Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 312 https://internationalpubls.com Pythagorean Fuzzy Ideals in Semiring 1 J.Jayaraj, 2 R.Raghu, 3R.Ezhilarasi 1 Assistant Professor, Department of Mathematics, Dr. Puratchithalaivar M.G.R Government Arts and Science College, Kudavasal- 612601, India. E.mail: joe.jayaraj@gmail.com 2 Department of Mathematics, Thiru. A. Govindasamy Government Arts College, Tindivanam-604307, India. E.mail: c.r.raghu89@gmail.com 1,2Department of Mathematics, Annamalai University, Annamalainagar, 608002, 3Asscociate Professor, Department of Mathematics, Arignar Anna Government Arts College,Villupuram-605 602, India. E.Mail: rearasi@gmail.com Article History: Received: 15-09-2024 Revised: 23-10-2024 Accepted: 03-11-2024 Abstract: In this paper, we introduce the notion of Pythagorean fuzzy ideals in semiring some interesting properties, results are discussed in this paper. Keywords: Pythagorean, Fuzzy set, Semiring 1 INTRODUCTION Nobusawa[5] studied the concept of gamma semiring as a generalization of ring after that Sen introduced the gamma semigroups as a generalization of gamma groups. Murali Krishna Rao[6] in 1995 introduced the notion of gamma semiring as a generalization of gamma ring, ring, ternary semiring and semiring.The important reason for development of gamma semiring is a generalization of results of rings, gamma rings, semirings, semigroup and ternary semirings. Zadeh[12] studied the notion of fuzzy set theory. Atanassov [2] introduced intuitionistic fuzzy sets as a generalization of fuzzy sets. In intuitionistic, the sum of membership degree and non-membership degree should not exceed one. Yager [10] initially introduced the concept of Pythagorean fuzzy sets. In a Pythagorean fuzzy sets, the sum of the squared membership and non-membership degrees satisfies the condition. More recently, Yager [10, 11] proposed Pythagorean fuzzy sets as a powerful tool for effectively managing uncertainty or imprecise information in real-world scenarios. These sets enforce a constraint where the sum of squares of membership and non-membership degrees is less than or equal to 1. Pythagorean fuzzy sets have showcased remarkable efficacy in navigating uncertainties, prompting a surge of scholarly exploration across diverse research avenues, resulting in significant progress. The conceptualization of Pythagorean fuzzy sets facilitates a more comprehensive and accurate portrayal of uncertain information when juxtaposed with intuitionistic fuzzy sets. Across various disciplines, academics have meticulously examined the algebraic attributes of Pythagorean fuzzy sets, shedding light on their practical applications and foundational theoretical constructs. Many authors studied the algebraic structures of Pythagorean fuzzy sets This paper is structured into three sections. The first and second sections serve as the introduction and cover basic results pertinent to the paper’s topic. In the third section, we introduce Pythagorean fuzzy Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 313 https://internationalpubls.com ideals in semirings and some interesting properties this ideals are discussed. 2 Preliminaries In this section we present the basic concepts related to this paper. Definition 2.1 A nonempty set 𝑆 is said to be a semi-ring with respect to two binary compositions, addition and multiplication defined on it, if the following conditions are satisfied: 1. (𝑆, +) is a commutative semigroup with zero. 2. (𝑆, . ) is a semigroup. 3. for any three elements π‘Ž, 𝑏, 𝑐 ∈ 𝑆, the left distributive law π‘Ž. (𝑏 + 𝑐) = π‘Ž. 𝑏 + π‘Ž. 𝑐 and the right distributive law (𝑏 + 𝑐). π‘Ž = 𝑏. π‘Ž + 𝑐. π‘Ž. 4. 𝑠. 0 = 0. 𝑠, for all 𝑠 ∈ 𝑆. Definition 2.2 A nonempty subset ℐ of a semi-ring 𝑆 is called an ideal if 1. π‘Ž, 𝑏 ∈ ℐ implies π‘Ž + 𝑏 ∈ ℐ 2. π‘Ž ∈ ℐ, 𝑠 ∈ 𝑆 implies 𝑠. π‘Ž ∈ ℐ and π‘Ž. 𝑠 ∈ ℐ Definition 2.3 Let πœ‡ be a nonempty fuzzy subset of a semi-ring 𝑆. Then πœ‡ is called a fuzzy left(right) ideal of 𝑆 if for all 𝑖, 𝑗 ∈ 𝑆. 1. πœ‡(𝑖 + 𝑗) β‰₯ min{πœ‡(𝑖), πœ‡(𝑗)} 2. πœ‡(𝑖𝑗) β‰₯ πœ‡(𝑗)(π‘Ÿπ‘’π‘ π‘. π‘Ÿπ‘–π‘”β„Žπ‘‘) πœ‡(𝑖𝑗) β‰₯ πœ‡(𝑖)) A fuzzy ideal of a semi-ring 𝑆 is a nonempty fuzzy subset of 𝑆 which is both a fuzzy left ideal and a fuzzy right ideal of 𝑆. 3 Pythagorean fuzzy ideals in semiring In this section 𝑆 denotes Semiring(S). Definition 3.1 Let 𝑃 = (πœ‡π‘ƒ , πœ—π‘ƒ) be a Pythagorean fuzzy subset of a semiring 𝑆 and βˆ€π‘₯, 𝑦 ∈ 𝑆. (𝑖) πœ‡π‘ƒ(π‘₯ + 𝑦) β‰₯ min{πœ‡π‘ƒ(π‘₯), πœ‡π‘ƒ(𝑦)}; πœ—π‘ƒ(π‘₯ + 𝑦) ≀ max{πœ—π‘ƒ(π‘₯), πœ—π‘ƒ(𝑦)} (𝑖𝑖) πœ‡π‘ƒ(π‘₯𝑦) β‰₯ min{πœ‡π‘ƒ(π‘₯), πœ‡π‘ƒ(𝑦)}; πœ—π‘ƒ(π‘₯𝑦) ≀ max{πœ—π‘ƒ(π‘₯), πœ—π‘ƒ(𝑦)} Then 𝑃 = (πœ‡π‘ƒ, πœ—π‘ƒ) is called a Pythagorean fuzzy subsemiring of 𝑅. Definition 3.2 Let 𝑃 = (πœ‡π‘ƒ , πœ—π‘ƒ) of 𝑆 is called a Pythagorean fuzzy left ideal of 𝑆, if 𝑃 satisfies the following conditions (𝑖) πœ‡π‘ƒ(π‘₯ + 𝑦) β‰₯ min{πœ‡π‘ƒ(π‘₯), πœ‡π‘ƒ(𝑦)}; πœ—π‘ƒ(π‘₯ + 𝑦) ≀ max{πœ—π‘ƒ(π‘₯), πœ—π‘ƒ(𝑦)} (𝑖𝑖) πœ‡π‘ƒ(π‘₯𝑦) β‰₯ πœ‡π‘ƒ(𝑦); πœ—π‘ƒ(π‘₯𝑦) ≀ πœ—π‘ƒ(𝑦) Definition 3.3 Let 𝑃 = (πœ‡π‘ƒ , πœ—π‘ƒ) of 𝑆 is called a Pythagorean fuzzy right ideal of 𝑆, if 𝑃 satisfies the following conditions (𝑖) πœ‡π‘ƒ(π‘₯ + 𝑦) β‰₯ min{πœ‡π‘ƒ(π‘₯), πœ‡π‘ƒ(𝑦)}; πœ—π‘ƒ(π‘₯ + 𝑦) ≀ max{πœ—π‘ƒ(π‘₯), πœ—π‘ƒ(𝑦)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 314 https://internationalpubls.com (𝑖𝑖) πœ‡π‘ƒ(π‘₯𝑦) β‰₯ πœ‡π‘ƒ(π‘₯); πœ—π‘ƒ(π‘₯𝑦) ≀ πœ—π‘ƒ(π‘₯). Theorem 3.4 Intersection of a non empty collection of Pythagorean fuzzy right (resp. left) ideals is also a Pythagorean fuzzy right (resp. left) ideal of 𝑆. Proof. Let {𝑃𝑖 = (πœ‡π‘–, πœ—π‘–)|𝑖 ∈ 𝐼} be a non empty family of Pythagorean fuzzy right ideals of 𝑆 and π‘₯, 𝑦 ∈ 𝑆. Then β‹‚ π‘–βˆˆπΌ πœ‡π‘–(π‘₯ + 𝑦) = inf π‘–βˆˆπΌ {πœ‡π‘–(π‘₯ + 𝑦)} β‰₯ inf π‘–βˆˆπΌ {min{πœ‡π‘–(π‘₯), πœ‡π‘–(𝑦)}} = min{inf π‘–βˆˆπΌ πœ‡π‘–(π‘₯), inf π‘–βˆˆπΌ πœ‡π‘–(𝑦)} = min{β‹‚π‘–βˆˆπΌ πœ‡π‘–(π‘₯), β‹‚π‘–βˆˆπΌ πœ‡π‘–(𝑦)}. Also β‹‚ π‘–βˆˆπΌ πœ—π‘–(π‘₯ + 𝑦) = sup π‘–βˆˆπΌ {πœ—π‘–(π‘₯ + 𝑦)} ≀ sup π‘–βˆˆπΌ {max{πœ—π‘–(π‘₯), πœ—π‘–(𝑦)}} = max{sup π‘–βˆˆπΌ πœ—π‘–(π‘₯), sup π‘–βˆˆπΌ πœ—π‘–(𝑦)} = max{β‹‚π‘–βˆˆπΌ πœ—π‘–(π‘₯), β‹‚π‘–βˆˆπΌ πœ—π‘–(𝑦)}. Moreover β‹‚ π‘–βˆˆπΌ πœ‡π‘–(π‘₯𝑦) = inf π‘–βˆˆπΌ {πœ‡π‘–(π‘₯𝑦)} β‰₯ inf π‘–βˆˆπΌ {πœ‡π‘–(π‘₯)} = β‹‚π‘–βˆˆπΌ πœ‡π‘–(π‘₯). Finally β‹‚ π‘–βˆˆπΌ πœ—π‘–(π‘₯𝑦) = sup π‘–βˆˆπΌ {πœ—π‘–(π‘₯𝑦)} ≀ sup π‘–βˆˆπΌ {πœ—π‘–(π‘₯)} = β‹‚π‘–βˆˆπΌ πœ—π‘–(π‘₯). Hence β‹‚π‘–βˆˆπΌ 𝑃𝑖 is a Pythagorean fuzzy right ideals of 𝑆. Similarly, we can prove the result for Pythagorean fuzzy left ideal also. Theorem 3.5 Union of a non empty collection of Pythagorean fuzzy right (resp. left) ideals is also a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 315 https://internationalpubls.com Pythagorean fuzzy right (resp. left) ideal of 𝑆. Proof. Let {𝑃𝑖 = (πœ‡π‘–, πœ—π‘–)|𝑖 ∈ 𝐼} be a non empty family of Pythagorean fuzzy right ideals of 𝑆 and π‘₯, 𝑦 ∈ 𝑆. Then ⋃ π‘–βˆˆπΌ πœ‡π‘–(π‘₯ + 𝑦) = sup π‘–βˆˆπΌ {πœ‡π‘–(π‘₯ + 𝑦)} ≀ sup π‘–βˆˆπΌ {max{πœ‡π‘–(π‘₯), πœ‡π‘–(𝑦)}} = max{sup π‘–βˆˆπΌ πœ‡π‘–(π‘₯), sup π‘–βˆˆπΌ πœ‡π‘–(𝑦)} = max{β‹ƒπ‘–βˆˆπΌ πœ‡π‘–(π‘₯), β‹ƒπ‘–βˆˆπΌ πœ‡π‘–(𝑦)}. Also ⋃ π‘–βˆˆπΌ πœ—π‘–(π‘₯ + 𝑦) = inf π‘–βˆˆπΌ {πœ—π‘–(π‘₯ + 𝑦)} β‰₯ inf π‘–βˆˆπΌ {min{πœ—π‘–(π‘₯), πœ—π‘–(𝑦)}} = min{inf π‘–βˆˆπΌ πœ—π‘–(π‘₯), inf π‘–βˆˆπΌ πœ—π‘–(𝑦)} = min{β‹ƒπ‘–βˆˆπΌ πœ—π‘–(π‘₯), β‹ƒπ‘–βˆˆπΌ πœ—π‘–(𝑦)}. Moreover ⋃ π‘–βˆˆπΌ πœ‡π‘–(π‘₯𝑦) = sup π‘–βˆˆπΌ {πœ‡π‘–(π‘₯𝑦)} ≀ sup π‘–βˆˆπΌ {πœ‡π‘–(π‘₯)} = β‹ƒπ‘–βˆˆπΌ πœ‡π‘–(π‘₯). Finally ⋃ π‘–βˆˆπΌ πœ—π‘–(π‘₯𝑦) = inf π‘–βˆˆπΌ {πœ—π‘–(π‘₯𝑦)} β‰₯ inf π‘–βˆˆπΌ {πœ—π‘–(π‘₯)} = β‹ƒπ‘–βˆˆπΌ πœ—π‘–(π‘₯). Hence β‹ƒπ‘–βˆˆπΌ 𝑃𝑖 is a Pythagorean fuzzy right ideals of 𝑆. Similarly, we can prove the result for Pythagorean fuzzy left ideal also. Definition 3.6 Let 𝑃1 = (πœ‡1, πœ—1) and 𝑃2 = (πœ‡2, πœ—2) Pythagorean fuzzy subsets of 𝑆. The cartesian product of 𝑃1 and 𝑃2 is defined by (i) πœ‡1 Γ— πœ‡2(π‘₯, 𝑦) = min{πœ‡1(π‘₯), πœ‡2(π‘₯)} (ii) πœ—1 Γ— πœ—2(π‘₯, 𝑦) = max{πœ—1(π‘₯), πœ—2(π‘₯)}, for all π‘₯, 𝑦 ∈ 𝑆. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 316 https://internationalpubls.com Theorem 3.7 Let 𝑃1 and 𝑃2 be a Pythagorean fuzzy left ideals of semiring 𝑆. Then 𝑃1 Γ— 𝑃2 is a Pythagorean fuzzy left ideal of 𝑆 Γ— 𝑆. Proof. Let (π‘₯1, π‘₯2), (𝑦1, 𝑦2) ∈ 𝑆 Γ— 𝑆. Then (πœ‡1 Γ— πœ‡2)((π‘₯1, π‘₯2) + (𝑦1, 𝑦2)) = (πœ‡1 Γ— πœ‡2)(π‘₯1 + 𝑦1, π‘₯2 + 𝑦2) = min{πœ‡1(π‘₯1 + 𝑦1), πœ‡2(π‘₯2 + 𝑦2)} β‰₯ min{min{πœ‡1(π‘₯1), πœ‡1(𝑦1)}, min{πœ‡2(π‘₯2), πœ‡2(𝑦2)}} = min{min{πœ‡1(π‘₯1), πœ‡2(π‘₯2)}, min{πœ‡1(𝑦1), πœ‡2(𝑦2)}} = min{(πœ‡1 Γ— πœ‡2)(π‘₯1, π‘₯2), (πœ‡1 Γ— πœ‡2)(𝑦1, 𝑦2)}. (πœ‡1 Γ— πœ‡2)((π‘₯1, π‘₯2)(𝑦1, 𝑦2)) = (πœ‡1 Γ— πœ‡2)(π‘₯1𝑦1, π‘₯2𝑦2) = min{πœ‡1(π‘₯1𝑦1), πœ‡2(π‘₯2𝑦2)} β‰₯ min{πœ‡1(𝑦1), πœ‡2(𝑦2)} = (πœ‡1 Γ— πœ‡2)(𝑦1, 𝑦2). (πœ—1 Γ— πœ—2)((π‘₯1, π‘₯2) + (𝑦1, 𝑦2)) = (πœ—1 Γ— πœ—2)(π‘₯1 + 𝑦1, π‘₯2 + 𝑦2) = max{πœ—1(π‘₯1 + 𝑦1), πœ—2(π‘₯2 + 𝑦2)} ≀ max{max{πœ—1(π‘₯1), πœ—1(𝑦1)}, max{πœ—2(π‘₯2), πœ—2(𝑦2)}} = max{max{πœ—1(π‘₯1), πœ—2(π‘₯2)}, max{πœ—1(𝑦1), πœ—2(𝑦2)}} = max{(πœ—1 Γ— πœ—2)(π‘₯1, π‘₯2), (πœ—1 Γ— πœ—2)(𝑦1, 𝑦2)}. (πœ—1 Γ— πœ—2)((π‘₯1, π‘₯2)(𝑦1, 𝑦2)) = (πœ—1 Γ— πœ—2)(π‘₯1𝑦1, π‘₯2𝑦2) = max{πœ—1(π‘₯1𝑦1), πœ—2(π‘₯2𝑦2)} ≀ max{πœ—1(𝑦1), πœ—2(𝑦2)} = (πœ—1 Γ— πœ—2)(𝑦1, 𝑦2). Therefore 𝑃1 Γ— 𝑃2 is a Pythagorean fuzzy left ideal of 𝑆 Γ— 𝑆. Theorem 3.8 Let 𝑃 be a Pythagorean fuzzy subset of semiring. Then 𝑃 is a Pythagorean fuzzy left ideal of 𝑆 if and only if 𝑃 Γ— 𝑃 is a Pythagorean fuzzy left ideal of 𝑆 Γ— 𝑆. Proof. Consider 𝑃 is a Pythagorean fuzzy left ideal of 𝑆. Then by Previous theorem 𝑃 Γ— 𝑆. Conversely 𝑃 Γ— 𝑃 is a Pythagorean fuzzy left ideal of 𝑆 Γ— 𝑆, for all π‘₯1, π‘₯2, 𝑦1, 𝑦2 ∈ 𝑆. Then min{πœ‡(π‘₯1 + 𝑦1), πœ‡(π‘₯2 + 𝑦2)} = πœ‡ Γ— πœ‡(π‘₯1 + 𝑦1, π‘₯2 + 𝑦2) = (πœ‡ Γ— πœ‡){(π‘₯1, π‘₯2) + (𝑦1, 𝑦2)} β‰₯ min{(πœ‡ Γ— πœ‡)(π‘₯1, π‘₯2), (πœ‡ Γ— πœ‡)(𝑦1, 𝑦2)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 317 https://internationalpubls.com = min{min{πœ‡(π‘₯1), πœ‡(π‘₯2)}, min{πœ‡(𝑦1), πœ‡(𝑦2)}} Next,we have min{πœ‡(π‘₯1𝑦1), πœ‡(π‘₯2𝑦2)} = (πœ‡ Γ— πœ‡)(π‘₯1𝑦1, π‘₯2𝑦2) = (πœ‡ Γ— πœ‡){(π‘₯1, π‘₯2)(𝑦1, 𝑦2)} = (πœ‡ Γ— πœ‡){𝑦1, 𝑦2} = min{πœ‡(𝑦1), πœ‡(𝑦2)} Also max{πœ—(π‘₯1 + 𝑦1), πœ—(π‘₯2 + 𝑦2)} = πœ— Γ— πœ—(π‘₯1 + 𝑦1, π‘₯2 + 𝑦2) = (πœ— Γ— πœ—){(π‘₯1, π‘₯2) + (𝑦1, 𝑦2)} ≀ max{(πœ— Γ— πœ—)(π‘₯1, π‘₯2), (πœ— Γ— πœ—)(𝑦1, 𝑦2)} = max{max{πœ—(π‘₯1), πœ—(π‘₯2)}, max{πœ—(𝑦1), πœ—(𝑦2)}} and max{πœ—(π‘₯1𝑦1), πœ—(π‘₯2𝑦2)} = (πœ— Γ— πœ—)(π‘₯1𝑦1, π‘₯2𝑦2) = (πœ— Γ— πœ—){(π‘₯1, π‘₯2)(𝑦1, 𝑦2)} = (πœ— Γ— πœ—){𝑦1, 𝑦2} = max{πœ—(𝑦1), πœ—(𝑦2)} Hence 𝑃 is a Pythagorean fuzzy left ideal of 𝑆. Theorem 3.9 If 𝑃1, 𝑃2 be any two Pythagorean fuzzy ideals of semiring 𝑆, then 𝑃1 + 𝑃2 is also so. Proof. Consider 𝑃1, 𝑃2 are any two Pythagorean fuzzy ideals of semiring 𝑆 and π‘₯, 𝑦 ∈ 𝑆. Then (πœ‡1 + πœ‡2)(π‘₯ + 𝑦) = sup π‘₯+𝑦≀𝑐+𝑑 {min{πœ‡1(𝑐), πœ‡2(𝑑)}} β‰₯ sup π‘₯+𝑦≀(π‘Ž1+𝑏1)+(π‘Ž2+𝑏2)=(π‘Ž1+π‘Ž2)+(𝑏1+𝑏2) {min{πœ‡1(π‘Ž1 + π‘Ž2), πœ‡2(𝑏1 + 𝑏2)}} β‰₯ sup{min{πœ‡1(π‘Ž1), πœ‡2(π‘Ž2)}, min{πœ‡2(𝑏1), πœ‡2(𝑏2)}} = min{ sup π‘₯β‰€π‘Ž1+𝑏1 {min{πœ‡1(π‘Ž1), πœ‡2(𝑏1)}}, sup π‘¦β‰€π‘Ž2+𝑏2 {min{πœ‡1(π‘Ž2), πœ‡2(𝑏2)}}} = min{(πœ‡1 + πœ‡2)(π‘₯), (πœ‡1 + πœ‡2)(𝑦)} Also (πœ—1 + πœ—2)(π‘₯ + 𝑦) = inf π‘₯+𝑦≀𝑐+𝑑 {max{πœ—1(𝑐), πœ—2(𝑑)}} ≀ inf π‘₯+𝑦≀(π‘Ž1+𝑏1)+(π‘Ž2+𝑏2)=(π‘Ž1+π‘Ž2)+(𝑏1+𝑏2) {max{πœ—1(π‘Ž1 + π‘Ž2), πœ—2(𝑏1 + 𝑏2)}} ≀ inf{max{πœ—1(π‘Ž1), πœ—2(π‘Ž2)}, max{πœ—2(𝑏1), πœ—2(𝑏2)}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 318 https://internationalpubls.com = max{ inf π‘₯β‰€π‘Ž1+𝑏1 {max{πœ—1(π‘Ž1), πœ—2(𝑏1)}}, inf π‘¦β‰€π‘Ž2+𝑏2 {max{πœ—1(π‘Ž2), πœ—2(𝑏2)}}} = max{(πœ—1 + πœ—2)(π‘₯), (πœ—1 + πœ—2)(𝑦)} Now let as consider 𝑃1, 𝑃2 are Pythagorean fuzzy right ideals and we have (πœ‡1 + πœ‡2)(π‘₯𝑦) = sup π‘₯+𝑦≀𝑐+𝑑 {min{πœ‡1(𝑐), πœ‡2(𝑑)}} β‰₯ sup π‘₯𝑦≀(π‘₯1+π‘₯2)𝑦 {min{πœ‡1(π‘₯1𝑦), πœ‡2(π‘₯2𝑦)}} β‰₯ sup π‘₯≀(π‘₯1+π‘₯2) {min{πœ‡1(π‘₯1), πœ‡2(π‘₯2)}} = (πœ‡1 + πœ‡2)(π‘₯). and (πœ—1 + πœ—2)(π‘₯𝑦) = inf π‘₯+𝑦≀𝑐+𝑑 {max{πœ—1(𝑐), πœ—2(𝑑)}} ≀ inf π‘₯𝑦≀(π‘₯1+π‘₯2)𝑦 {max{πœ—1(π‘₯1𝑦), πœ—2(π‘₯2𝑦)}} ≀ inf π‘₯≀(π‘₯1+π‘₯2) {max{πœ—1(π‘₯1), πœ—2(π‘₯2)}} = (πœ—1 + πœ—2)(π‘₯). Similarly assuming 𝑃1, 𝑃2 are Pythagorean fuzzy left ideal, we can show that (𝑃1 + 𝑃2)(π‘₯𝑦) β‰₯ (𝑃1 + 𝑃2)(𝑦) Also (πœ‡1 + πœ‡2)(π‘₯) = sup π‘₯≀π‘₯1+π‘₯2 {min{πœ‡1(π‘₯1), πœ‡2(π‘₯2)}} β‰₯ sup π‘₯≀𝑦≀𝑦1+𝑦2 {min{πœ‡1(𝑦1), πœ‡2(𝑦2)}} = sup 𝑦≀𝑦1+𝑦2 {min{πœ‡1(𝑦1), πœ‡2(𝑦2)}} = (πœ‡1 + πœ‡2)(𝑦) and (πœ—1 + πœ—2)(π‘₯) = inf π‘₯≀π‘₯1+π‘₯2 {max{πœ—1(π‘₯1), πœ—2(π‘₯2)}} ≀ inf π‘₯≀𝑦≀𝑦1+𝑦2 {max{πœ—1(𝑦1), πœ—2(𝑦2)}} = inf 𝑦≀𝑦1+𝑦2 {max{πœ—1(𝑦1), πœ—2(𝑦2)}} = (πœ—1 + πœ—2)(𝑦) Hence 𝑃1 + 𝑃2 is a Pythagorean fuzzy ideal of 𝑆. Theorem 3.10 If 𝑃1, 𝑃2 be any two Pythagorean fuzzy ideals of semiring 𝑆, then 𝑃1 ∘ 𝑃2 is also so. Proof. Let 𝑃1, 𝑃2 are any two Pythagorean fuzzy ideals of semiring 𝑆 and π‘₯, 𝑦 ∈ 𝑆. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 319 https://internationalpubls.com Then (πœ‡1 ∘ πœ‡2)(π‘₯ + 𝑦) = sup π‘₯+𝑦≀𝑐+𝑑 {min{πœ‡1(𝑐), πœ‡2(𝑑)}} β‰₯ sup π‘₯+𝑦≀(𝑐1𝑑1)+(𝑐2𝑑2)≀(𝑐1+𝑐2)(𝑑1+𝑑2) {min{πœ‡1(𝑐1 + 𝑐2), πœ‡2(𝑑1 + 𝑑2)}} β‰₯ sup{min{πœ‡1(𝑐1), πœ‡1(𝑐2)}, min{πœ‡2(𝑑1), πœ‡2(𝑑2)}} = min{ sup π‘₯≀𝑐1𝑑1 {min{πœ‡1(𝑐1), πœ‡2(𝑑1)}}, sup 𝑦≀𝑐2𝑑2 {min{πœ‡1(𝑐2), πœ‡2(𝑑2)}}} = min{(πœ‡1 ∘ πœ‡2)(π‘₯), (πœ‡1 ∘ πœ‡2)(𝑦)} Also (πœ—1 ∘ πœ—2)(π‘₯ + 𝑦) = inf π‘₯+𝑦≀𝑐+𝑑 {max{πœ—1(𝑐), πœ—2(𝑑)}} ≀ inf π‘₯+𝑦≀(𝑐1𝑑1)+(𝑐2𝑑2)≀(𝑐1+𝑐2)(𝑑1+𝑑2) {max{πœ—1(𝑐1 + 𝑐2), πœ—2(𝑑1 + 𝑑2)}} ≀ inf{max{πœ—1(𝑐1), πœ—1(𝑐2)}, max{πœ—2(𝑑1), πœ—2(𝑑2)}} = max{ inf π‘₯≀𝑐1𝑑1 {max{πœ—1(𝑐1), πœ—2(𝑑1)}}, inf 𝑦≀𝑐2𝑑2 {max{πœ—1(𝑐2), πœ—2(𝑑2)}}} = max{(πœ—1 ∘ πœ—2)(π‘₯), (πœ—1]π‘π‘–π‘Ÿπ‘πœ—2)(𝑦)} Now let as consider 𝑃1, 𝑃2 are Pythagorean fuzzy right ideals and we have (πœ‡1 ∘ πœ‡2)(π‘₯𝑦) = sup π‘₯𝑦≀𝑐𝑑 {min{πœ‡1(𝑐), πœ‡2(𝑑)}} β‰₯ sup π‘₯𝑦≀(π‘₯1π‘₯2)𝑦 {min{πœ‡1(π‘₯1𝑦), πœ‡2(π‘₯2𝑦)}} β‰₯ sup π‘₯≀(π‘₯1π‘₯2) {min{πœ‡1(π‘₯1), πœ‡2(π‘₯2)}} = (πœ‡1 ∘ πœ‡2)(π‘₯). and (πœ—1 ∘ πœ—2)(π‘₯𝑦) = inf π‘₯𝑦≀𝑐𝑑 {max{πœ—1(𝑐), πœ—2(𝑑)}} ≀ inf π‘₯𝑦≀(π‘₯1π‘₯2)𝑦 {max{πœ—1(π‘₯1𝑦), πœ—2(π‘₯2𝑦)}} ≀ inf π‘₯≀(π‘₯1π‘₯2) {max{πœ—1(π‘₯1), πœ—2(π‘₯2)}} = (πœ—1 ∘ πœ—2)(π‘₯). Similarly assuming 𝑃1, 𝑃2 are Pythagorean fuzzy left ideal, we can show that (𝑃1 ∘ 𝑃2)(π‘₯𝑦) β‰₯ (𝑃1 ∘ 𝑃2)(𝑦) Also (πœ‡1 ∘ πœ‡2)(π‘₯) = sup π‘₯≀π‘₯1π‘₯2 {min{πœ‡1(π‘₯1), πœ‡2(π‘₯2)}} β‰₯ sup π‘₯≀𝑦≀𝑦1𝑦2 {min{πœ‡1(𝑦1), πœ‡2(𝑦2)}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 320 https://internationalpubls.com = sup 𝑦≀𝑦1𝑦2 {min{πœ‡1(𝑦1), πœ‡2(𝑦2)}} = (πœ‡1 ∘ πœ‡2)(𝑦) and (πœ—1 ∘ πœ—2)(π‘₯) = inf π‘₯≀π‘₯1π‘₯2 {max{πœ—1(π‘₯1), πœ—2(π‘₯2)}} ≀ inf π‘₯≀𝑦≀𝑦1𝑦2 {max{πœ—1(𝑦1), πœ—2(𝑦2)}} = inf 𝑦≀𝑦1𝑦2 {max{πœ—1(𝑦1), πœ—2(𝑦2)}} = (πœ—1 ∘ πœ—2)(𝑦) Hence 𝑃1 ∘ 𝑃2 is a Pythagorean fuzzy ideal of 𝑆. Definition 3.11 A Pythagorean fuzzy subset 𝑃 = (πœ‡, πœ—) is called a Pythagorean fuzzy bi-ideal of 𝑆, for all π‘₯, 𝑦, 𝑧 ∈ 𝑆. (i) πœ‡(π‘₯ + 𝑦) β‰₯ min{πœ‡(π‘₯), πœ‡(𝑦)};πœ—(π‘₯ + 𝑦) ≀ max{πœ—(π‘₯), πœ—(𝑦)} (ii) πœ‡(π‘₯𝑦) β‰₯ min{πœ‡(π‘₯), πœ‡(𝑦)};πœ—(π‘₯𝑦) ≀ max{πœ—(π‘₯), πœ—(𝑦)} (iii) πœ‡(π‘₯𝑦𝑧) β‰₯ min{πœ‡(π‘₯), πœ‡(𝑧)};πœ—(π‘₯𝑦𝑧) ≀ max{πœ—(π‘₯), πœ—(𝑧)} Theorem 3.12 Intersection of a non empty collection of Pythagorean fuzzy bi-ideals is also Pythagorean fuzzy bi-ideal of 𝑆. Proof. Let {𝑃𝑖 = (πœ‡π‘–, πœ—π‘–)|𝑖 ∈ 𝐼} be a family of Pythagorean fuzzy bi-ideals of 𝑆 and π‘₯, 𝑦 ∈ 𝑆. Then β‹‚ π‘–βˆˆπΌ πœ‡π‘–(π‘₯𝑦𝑧) = inf π‘–βˆˆπΌ {πœ‡π‘–(π‘₯𝑦𝑧)} β‰₯ inf π‘–βˆˆπΌ {min{πœ‡π‘–(π‘₯), πœ‡π‘–(𝑧)}} = min{inf π‘–βˆˆπΌ πœ‡π‘–(π‘₯), inf π‘–βˆˆπΌ πœ‡π‘–(𝑧)} = min{β‹‚π‘–βˆˆπΌ πœ‡π‘–(π‘₯), β‹‚π‘–βˆˆπΌ πœ‡π‘–(𝑧)}. Finally β‹‚ π‘–βˆˆπΌ πœ—π‘–(π‘₯𝑦𝑧) = sup π‘–βˆˆπΌ {πœ—π‘–(π‘₯𝑦𝑧)} ≀ sup π‘–βˆˆπΌ {max{πœ—π‘–(π‘₯), πœ—π‘–(𝑧)}} = max{sup π‘–βˆˆπΌ πœ—π‘–(π‘₯), sup π‘–βˆˆπΌ πœ—π‘–(𝑧)} = max{β‹‚π‘–βˆˆπΌ πœ—π‘–(π‘₯), β‹‚π‘–βˆˆπΌ πœ—π‘–(𝑧)}. Hence 𝑃𝑖 is a Pythagorean fuzzy bi-ideal of 𝑆. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 321 https://internationalpubls.com Theorem 3.13 Let 𝑃1 and 𝑃2 be two Pythagorean fuzzy bi-ideal of 𝑆. Then 𝑃 is a Pythagorean fuzzy bi-ideal of 𝑆. Proof. Let π‘₯, 𝑦, 𝑧 ∈ 𝑆 (πœ‡1 β‹… πœ‡2)(π‘₯ + 𝑦) = min{πœ‡1(π‘₯ + 𝑦), πœ‡2(π‘₯ + 𝑦)} β‰₯ min{min{πœ‡1(π‘₯), πœ‡1(𝑦)}, min{πœ‡2(π‘₯), πœ‡2(𝑦)}} = min{min{πœ‡1(π‘₯), πœ‡2(π‘₯)}, min{πœ‡1(𝑦), πœ‡2(𝑦)}} = min{(πœ‡1 β‹… πœ‡2)(π‘₯), (πœ‡1 β‹… πœ‡2)(𝑦)} and (πœ—1 β‹… πœ—2)(π‘₯ + 𝑦) = max{πœ—1(π‘₯ + 𝑦), πœ—2(π‘₯ + 𝑦)} ≀ max{max{πœ—1(π‘₯), πœ—1(𝑦)}, max{πœ—2(π‘₯), πœ—2(𝑦)}} = max{max{πœ—1(π‘₯), πœ—2(π‘₯)}, max{πœ—1(𝑦), πœ—2(𝑦)}} = max{(πœ—1 β‹… πœ—2)(π‘₯), (πœ—1 β‹… πœ—2)(𝑦)} Next (πœ‡1 β‹… πœ‡2)(π‘₯𝑦) = min{πœ‡1(π‘₯𝑦), πœ‡2(π‘₯𝑦)} β‰₯ min{min{πœ‡1(π‘₯), πœ‡1(𝑦)}, min{πœ‡2(π‘₯), πœ‡2(𝑦)}} = min{min{πœ‡1(π‘₯), πœ‡2(π‘₯)}, min{πœ‡1(𝑦), πœ‡2(𝑦)}} = min{(πœ‡1 β‹… πœ‡2(π‘₯)), (πœ‡1 β‹… πœ‡2(𝑦))} and (πœ—1 β‹… πœ—2)(π‘₯𝑦) = max{πœ—1(π‘₯𝑦), πœ—2(π‘₯𝑦)} ≀ max{max{πœ—1(π‘₯), πœ—1(𝑦)}, max{πœ—2(π‘₯), πœ—2(𝑦)}} = max{max{πœ—1(π‘₯), πœ—2(π‘₯)}, max{πœ—1(𝑦), πœ—2(𝑦)}} = max{(πœ—1 β‹… πœ—2(π‘₯)), (πœ—1 β‹… πœ—2(𝑦))} Also (πœ‡1 β‹… πœ‡2)(π‘₯𝑦𝑧) = min{πœ‡1(π‘₯𝑦𝑧), πœ‡2(π‘₯𝑦𝑧)} β‰₯ min{min{πœ‡1(π‘₯), πœ‡1(𝑧)}, min{πœ‡2(π‘₯), πœ‡2(𝑧)}} = min{min{πœ‡1(π‘₯), πœ‡2(π‘₯)}, min{πœ‡1(𝑧), πœ‡2(𝑧)}} = min{(πœ‡1 β‹… πœ‡2(π‘₯)), (πœ‡1 β‹… πœ‡2(π‘₯))} and (πœ—1 β‹… πœ—2)(π‘₯𝑦𝑧) = max{πœ—1(π‘₯𝑦𝑧), πœ—2(π‘₯𝑦𝑧)} ≀ max{max{πœ—1(π‘₯), πœ—1(𝑧)}, max{πœ—2(π‘₯), πœ—2(𝑧)}} = max{max{πœ—1(π‘₯), πœ—2(π‘₯)}, max{πœ—1(𝑧), πœ—2(𝑧)}} = max{(πœ—1 β‹… πœ—2(π‘₯)), (πœ—1 β‹… πœ—2(𝑧))} Hence 𝑃1 and 𝑃2 is a Pythagorean fuzzy bi-ideal of 𝑆. Definition 3.14 The product of 𝑃1 and 𝑃2 is a Pythagorean fuzzy subset 𝑃1 ∘ 𝑃2: 𝑆 β†’ [0,1] by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 322 https://internationalpubls.com (πœ‡1 ∘ πœ‡2)(π‘Ž) = sup π‘Ž=𝑏𝑐 {min{πœ‡1(𝑏), πœ‡2(𝑐)}} (πœ—1 ∘ πœ—2)(π‘Ž) = inf π‘Ž=𝑏𝑐 {max{πœ—1(𝑏), πœ—2(𝑐)}} Theorem 3.15 If 𝑃1, 𝑃2 be any two Pythagorean fuzzy bi-ideals of semiring 𝑆, then 𝑃1 ∘ 𝑃2 is a Pythagorean fuzzy bi-ideal of 𝑆. Proof. Let 𝑃1, 𝑃2 are any two Pythagorean fuzzy ideals of semiring 𝑆 and π‘₯, 𝑦 ∈ 𝑆. Then (πœ‡1 ∘ πœ‡2)(π‘₯ + 𝑦) = sup π‘₯+𝑦≀𝑐+𝑑 {min{πœ‡1(𝑐), πœ‡2(𝑑)}} β‰₯ sup π‘₯+𝑦≀(𝑐1𝑑1)+(𝑐2𝑑2)≀(𝑐1+𝑐2)(𝑑1+𝑑2) {min{πœ‡1(𝑐1 + 𝑐2), πœ‡2(𝑑1 + 𝑑2)}} β‰₯ sup{min{πœ‡1(𝑐1), πœ‡1(𝑐1)}, min{πœ‡2(𝑑1), πœ‡2(𝑑2)}} = min{ sup π‘₯≀𝑐1𝑑1 {min{πœ‡1(𝑐1), πœ‡2(𝑑1)}}, sup 𝑦≀𝑐2𝑑2 {min{πœ‡1(𝑐2), πœ‡2(𝑑2)}}} = min{(πœ‡1 ∘ πœ‡2)(π‘₯), (πœ‡1 ∘ πœ‡2)(𝑦)} Also (πœ—1 ∘ πœ—2)(π‘₯ + 𝑦) = inf π‘₯+𝑦≀𝑐+𝑑 {max{πœ—1(𝑐), πœ—2(𝑑)}} ≀ inf π‘₯+𝑦≀(𝑐1𝑑1)+(𝑐2𝑑2)≀(𝑐1+𝑐2)(𝑑1+𝑑2) {max{πœ—1(𝑐1 + 𝑐2), πœ—2(𝑑1 + 𝑑2)}} ≀ inf{max{πœ—1(𝑐1), πœ—1(𝑐2)}, max{πœ—2(𝑑1), πœ—2(𝑑2)}} = max{ inf π‘₯≀𝑐1𝑑1 {max{πœ—1(𝑐1), πœ—2(𝑑1)}}, inf 𝑦≀𝑐2𝑑2 {max{πœ—1(𝑐2), πœ—2(𝑑2)}}} = max{(πœ—1 ∘ πœ—2)(π‘₯), (πœ—1]π‘π‘–π‘Ÿπ‘πœ—2)(𝑦)} Now let as consider 𝑃1, 𝑃2 are Pythagorean fuzzy right ideals and we have (πœ‡1 ∘ πœ‡2)(π‘₯𝑦) = sup π‘₯𝑦≀𝑐𝑑 {min{πœ‡1(𝑐), πœ‡2(𝑑)}} β‰₯ sup π‘₯𝑦≀(π‘₯1π‘₯2)𝑦 {min{πœ‡1(π‘₯1𝑦), πœ‡2(π‘₯2𝑦)}} β‰₯ sup π‘₯≀(π‘₯1π‘₯2) {min{πœ‡1(π‘₯1), πœ‡2(π‘₯2)}} = (πœ‡1 ∘ πœ‡2)(π‘₯). and (πœ—1 ∘ πœ—2)(π‘₯𝑦) = inf π‘₯𝑦≀𝑐𝑑 {max{πœ—1(𝑐), πœ—2(𝑑)}} ≀ inf π‘₯𝑦≀(π‘₯1π‘₯2)𝑦 {max{πœ—1(π‘₯1𝑦), πœ—2(π‘₯2𝑦)}} ≀ inf π‘₯≀(π‘₯1π‘₯2) {max{πœ—1(π‘₯1), πœ—2(π‘₯2)}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 323 https://internationalpubls.com = (πœ—1 ∘ πœ—2)(π‘₯). Similarly assuming 𝑃1, 𝑃2 are Pythagorean fuzzy left ideal, we can show that (𝑃1 ∘ 𝑃2)(π‘₯𝑦) β‰₯ (𝑃1 ∘ 𝑃2)(𝑦) Also (πœ‡1 ∘ πœ‡2)(π‘₯) = sup π‘₯≀π‘₯1π‘₯2 {min{πœ‡1(π‘₯1), πœ‡2(π‘₯2)}} β‰₯ sup π‘₯≀𝑦≀𝑦1𝑦2 {min{πœ‡1(𝑦1), πœ‡2(𝑦2)}} = sup 𝑦≀𝑦1𝑦2 {min{πœ‡1(𝑦1), πœ‡2(𝑦2)}} = (πœ‡1 ∘ πœ‡2)(𝑦) and (πœ—1 ∘ πœ—2)(π‘₯) = inf π‘₯≀π‘₯1π‘₯2 {max{πœ—1(π‘₯1), πœ—2(π‘₯2)}} ≀ inf π‘₯≀𝑦≀𝑦1𝑦2 {max{πœ—1(𝑦1), πœ—2(𝑦2)}} = inf 𝑦≀𝑦1𝑦2 {max{πœ—1(𝑦1), πœ—2(𝑦2)}} = (πœ—1 ∘ πœ—2)(𝑦) Hence 𝑃1 ∘ 𝑃2 is a Pythagorean fuzzy ideal of 𝑆. References [1] Ahsan, J. Saifullah, K. and Khan, M.F.(1993), Fuzzy Semirings, Fuzzy Sets and Systems, 302-309. [2] Atanassov, K. T. (1986) Intuitionistic fuzzy sets. Fuzzy Sets and Systems,20, 87-96. [3] Bhargavi Y and Eswarlal T (2015), Fuzzy Ξ“ -semirings, international Journal of Pure and Applied Mathematics,98,339-349. [4] Dutta, T. K., Sardar, S. K. and Goswami, S. 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