝑆 and π‘₯ ∈ 𝑆,𝛼, 𝛽 ∈ Ξ“. πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(𝑏)} β‰₯ sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘Žπ›Όπ‘)} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯) π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑒𝛼𝑣) ∘ πœ’π‘†, π΄πœ‡(𝑠)}} β‰₯ sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑒𝛼𝑣), π΄πœ‡(𝑠)}} = π΄πœ‡(π‘₯) Hence πœ’π‘† ∘ π΄πœ‡ ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡. Next we have to prove for non membership function Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 160 https://internationalpubls.com πœ’π‘† ∘ 𝐴𝜈(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(𝑏)} ≀ inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘Žπ›Όπ‘)} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘₯)} = 𝐴𝜈(π‘₯) 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯) = inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈(𝑒), πœ’π‘† ∘ 𝐴𝜈(𝑣𝛽𝑠)}} ≀ inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈(𝑒), 𝐴𝜈(𝑣𝛽𝑠)}} = 𝐴𝜈(π‘₯) Hence πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 βŠ† 𝐴𝜈. Theorem 4.5 Every PF right ideal of 𝑆 is a PFRBQI of 𝑆. Proof. Let 𝐴 be a 𝑃𝐹 right ideal of 𝑆 and π‘₯ ∈ 𝑆,𝛼, 𝛽 ∈ Ξ“. π΄πœ‡ ∘ πœ’π‘†(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{π΄πœ‡(π‘Ž), πœ’π‘†(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘Ž)} β‰₯ sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘Žπ›Όπ‘)} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯) π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡ ∘ πœ’π‘†(𝑒𝛼𝑣), π΄πœ‡(𝑠)}} β‰₯ sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑒𝛼𝑣), π΄πœ‡(𝑠)}} = π΄πœ‡(π‘₯) Hence π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ βŠ‡ π΄πœ‡. Next we have to prove for non membership function 𝐴𝜈 ∘ πœ’π‘†(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{π΄πœ‡(π‘Ž), πœ’π‘†(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘Ž)} ≀ inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘Žπ›Όπ‘)} = inf π‘₯=π‘Žπ›Όπ‘ {𝐴𝜈(π‘₯)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 161 https://internationalpubls.com = 𝐴𝜈(π‘₯) 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯) = inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈 ∘ πœ’π‘†(𝑒𝛼𝑣), 𝐴𝜈(𝑠)}} ≀ inf π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘Žπ‘₯{𝐴𝜈(𝑒𝛼𝑣), 𝐴𝜈(𝑠)}} = 𝐴𝜈(π‘₯). Hence 𝐴𝜈 ∘ πœ’π‘† ∩ 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 βŠ† 𝐴𝜈. Theorem 4.6 Every PF left ideal of 𝑆 is a PFRBQI of 𝑆. Proof. Let 𝐴 be a 𝑃𝐹 left ideal of 𝑆 and π‘₯ ∈ 𝑆,𝛼, 𝛽 ∈ Ξ“. πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(𝑏)} β‰₯ sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘Žπ›Όπ‘)} = sup π‘₯=π‘Žπ›Όπ‘ {π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯) π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑠), πœ’π‘† ∘ π΄πœ‡(𝑣𝛽𝑠)}} β‰₯ sup π‘₯=𝑒𝛼𝑣𝛽𝑠 {π‘šπ‘–π‘›{π΄πœ‡(𝑠), π΄πœ‡(𝑣𝛽𝑠)}} = π΄πœ‡(π‘₯) Now π΄πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯) = π‘šπ‘–π‘›{π΄πœ‡ ∘ πœ’π‘†, π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯)} β‰₯ π‘šπ‘–π‘›{π΄πœ‡ ∘ πœ’π‘†(π‘₯), π΄πœ‡(π‘₯)} = π΄πœ‡(π‘₯). Hence π΄πœ‡ is a 𝑃𝐹𝑅𝐡𝑄𝐼 of 𝑆. Similarly we can prove 𝐴𝜈 is a 𝑃𝐹𝑅𝐡𝑄𝐼 of 𝑆. Theorem 4.7 Every PF right ideal of 𝑆 is a PFLBQI of 𝑆. Proof. Proof is straight forward. Corollary 4.8 Every PF left(right) ideal of 𝑆 is a PFRBQI of 𝑆. Theorem 4.9 Let 𝐡 be a nonempty subset of 𝑆. Then 𝐡 is a right bi-quasi-ideal of 𝑆 ⟺ πœ’π΅ is an PFRBQI of 𝑆. Proof. Assume that 𝐡 is a right bi-quasi-ideal of 𝑆. Then πœ’π΅ is an 𝑃𝐹 subsemiring of 𝑆. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 162 https://internationalpubls.com By hypothesis we’ve 𝑆Γ𝐡 ∩ 𝐡Γ𝑆Γ𝐡 βŠ† 𝐡. Then, πœ’π‘† ∘ πœ’π΅ ∩ πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ = πœ’π‘†Ξ“π΅ ∩ πœ’π΅Ξ“π‘†Ξ“π΅ = πœ’π‘†Ξ“π΅βˆ©π΅Ξ“π‘†Ξ“π΅ βŠ† πœ’π΅ Hence πœ’π΅ is an 𝑃𝐹𝑅𝐡𝑄𝐼 of 𝑆. Conversely, let us assume that πœ’π΅ is a 𝑃𝐹𝑅𝐡𝑄𝐼 of 𝑆. Then 𝐡 is a subsemiring of 𝑆. We have πœ’π‘† ∘ πœ’π΅ ∩ πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ βŠ† πœ’π΅ πœ’π‘†Ξ“π΅ ∩ πœ’π΅Ξ“π‘†Ξ“π΅ βŠ† πœ’π΅ Hence 𝐡 is a bi-quasi-ideal of 𝑆. Theorem 4.10 Let 𝐡 be a nonempty subset of 𝑆. Then 𝐡 is a left bi-quasi-ideal of 𝑆 ⟺ πœ’π΅ is an PFLBQI of 𝑆. Proof. Assume that 𝐡 is a left bi-quasi-ideal of 𝑆. Then πœ’π΅ is an 𝑃𝐹 sub Ξ“ semi ring of 𝑆. By hypothesis we’ve 𝐡Γ𝑆 ∩ 𝐡Γ𝑆Γ𝐡 βŠ† 𝐡. Then, πœ’π΅ ∘ πœ’π‘† ∩ πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ = πœ’π΅Ξ“π‘† ∩ πœ’π΅Ξ“π‘†Ξ“π΅ = πœ’π΅Ξ“π‘†βˆ©π΅Ξ“π‘†Ξ“π΅ βŠ† πœ’π΅ Hence πœ’π΅ is an 𝑃𝐹𝑅𝐡𝑄𝐼 of 𝑆. Conversely, let us assume that πœ’π΅ is a 𝑃𝐹𝑅𝐡𝑄𝐼 of 𝑆. Then 𝐡 is a sub Ξ“ semi ring of 𝑆. We have πœ’π΅ ∘ πœ’π‘† ∩ πœ’π΅ ∘ πœ’π‘† ∘ πœ’π΅ βŠ† πœ’π΅ πœ’π΅Ξ“π‘† ∩ πœ’π΅Ξ“π‘†Ξ“π΅ βŠ† πœ’π΅ Hence 𝐡 is a bi-quasi-ideal of 𝑆. Theorem 4.11 If 𝐴 and 𝐡 are PFLBQI of 𝑆 then 𝐴 ∩ 𝐡 is PFLBQI of 𝑆. Proof. Let 𝐴 and 𝐡 are 𝑃𝐹𝐿𝐡𝑄𝐼 of 𝑆 and π‘₯, 𝑦 ∈ 𝑆 and 𝛼, 𝛽 ∈ Ξ“. (π΄πœ‡ ∩ π΅πœ‡)(π‘₯ + 𝑦) = π‘šπ‘–π‘›{π΄πœ‡(π‘₯ + 𝑦), π΅πœ‡(π‘₯ + 𝑦)} β‰₯ π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΄πœ‡(𝑦)}, π‘šπ‘–π‘›{π΅πœ‡(π‘₯), π΅πœ‡(𝑦)}} = π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΅πœ‡(π‘₯)}, π‘šπ‘–π‘›{π΄πœ‡(𝑦), π΅πœ‡(𝑦)}} = π‘šπ‘–π‘›{(π΄πœ‡ ∩ π΅πœ‡)(π‘₯), (π΄πœ‡ ∩ π΅πœ‡)(𝑦)} πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡)(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡ ∩ π΅πœ‡(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘› {πœ’π‘†(π‘Ž), π‘šπ‘–π‘›{π΄πœ‡(𝑏), π΅πœ‡(𝑏)}}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 163 https://internationalpubls.com = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘› {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}, π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΅πœ‡(𝑏)}}} = π‘šπ‘–π‘› { sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}} , sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΅πœ‡(𝑏)}}} = π‘šπ‘–π‘›{πœ’π‘† ∘ π΄πœ‡(π‘₯), πœ’π‘† ∘ π΅πœ‡(π‘₯)} = (πœ’π‘† ∘ π΄πœ‡) ∩ (πœ’π‘† ∘ π΅πœ‡)(π‘₯) (π΄πœ‡ ∩ π΅πœ‡) ∘ πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡)(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘›{π΄πœ‡ ∩ π΅πœ‡(π‘Ž), πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡(𝑏𝛽𝑐)}} = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘›{(π΄πœ‡ ∩ π΅πœ‡)(π‘Ž), πœ’π‘† ∘ π΄πœ‡ ∩ πœ’π‘† ∘ π΅πœ‡(𝑏𝛽𝑐)}} = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘Ž), π΅πœ‡(π‘Ž)}, π‘šπ‘–π‘›{(πœ’π‘† ∘ π΄πœ‡)(𝑏𝛽𝑐), (πœ’π‘† ∘ π΅πœ‡)(𝑏𝛽𝑐)}}} = π‘šπ‘–π‘›{π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯), π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡(π‘₯)} Therefore (π΄πœ‡ ∩ π΅πœ‡) ∘ πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡) = π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡ Hence πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ = (πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘†) ∩ (π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡) ∩ (πœ’π‘† ∘ π΅πœ‡ ∘ πœ’π‘†) ∩ (π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡) βŠ‡ π΄πœ‡ ∩ π΅πœ‡ Similarly we can prove for non membership (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯ + 𝑦) = π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯ + 𝑦), 𝐡𝜈(π‘₯ + 𝑦)} ≀ π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐴𝜈(𝑦)}, π‘šπ‘Žπ‘₯{𝐡𝜈(π‘₯), 𝐡𝜈(𝑦)}} = π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐡𝜈(π‘₯)}, π‘šπ‘Žπ‘₯{𝐴𝜈(𝑦), 𝐡𝜈(𝑦)}} = π‘šπ‘Žπ‘₯{(𝐴𝜈 ∩ 𝐡𝜈)(π‘₯), (𝐴𝜈 ∩ 𝐡𝜈)(𝑦)} πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈 ∩ π΅πœ‡(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯ {πœ’π‘†(π‘Ž), π‘šπ‘Žπ‘₯{𝐴𝜈(𝑏), π΅πœ‡(𝑏)}}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}, π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐡𝜈(𝑏)}}} = π‘šπ‘Žπ‘₯ { inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}}, inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐡𝜈(𝑏)}}} = π‘šπ‘Žπ‘₯{πœ’π‘† ∘ 𝐴𝜈(π‘₯), πœ’π‘† ∘ 𝐡𝜈(π‘₯)} = (πœ’π‘† ∘ 𝐴𝜈) ∩ (πœ’π‘† ∘ 𝐡𝜈)(π‘₯) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 164 https://internationalpubls.com (𝐴𝜈 ∩ 𝐡𝜈) ∘ πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{𝐴𝜈 ∩ 𝐡𝜈(π‘Ž), πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈(𝑏𝛽𝑐)}} = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{(𝐴𝜈 ∩ 𝐡𝜈)(π‘Ž), πœ’π‘† ∘ 𝐴𝜈 ∩ πœ’π‘† ∘ 𝐡𝜈(𝑏𝛽𝑐)}} = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘Ž), 𝐡𝜈(π‘Ž)}, π‘šπ‘Žπ‘₯{(πœ’π‘† ∘ 𝐴𝜈)(𝑏𝛽𝑐), (πœ’π‘† ∘ 𝐡𝜈)(𝑏𝛽𝑐)}}} = π‘šπ‘Žπ‘₯{𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯), 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈(π‘₯)} Therefore (𝐴𝜈 ∩ 𝐡𝜈) ∘ πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈) = 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈 Hence πœ’π‘† ∘ 𝐴𝜈 ∩ π΅πœ‡ ∘ πœ’π‘† ∩ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 ∩ π΅πœ‡ = (πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘†) ∩ (𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈) ∩ (πœ’π‘† ∘ 𝐡𝜈 ∘ πœ’π‘†) ∩ (𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈) βŠ† 𝐴𝜈 ∩ 𝐡𝜈. Theorem 4.12 If 𝐴 and 𝐡 are PFRBQI of 𝑆 then 𝐴 ∩ 𝐡 is PFRBQI of 𝑆. Proof. Let 𝐴 and 𝐡 are 𝑃𝐹𝑅𝐡𝑄𝐼 of 𝑆 and π‘₯, 𝑦 ∈ 𝑆 and 𝛼, 𝛽 ∈ Ξ“. (π΄πœ‡ ∩ π΅πœ‡)(π‘₯ + 𝑦) = π‘šπ‘–π‘›{π΄πœ‡(π‘₯ + 𝑦), π΅πœ‡(π‘₯ + 𝑦)} β‰₯ π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΄πœ‡(𝑦)}, π‘šπ‘–π‘›{π΅πœ‡(π‘₯), π΅πœ‡(𝑦)}} = π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘₯), π΅πœ‡(π‘₯)}, π‘šπ‘–π‘›{π΄πœ‡(𝑦), π΅πœ‡(𝑦)}} = π‘šπ‘–π‘›{(π΄πœ‡ ∩ π΅πœ‡)(π‘₯), (π΄πœ‡ ∩ π΅πœ‡)(𝑦)} πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡)(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡ ∩ π΅πœ‡(𝑏)}} = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘› {πœ’π‘†(π‘Ž), π‘šπ‘–π‘›{π΄πœ‡(𝑏), π΅πœ‡(𝑏)}}} = sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘› {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}, π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΅πœ‡(𝑏)}}} = π‘šπ‘–π‘› { sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΄πœ‡(𝑏)}} , sup π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘–π‘›{πœ’π‘†(π‘Ž), π΅πœ‡(𝑏)}}} = π‘šπ‘–π‘›{πœ’π‘† ∘ π΄πœ‡(π‘₯), πœ’π‘† ∘ π΅πœ‡(π‘₯)} = (πœ’π‘† ∘ π΄πœ‡) ∩ (πœ’π‘† ∘ π΅πœ‡)(π‘₯). (π΄πœ‡ ∩ π΅πœ‡) ∘ πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡)(π‘₯) = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘›{π΄πœ‡ ∩ π΅πœ‡(π‘Ž), πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡(𝑏𝛽𝑐)}} = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘›{(π΄πœ‡ ∩ π΅πœ‡)(π‘Ž), πœ’π‘† ∘ π΄πœ‡ ∩ πœ’π‘† ∘ π΅πœ‡(𝑏𝛽𝑐)}} = sup π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘–π‘› {π‘šπ‘–π‘›{π΄πœ‡(π‘Ž), π΅πœ‡(π‘Ž)}, π‘šπ‘–π‘›{(πœ’π‘† ∘ π΄πœ‡)(𝑏𝛽𝑐), (πœ’π‘† ∘ π΅πœ‡)(𝑏𝛽𝑐)}}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 165 https://internationalpubls.com = π‘šπ‘–π‘›{π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡(π‘₯), π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡(π‘₯)} Therefore (π΄πœ‡ ∩ π΅πœ‡) ∘ πœ’π‘† ∘ (π΄πœ‡ ∩ π΅πœ‡) = π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡ Hence πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∩ π΄πœ‡ ∩ π΅πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡ ∩ π΅πœ‡ = (πœ’π‘† ∘ π΄πœ‡ ∘ πœ’π‘†) ∩ (π΄πœ‡ ∘ πœ’π‘† ∘ π΄πœ‡) ∩ (πœ’π‘† ∘ π΅πœ‡ ∘ πœ’π‘†) ∩ (π΅πœ‡ ∘ πœ’π‘† ∘ π΅πœ‡) βŠ‡ π΄πœ‡ ∩ π΅πœ‡ Similarly we can prove for non membership (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯ + 𝑦) = π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯ + 𝑦), 𝐡𝜈(π‘₯ + 𝑦)} ≀ π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐴𝜈(𝑦)}, π‘šπ‘Žπ‘₯{𝐡𝜈(π‘₯), 𝐡𝜈(𝑦)}} = π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘₯), 𝐡𝜈(π‘₯)}, π‘šπ‘Žπ‘₯{𝐴𝜈(𝑦), 𝐡𝜈(𝑦)}} = π‘šπ‘Žπ‘₯{(𝐴𝜈 ∩ 𝐡𝜈)(π‘₯), (𝐴𝜈 ∩ 𝐡𝜈)(𝑦)} πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈 ∩ π΅πœ‡(𝑏)}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯ {πœ’π‘†(π‘Ž), π‘šπ‘Žπ‘₯{𝐴𝜈(𝑏), π΅πœ‡(𝑏)}}} = inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}, π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐡𝜈(𝑏)}}} = π‘šπ‘Žπ‘₯ { inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐴𝜈(𝑏)}}, inf π‘₯=π‘Žπ›Όπ‘ {π‘šπ‘Žπ‘₯{πœ’π‘†(π‘Ž), 𝐡𝜈(𝑏)}}} = π‘šπ‘Žπ‘₯{πœ’π‘† ∘ 𝐴𝜈(π‘₯), πœ’π‘† ∘ 𝐡𝜈(π‘₯)} = (πœ’π‘† ∘ 𝐴𝜈) ∩ (πœ’π‘† ∘ 𝐡𝜈)(π‘₯). (𝐴𝜈 ∩ 𝐡𝜈) ∘ πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈)(π‘₯) = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{𝐴𝜈 ∩ 𝐡𝜈(π‘Ž), πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈(𝑏𝛽𝑐)}} = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{(𝐴𝜈 ∩ 𝐡𝜈)(π‘Ž), πœ’π‘† ∘ 𝐴𝜈 ∩ πœ’π‘† ∘ 𝐡𝜈(𝑏𝛽𝑐)}} = inf π‘₯=π‘Žπ›Όπ‘π›½π‘ {π‘šπ‘Žπ‘₯{π‘šπ‘Žπ‘₯{𝐴𝜈(π‘Ž), 𝐡𝜈(π‘Ž)}, π‘šπ‘Žπ‘₯{(πœ’π‘† ∘ 𝐴𝜈)(𝑏𝛽𝑐), (πœ’π‘† ∘ 𝐡𝜈)(𝑏𝛽𝑐)}}} = π‘šπ‘Žπ‘₯{𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈(π‘₯), 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈(π‘₯)} Therefore (𝐴𝜈 ∩ 𝐡𝜈) ∘ πœ’π‘† ∘ (𝐴𝜈 ∩ 𝐡𝜈) = 𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈 Hence πœ’π‘† ∘ 𝐴𝜈 ∩ π΅πœ‡ ∘ πœ’π‘† ∩ 𝐴𝜈 ∩ 𝐡𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈 ∩ π΅πœ‡ = (πœ’π‘† ∘ 𝐴𝜈 ∘ πœ’π‘†) ∩ (𝐴𝜈 ∘ πœ’π‘† ∘ 𝐴𝜈) ∩ (πœ’π‘† ∘ 𝐡𝜈 ∘ πœ’π‘†) ∩ (𝐡𝜈 ∘ πœ’π‘† ∘ 𝐡𝜈) βŠ† 𝐴𝜈 ∩ 𝐡𝜈 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 166 https://internationalpubls.com 5 Conclusion This paper deals with the concept of Pythagorean fuzzy bi-interior-ideal, Pythagorean fuzzy soft bi- interior-ideal and Pythagorean fuzzy bi-quasi-ideals in Ξ“-semirings. 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