Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 635 https://internationalpubls.com Impact Fuzzy Ideal Extension in Terms of Gamma Semigroup 1Vasantha.Gadipally, 2Dr. Sri Lakshmi T 1GITAM, Vishakapatnam, AP, INDIA, vgadipal@gitam.in 2GITAM, Vishakapatnam, AP, INDIA, stalasil@gitam.in Article History: Received: 12-11-2024 Revised: 24-12-2024 Accepted: 16-01-2025 Abstract: Our exploration into the properties of some fuzzy Semigroups and ideals is a deeply collaborative effort involving the contributions of many researchers in the field. This collective endeavour, which builds on the work of our peers, aims to deepen our understanding of fuzzy ideal semigroup and fuzzy extension in terms of gamma semigroups. Using a star zeta, we define, characterise and describe the different classes of fuzzy ideal extension in the gamma Semigroup, resulting from our shared research efforts. Investigating star zeta and fuzzy ideal properties and their results is a testament to the power of shared knowledge in our academic community. The description of many properties of fuzzy prime ideal and fuzzy semiprime in the gamma function further underscores the collaborative nature of scholarly research, making each member of our community feel included and valued. Keywords: Fuzzy subsemigroups, Fuzzy ideal, fuzzy prime ideal, fuzzy ideal extension Semigroup, star zeta of fuzzy Semigroup. 1. Introduction The work aims to explain the terms used by the authors and a quick review of semigroup theory. This study examined the fundamental properties and variations of semigroups. Early authors described an extensive range of semigroup kinds and verified various procedures. We studied the roots of semigroups and worked on perplexing assertions in regular semigroups. We found properties related to the semigroup theory written by four critical writers. Anton Schushewitch is the first semigroup theorist in history. In 1941, Clifford proved that 'if S is a collection of groups, then it is a semilattice of completely simple semigroups [1-3].' Furthermore, he proved that 'a band is a semilattice of rectangular bands.' Vagner did, however, provide the opposite semigroup [4-7]. This paper introduces and defines a new concept of Ṱ-norms and Ṧ-norms (notations used to represent certain operations in fuzzy set theory) and their properties, which we denote throughout the paper, bringing a novel perspective to the field. Zadeh introduced the essential concept of a fuzzy set in 1965 [8,9], leading to insightful discoveries and practical uses in various scientific fields. This seminal work paved the way for numerous writers who wrote after that time, confirming the necessity and value of the concept. Rosenfeld extended several group results to include ambiguous groupings. Additionally, he suggested noting fuzzy groups [10]. Wu studied and introduced traditional fuzzy subgroups. Rosenfeld also demonstrated that the fuzzy subgroup's homomorphic image equals one. Therefore, Anthony and Sherwood should have utilised these properties to study fuzzy homomorphisms. Initially, Authors Imtiaz, A., Alolaiyan, H., and Shuaib investigated the idea of Applications of conjunctive complex fuzzy or vague cosets and mailto:vgadipal@gitam.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 636 https://internationalpubls.com their interrelation within fuzzy normal subgroups, proving how impact in Sylow theory[11-19]. These historical developments in the field of fuzzy set theory and algebraic structures have laid the foundation for our current research. For example, the introduction of fuzzy (subgroupoids) subgroups and fuzzy (left, fight) ideals in the seminal publication of [ 20-24] Yiarayong marked the beginning of the study of fuzzy algebraic structures. Fuzzy set theory has now been extended to semigroups by several writers. For instance, Mursaleen, Srivastava, and Sunil [32] investigated particular novel spaces of statistically convergent and strongly summable sequences of fuzzy numbers.[25-31] Jun, Song, and Muhiuddin established the concept of hybrid structure in a set of parameters over an initial universe set as a parallel circuit of fuzzy sets and soft sets (or hesitant fuzzy sets) [34-38]. They used it on linear spaces and BCK/BCI algebras. A Russian scientist (1999) presented the soft set theory as a novel mathematical technique for handling uncertainties.[33] Torra, 2010; Torra & Narukawa, a generalisation of Zadeh’s fuzzy or vague set. The hesitant fuzzy set is handy for expressing people's hesitancy in daily life, and it is a convenient tool to deal with uncertainty, which can be accurately and perfectly described in terms of decision-makers' opinions. We introduce and define a new concept of Ṱ-norms and Ṧ-norms (notations used to represent certain operations in fuzzy set theory) and their properties, which we denote throughout the paper, bringing a novel perspective to the field. This paper introduces the star of zeta, a key concept in our research, and its application to fuzzy ideal extension in gamma Semigroups. This extension is significant as it provides a new perspective on the properties and extensions of fuzzy semigroups. It has practical applications in various scientific fields, such as data analysis, pattern recognition, and decision-making under uncertainty. We aim to inspire and motivate further research and innovation by highlighting these practical applications. We present the notions of fuzzy subsemigroups and discuss their potential real-world applications. Using these notions, we consider characterisations of sub-semigroups and extensions of fuzzy semigroups. We also introduce the concept of the dot(product) with properties and star zeta with properties developed in this paper and discuss characterisations of the fuzzy ideal of gamma semigroups. 2. Objectives This study comprehensively explores semigroups' fundamental properties and variations to review semigroup theory and its historical evolution. By examining the contributions of significant theorists, this study aims to comprehend the evolution of semigroups and their structural classifications. Fuzzy algebraic structures are also analysed in the research, including fuzzy subgroups, fuzzy ideals and their extensions, and the use of fuzzy set theory with semigroups. This study provides fresh insights into semigroup features by presenting and describing new concepts, including fuzzy ideal extension, star zeta, Ṱ-norms, Ṧ-norms, and the dot product in the context of gamma semigroups. With two theorems and eleven lemmas that expand our theoretical and applied understanding of semigroup structures, the study provides strong mathematical evidence in favour of these concepts. The paper also highlights the valuable applications of fuzzy semigroups, emphasising how they can be used in various scientific domains. This work intends to stimulate additional research and creativity by characterising sub-semigroups and expanding fuzzy semigroup theory, promoting a deeper investigation of algebraic structures and their computing applications. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 637 https://internationalpubls.com 3. Preliminaries Definition 3.1: [10] A non-empty set Ꟊ is called a Ternary Semigroup if there exists a mapping ꟉxꟉxꟉ → Ꟊ is defined by 𝑎𝑏𝑐 ∈ Ꟊ and (𝑎𝑏𝑐)𝑑𝑒 = 𝑎(𝑏𝑐𝑑)𝑒 = 𝑎𝑏(𝑐𝑑𝑒) For all a,b,c,d,e ∈ Ꟊ. Example: Let Ꟊ = { x√5 / x∈ Z- } where Z- is the set of negative odd integers. Then Ꟊ is a ternary Semigroup under usual multiplication. Definition 3.2: [12] A non-empty set L of a ternary Semigroup Ꟊ is called (i) A left ideal of Ꟊ if ꟉꟉL ⊆ L (ii) An interior ideal of Ꟊ if ꟉLꟉ ⊆ L (iii) A proper ideal of Ꟊ LꟉꟉ ⊆ L (iv) An ideal of Ꟊ if L is a left ideal, a right ideal and an interior ideal of Ꟊ An ideal L of a ternary Semigroup Ꟊ is a proper ideal if L. ≠ Ꟊ. Definition 3.3: Let ζ be a fuzzy subset of Ternary Semigroup Ꟊ, then (Ꟊ,ζ) is called fuzzy Ternary Semigroup if ζ(pqr) ≥ min{ζ(p),ζ(q),ζ(r) } for all p,q,r,∈ Ꟊ. Example: Consider set Ꟊ = {p,q,r} with the ternary operation ʘ and assigned membership values of fuzzy set ζ as follows. Table of ternary operation: ʘ p q r p p q r q q r p r r p q Table-1(Caylley’s table of p, q r) And ζ(p) = 0.8 ; ζ(q) = 0.6; ζ(r) = 0.4 The graph shows this ternary operation and membership function as visualisation. Let us assume that nodes represent elements p,q,r, and edges represent the ternary operation; the membership values can be indicated next to the nodes. A textual representation of the fuzzy ternary semigroup is below. Nodes: p (0.8); q (0.6); r (0.4) Edges (ternary operation) : (p, p, p) it tends to p; (p, q, r) it tends to q; (p, r, q) is tends to r; (q, q, q) is tends to r; (r, r, r) is tends to p (q, r, p) is tends to q and so on. Using the above steps, we can draw nodes and edges with labelled membership values and operation results. This visualisation technique is a helpful tool for understanding the relationships and operations within a fuzzy Ternary Semigroup. This Section provides a step-by-step guide on creating such visualisations and a detailed example of the algorithm. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 638 https://internationalpubls.com Figure (Algorithm) Definition 3.4: A non-empty set Ꟊ is called ternary gamma semigroup if there exists a map ꟉΓꟉΓꟉ. → Ꟊ is ternary gamma semigroup; then it is satisfies (i) (p αq β r )γ s δ t = p α(q β r γ s )δ t = p αq β( r γ s δ t) (ii) p α q β r ∈ Ꟊ for all p ,q , r, s, t ∈ Ꟊ and α,β, γ,δ ∈ Γ. Definition 3.5: Let ζ be a fuzzy subset of the ternary gamma semigroup Ꟊ, then it is called fuzzy ternary gamma semigroup if 𝜁(𝑝𝛼𝑞𝛽𝑟) ≥ ⋁(𝑝𝛼𝑞𝛽𝑟)𝑚𝑖𝑛 {𝜁(𝑝), 𝜁(𝑞), 𝜁(𝑟)}.If ζ is the fuzzy interior ideal of ternary gamma Semigroup Ꟊ, then ζ(pαqβr) ≥ ζ(q); if ζ is the fuzzy left ideal of ternary gamma Semigroup Ꟊ, then ζ(pαqβr) ≥ ζ(r); if ζ is the fuzzy right ideal of ternary gamma Semigroup Ꟊ, then ζ(pαqβr) ≥ ζ(p). Definition 3.6: A non-empty set Ꞗ of Ꟊ is called generalised bi-ideal if ꞖꟉꞖꟉꞖ ⊆ Ꞗ.The generalised bi-ideal Ꞗ of Ꟊ is called bi-ideal if ꞖꞖꞖ ⊆ Ꞗ. A mapping ζ:Ӽ → [0,1] is called a fuzzy set of Ӽ. The fuzzy set ζ of Ꟊ is called generalised fuzzy bi-ideal if ζ(pαqβr) ≥ min{ζ(p),ζ(q),ζ(r)}. The generalised fuzzy bi-ideal ζ of Ꟊ is called a fuzzy bi-ideal if ζ(pαqβr) ≥ min{ζ(p),ζ(q),ζ(r)}. The fuzzy set ζ of Ꟊ is called generalised anti-fuzzy bi-ideal if ζ(pαqβr) ≤ max{ζ(p),ζ(q),ζ(r)}. The generalised fuzzy bi- ideal ζ of Ꟊ is called an anti-fuzzy bi-ideal if ζ(pαqβr) ≤ max{ζ(p),ζ(q),ζ(r)}. 4. Methods 4.1.Star fuzzing Semigroups: let ξ ∈ 𝐹𝑇 and x ∈ S then we defined ξ* = {x∈s such that ξ(x) = ξ(0)} 1. Lemma: Let ξ ∈ 𝐹𝑇 (S) then (i) (ξi )*⊆ξ* for all i ∈∞ (ii) (ξ(i) )* ⊆ ξ* Let x ∈ (ξi )* Then ξi (x) = ξi (0) = ξ(0) ∵ξi (x) ≤ ξ(1) ξ(x) = ξ(0) Then x∈ξ* Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 639 https://internationalpubls.com Here (ξi )* ⊆ ξ* Similarly we can prove (ii) (ξ(i) )* ⊆ ξ* 2. Lemma: Let ξ ∈ F(S) and K∈N then 𝜉𝐾+1(𝑥1, 𝑥2, … , 𝑥𝑘+1) = Λ𝑖=1 𝐾+1𝜉(𝑥𝑖) If x1,x2,…,xk ∈ S Then ξK (x1,x2,…,xk)≥𝛬(𝑖 = 1) 𝐾 ξ(xi) Clearly, the result is valid for K = 1 Assume that it is valid for K ≥ 1 Now ξ(K+1) (x1,x2,…,xk+1) = ξK. ξ(x1,x2,…,xk+1 ) ⋏ ξ(x(k+1)) = 𝛬(𝑖 = 1) 𝐾 ξ(xi)⋏ξ(x(k+1)) = 𝛬(𝑖 = 1) 𝐾 ξ(xi) 3. Lemma: Let ξ,µ∈F(S) Prove that 𝜉∗⋂µ ∗ ⊆ (𝜉 ∩ µ)∗ Let x∈𝜉∗⋂µ* Then ξ(x) = ξ(0) and µ(x) = µ(0) Now (ξ∩µ)(x) = ξ(x)∩µ(x) = ξ(0)∩µ(0) = (ξ∩µ)(0) x∈(ξ∩µ)* Hence 𝜉∗ ∩ µ ∗ ⊆ (𝜉 ∩ µ)∗ Definition 4.1: Let ξ, ζ∈FS define ζ⨁ξ as follows ( 𝜁⨁𝜉)(𝑥) = ∨ { 𝜁(𝑦) ∧ 𝜉(𝑧)/𝑦 ∈ 𝑆, 𝑦 + 𝑧 = 𝑥} ( 𝜁 ⊙ 𝜉)(𝑥) =∨ { Λ𝑖=1 𝑛 𝜁(𝑟𝑖) ⋏ 𝜉(𝑥𝑖)/𝑟𝑖 ∈ 𝑅1 𝑥𝑖 ∈ 𝑆, ℎ ≤ 𝑖 ≤ 𝑛, 𝑛 ∈ 𝑁, ∑ 𝑟𝑖𝑥𝑖 = 𝑥ℎ 𝑖=1 } 4. Lemma: 1. (𝑟𝜁)(𝑟𝑥) ≥ 𝜁(𝑥); ∀𝑥 ∈ 𝑆 2. 𝜉(𝑟𝑥) ≥ 𝜁(𝑥) ∀ 𝑥 ∈ 𝑆 = > 𝑟 𝜁 ⊆ 𝜉 3. (𝑟𝜁 + 𝑠𝜉)(𝑟𝑥 + 𝑠𝑦) ≥ 𝜁(𝑥) ∧ 𝜉(𝑦) ∀𝑥, 𝑦 ∈ 𝑆 4. 𝜉(𝑟𝑥 + 𝑠𝑦) ≥ 𝜁(𝑥) ∧ 𝜎(𝑦) ∀𝑥, 𝑦 ∈ 𝑆 = > 𝑟𝜁 + 𝑆𝜎 ⊆ 𝜉 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 640 https://internationalpubls.com Proof: (1). (𝑥𝜁)(𝑟𝑥) ≥ 𝜁(𝑥) ∀𝑥 ∈ 𝑆 (𝑥𝜁)(𝑟𝑥) = 𝑉 {𝜁(𝑦) ⁄ 𝑦 ∈ 𝑆, 𝑟𝑦 = 𝑟𝑥} ≥ ζ(x) ∀x∈S (2). 𝐼𝑓 𝜉(𝑟𝑥) ≥ 𝜁(𝑥) ∀𝑥 ∈ 𝑆 Then (𝑟𝜁)(𝑥) = 𝑉{𝜁(𝑦) ⁄ 𝑦 ∈ 𝑆, 𝑟𝑦 = 𝑥} ≤ 𝑉{𝜉(𝑟𝑦) ⁄ 𝑦 ∈ 𝑆, 𝑟𝑦 = 𝑥} ≤ 𝜉(𝑥) ∀ 𝑥 ∈ 𝑦 𝑟𝜁 ⊆ 𝜉 𝐼𝑓 𝑟𝜁 ⊆ 𝜉 Then 𝜉(𝑟𝑥) ≥ 𝑟𝜁(𝑟𝑥) ≥ 𝜁(𝑥) ∀𝑥 ∈ 𝑆 (3). By definition of rζ consider ⨁ (𝑟𝜁 + 𝑠𝜉)(𝑟𝑥 + 𝑠𝑦) ≥ (𝑟𝜁)(𝑟𝑥) ∧ (𝑠𝜉)(𝑠𝑦) ≥ 𝜁(𝑥) ∧ 𝜉(𝑦) ∀ 𝑥, 𝑦 ∈ 𝑆 (4). Suppose that 𝜉(𝑟𝑥 + 𝑠𝑦) ≥ 𝜁(𝑥) ∧ 𝜎(𝑦) ∀ 𝑥, 𝑦 ∈ 𝑆 Then (𝑟𝜁 + 𝑠𝜎)(𝑧) = 𝑉{(𝑟𝜁)(𝑢) ∧ (𝑠𝜎)((𝑣)) ⁄ (𝑢, 𝑣) ∈ 𝑆; 𝑢 + 𝑣 = 𝑧} = 𝑉{(𝑉{𝜁(𝑥)│𝑥 ∈ 𝑆, 𝑟𝑥 = 𝑢}) ∧ (𝑉{𝜎(𝑦 ) ⁄ 𝑦 ∈ 𝑆, 𝑠𝑦 = 𝑣}) ⁄ 𝑢, 𝑣 ∈ 𝑆, 𝑢 + 𝑣 = 𝑧} = 𝑉{𝜁(𝑥) ∧ 𝜎(𝑦)|𝑥, 𝑦 ∈ 𝑆, 𝑟𝑥 + 𝑠𝑦 = 𝑧} = 𝜉(𝑥) ∀ 𝑧 ∈ 𝑆 Hence 𝑟𝜁 + 𝑠𝜎 ⊆ 𝜉 Conversely, suppose that. 𝑟𝜁 + 𝑠𝜎 ⊆ 𝜉 𝜉(𝑟𝑥 + 𝑠𝑦) ≥ (𝑟𝜁 + 𝑠𝜎)(𝑟𝑥 + 𝑠𝑦) ≥ (𝑟𝜁)(𝑟𝑥) ∧ (𝑠𝜎)(𝑠𝑦) ≥ 𝜁(𝑥) ∧ 𝜎(𝑦)[∵ (1)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 641 https://internationalpubls.com ∀ 𝑥, 𝑦 ∈ 𝑆 5. Lemma: Prove that (𝜁 ⊙ 𝜎)(𝑥 + 𝑦) ≥ (𝜁 ⊙ 𝜎)(𝑥) ∧ (𝜁 ⊙ 𝜎)(𝑦)(𝜁 ⊙ 𝜎)(𝑟𝑥) = 𝑉{𝛬(𝑖 = 1) 𝑛 𝜁(𝑠𝑖) ∧ 𝜁(𝑧𝑖)𝑠𝑖, 𝑧 ∈ 𝑆; 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ∈ 𝑁, ∑ 𝑠𝑖 𝑧𝑖 𝑛 𝑖=1 = 𝑟𝑥} ≥ Ѵ{𝛬(𝑖 = 1) 𝑛 (𝜁(𝑟 𝑟𝑖) ∧ 𝜎(𝑥𝑖 ))𝑟𝑖 , 𝑥𝑖 ∈ 𝑆; 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ∈ 𝑁, ∑ (𝑟𝑟𝑖)𝑥𝑖 𝑛 𝑖=1 = 𝑟𝑥} ≥ Ѵ{𝛬(𝑖 = 1) 𝑛 (𝜁( 𝑟𝑖) ∧ 𝜎(𝑥𝑖 ))𝑟𝑖 , 𝑥𝑖 ∈ 𝑆; 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ∈ 𝑁, ∑ (𝑟𝑟𝑖)𝑥𝑖 𝑛 𝑖=1 = 𝑟𝑥} = (𝜁 ⊙ 𝜎)(𝑥) … … … . . (1) Similarly (𝜁 ⊙ 𝜎)(𝑟𝑦) ≥ (𝜁 ⊙ 𝜎)(𝑦) … … … … (2) By definition additive of (1) & (2) (𝜁 ⊙ 𝜎)(𝑟𝑥 + 𝑟𝑦) ≥ (𝜁 ⊙ 𝜎)(𝑥) ∧ (𝜁 ⊙ 𝜎)(𝑦) Put 𝑟 = 1 (𝜁 ⊙ 𝜎)(𝑥 + 𝑦) ≥ (𝜁 ⊙ 𝜎)(𝑥) ∧ (𝜁 ⊙ 𝜎)(𝑦) Definition 4.2: [13] A fuzzy ideal ζ of a Γ- Semigroup Ꟊ is called a fuzzy prime ideal if 𝜁(𝑥𝛾𝑦)(𝛾∈𝛤) 𝑖𝑛𝑓 = 𝑚𝑎𝑥 { 𝜁(𝑥), 𝜁(𝑦) } for all x,y ∈ Ꟊ and γ∈ Γ. Example: Let Ꟊ be the set of all 1x2 matrices, Γ be the set of all 2x1 matrices, then Ꟊ is Γ-Semigroup where p,q∈Ꟊ; α,β∈Γ Which denotes the usual matrix product. Let ζ: Ꟊ → [0,1] be defined by 𝜁(𝑝) = { 0.1 𝑖𝑓 [0 0] 0.3 0.𝑤 let 𝑝 = [1 0] ; 𝑞 = [0 0] and 𝛾 = [ 1 1 ] then 𝜁(𝑝𝛾𝑞) = 𝜁 ([1 0] [ 1 1 ] [0 0]) = 𝜁([1][0 0]) = 𝜁([0 0]) = 0.1 𝑀𝑎𝑥 { 𝜁(𝑝), 𝜁(𝑞) } = 𝑚𝑎𝑥 {0.3,0.1} = 0.3 𝜁(𝑥𝛾𝑦) ≤ 𝑚𝑎𝑥 { 𝜁(𝑥), 𝜁(𝑦) } Definition 4.3: A fuzzy ideal ζ of a ternary Γ- Semigroup Ꟊ is called a fuzzy prime ideal if inf¦(γ∈Γ) ζ(pαqβr) = max { ζ(p), ζ(q),ζ(r) } for all p,q,r ∈ Ꟊ and γ∈ Γ. Example: Let Ꟊ be the set of all 1x2 matrices, Γ be the set of all 2x1 matrices, then Ꟊ is ternary Γ- Semigroup where p,q∈Ꟊ; α,β∈Γ which denotes the usual matrix product. Let ζ: Ꟊ → [0,1] be defined by 𝜁(𝑝) = { 0.5 𝑖𝑓 [0 0] 0.9 𝑖𝑓 [1 2] 0.7 0. 𝑤 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 642 https://internationalpubls.com let 𝑝 = [0 0] ; 𝑞 = [1 2] ; 𝑟 = [2 1] 𝑎𝑛𝑑 𝛼 = [ 1 1 ] ; 𝛽 = [ 1 0 ] then 𝜁(𝑝𝛼𝑞𝛽𝑟) = 𝜁 ([0 0] [ 1 1 ] [1 2] [ 1 0 ] [2 1]) = 𝜁([0][1][2 1]) = 𝜁([0 0]) = 0.5 𝑚𝑎𝑥 { 𝜁(𝑝), 𝜁(𝑞), 𝜁(𝑟) } = 𝑚𝑎𝑥 {0.5,0.9,0.7} = 0.9 𝜁(𝑝𝛼𝑞𝛽𝑟) ≤ 𝑚𝑎𝑥 { 𝜁(𝑝), 𝜁(𝑞), 𝜁(𝑟) } Definition 4.4:[18] A fuzzy ideal ζ of a Γ- Semigroup Ꟊ is called a fuzzy Semiprime ideal if ζ(x) ≥ 𝑖𝑛𝑓 𝛾∈𝛤 ζ(xγx). Definition 4.5: A fuzzy ideal ζ of a ternary Γ- Semigroup Ꟊ is called a fuzzy Semiprime ideal if 𝜁(𝑥) ≥ 𝑖𝑛𝑓 𝛼,𝛽∈𝛤 𝜁(𝑥𝛼𝑥𝛽𝑥) . 6.Lemma: [19] Let Ꟊ be a Γ-Semigroup and ∅≠I⊆ Ꟊ. Then I is a prime ideal (Semiprime ideal) of Ꟊ iff 𝜁𝐼 is a fuzzy prime ideal (respectively fuzzy Semiprime ideal) of Ꟊ, where 𝜁𝐼 is the characteristic function of I. Definition 4.6: [35] Let Ꟊ be Γ- Semigroup, ζ be a fuzzy subset of Ꟊ and x∈ Ꟊ then the fuzzy subset < x, ζ > :Ꟊ→ [0,1] defined by < 𝑥, 𝜁 > (𝑦) = 𝑖𝑛𝑓 𝛾∈𝛤 𝜁(𝑥𝛾𝑦) is called the extension of ζ by x. Note: For a fuzzy subset ζ of ℛ(Ω of ℒ ) a fuzzy subset ζ* (Ωº) of Ꟊ by ζ* (𝑎) = 𝑖𝑛𝑓¦(𝛾 ∈ 𝛤) 𝜁 ([𝛾, 𝑎]) ; Ωº (𝑎) = 𝑖𝑛𝑓 𝛾∈𝛤 Ω ([𝑎, 𝛾]) 𝑎𝑛𝑑 ղ ∗′([𝛼,𝑎])= 𝑖𝑛𝑓 𝑠∈Ꟊ ղ (𝑠𝛼𝑎) ; ղ𝑜′([𝑎,𝛼]) = 𝑖𝑛𝑓 𝑠∈Ꟊ ղ (𝑎𝛼𝑠). 7. Lemma: Let ζ be a non-empty fuzzy subset of commutative Γ-Semigroup Ꟊ, then for all x∈Ꟊ (𝑖) < 𝑥, 𝜁 >∗′ ⊆ < [𝛼 , 𝑥], 𝜁∗1 > 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼𝜖𝛤 (𝑖𝑖) < 𝑥, 𝜁 >∗′ = 𝑖𝑛𝑓 𝛼𝜖𝛤 < [𝑥, 𝛼 ], 𝜁∗1 > 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼𝜖𝛤 (i) Let [β,y] ∈ Ꟊ then < 𝑥, 𝜁 >∗′ ([𝛽, 𝑦]) = 𝑖𝑛𝑓𝑠∈𝑆 < 𝑥, 𝜁 > (𝑠𝛽𝑦) = 𝑖𝑛𝑓𝑠∈𝑆 𝑖𝑛𝑓𝛾∈𝛤𝜁(𝑥𝛾𝑠𝛽𝑦) Again < [𝛼, 𝑥], 𝜁∗′ > ([𝛽, 𝑦]) = 𝜁∗([𝛼, 𝑥][𝛽, 𝑦]) = 𝜁∗′([𝛼, 𝑥𝛽𝑦]) = 𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑠𝛼𝑥𝛽𝑦) 𝑆𝑖𝑛𝑐𝑒 𝑖𝑛𝑓𝛾∈𝛤𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑥𝛾𝑠𝛽𝑦) ≤ 𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑥𝛼𝑠𝛽𝑦) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 643 https://internationalpubls.com < 𝑥, 𝜁 >∗′ ([𝛽, 𝑦]) ≤ < [𝛼, 𝑥], 𝜁∗′ > ([𝛽, 𝑦]) 𝐻𝑒𝑛𝑐𝑒, < 𝑥, 𝜁 >∗ ⊆ < [𝛼, 𝑥], 𝜁∗′ > (ii) Let [β,y]∈R then 𝑖𝑛𝑓 < [𝛼, 𝑥], 𝜁∗′ > ([𝛽, 𝑦]) = 𝑖𝑛𝑓𝛼∈𝛤𝜁∗′([𝛼, 𝑥] [𝛽, 𝑦]) = 𝑖𝑛𝑓𝛼∈𝛤𝜁∗′([𝛼, 𝑥𝛽𝑦]) = 𝑖𝑛𝑓𝛼∈𝛤 𝑖𝑛𝑓𝑠∈𝑆 𝜁([𝑠𝛼𝑥𝛽𝑦]) = 𝑖𝑛𝑓𝑠∈𝑆 < 𝑥, 𝜁 > (𝑠𝛽𝑦) =< 𝑥, 𝜁 >∗′ ([𝛽, 𝑦]) 𝑇ℎ𝑢𝑠 < 𝑥, 𝜁 >∗′= 𝑖𝑛𝑓𝛼∈𝛤 < [𝛼, 𝑥], 𝜁∗′ > 5. Results 1. let σ be a non-empty fuzzy subset of the right operator semi-groups of a Γ-Semigroup Ꟊ. then ∀ 𝑥 ∈ Ꟊ, < [𝛽, 𝑥], 𝜎 >∗ ≥ < 𝑥, 𝜎∗ > ∀𝛽 ∈ 𝛤 Let p∈Ꟊ then < [𝛽, 𝑥], 𝜎 >∗ (𝑝) = 𝑖𝑛𝑓𝛼∈𝛤 < [𝛽, 𝑥], 𝜎 > ([𝛾, 𝑝]) = 𝑖𝑛𝑓𝛼∈𝛤 𝜎([𝛽, 𝑥] [𝛾, 𝑝]) = 𝑖𝑛𝑓𝛼∈𝜎 𝜎([𝛽, 𝑥𝛾𝑝] ) 𝐴𝑔𝑎𝑖𝑛 < 𝑥, 𝜎∗ > (𝑝) = 𝑖𝑛𝑓𝛾∈𝛤𝜎∗(𝑥𝛾𝑝) = 𝑖𝑛𝑓𝛾∈𝛤 𝑖𝑛𝑓𝛽∈𝛤 𝜎[(𝛽, 𝑥𝛾𝑝)] = 𝑖𝑛𝑓𝛽∈𝛤 𝑖𝑛𝑓𝛾∈𝛤 𝜎[(𝛽, 𝑥𝛾𝑝)] ∵ 𝑖𝑛𝑓𝛾∈𝛤 𝜎[(𝛽, 𝑥𝛾𝑝)] ≥ 𝑖𝑛𝑓𝛽∈𝛤 𝑖𝑛𝑓𝛾∈𝛤 𝜎[(𝛽, 𝑥𝛾𝑝)] We have < [𝛽, 𝑥], 𝜎 >∗ (𝑝) ≥ < 𝑥, 𝜎∗ > (𝑝) Consequently < [𝛽, 𝑥], 𝜎 >∗ ⊇ < 𝑥, 𝜎∗ > 2. let {𝐴𝛼}𝛼∈𝐴 be a family of ideals of a Γ-semigroup Ꟊ. Then 〖(∩𝛼∈∧ 𝐴𝛼)∗′ = ∩𝛼∈∧ 𝐴𝛼 ∗ ′ Let [𝛼, 𝑥](∩𝛼∈∧ 𝐴𝛼)∗′ Then 𝑠𝛼𝑥 ∈ ∩𝛼∈∧ 𝐴𝛼 ∀ 𝑠 ∈ 𝑆 Here 𝑠 ∈ 𝑆 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 644 https://internationalpubls.com 𝑠𝛼𝑥 ∈ 𝐴𝛼 ∀ 𝛼 ∈∧ [𝛼, 𝑥] ∩𝛼∈∧ 𝐴𝛼 ∗′ Here (∩𝛼∈∧ 𝐴𝛼)∗′ ⊆ ∩𝛼∈∧ 𝐴𝛼 ∗′ − (1) We can deduce that ∩𝛼∈∧ 𝐴𝛼 ∗′ ⊆ (∩𝛼∈∧ 𝐴𝛼)∗′ [𝛼, 𝑥] ∈∩𝛼∈∧ 𝐴𝛼 ∗′ Then 𝑠𝛼𝑥 ∈ 𝐴𝛼 ∀ 𝛼 ∈∧ 𝑠𝛼𝑥 ∈ ∩𝛼∈∧ 𝐴𝛼 ∀ 𝑠 ∈ 𝑆 [𝛼, 𝑥] ∈ (∩𝛼∈∧ 𝐴𝛼)∗ ′ ∩𝛼∈∧ 𝐴𝛼 ∗′ ⊆ (∩𝛼∈∧ 𝐴𝛼 )∗′ − (2) From (1) & (2) we get ∩𝛼∈∧ 𝐴𝛼 ∗′ = (∩𝛼∈∧ 𝐴𝛼)∗′ 3. Let Ꟊ be a Γ-semigroups, R its right operator Semigroup and 𝜉 = 𝑖𝑛 𝑓{𝜉𝑖: 𝑖 ∈ 𝐼} a non- empty family of the fuzzy subset of Ꟊ. Then 𝜉∗′ = 𝑖𝑛 𝑓 {𝜉 𝑝∗′: 𝑖 ∈ 𝐼} fuzzy ideal extension of Γ-semigroup. Let [α,x]∈R then 𝜉∗′[𝛼, 𝑥] = 𝑖𝑛 𝑓{𝜉𝑖: 𝑖 ∈ 𝐼}∗′ [𝛼, 𝑥] = 𝑖𝑛𝑓𝑠∈𝑆(𝑖𝑛 𝑓{𝜉𝑖: 𝑖 ∈ 𝐼}(𝑠𝛼𝑥)) = 𝑖𝑛𝑓𝑠∈𝑆 𝑖𝑛𝑓𝑖∈𝐼𝜉𝑖(𝑠𝛼𝑥) Now 𝑖𝑛𝑓 {𝜉𝑖 ∗′ 𝑖 ∈ 𝐼} [𝛼, 𝑥] = 𝑖𝑛𝑓𝑖∈𝐼 (𝜉𝑖 ∗′[𝛼, 𝑥]) = 𝑖𝑛𝑓𝑖∈𝐼𝑖𝑛𝑓𝑠∈𝑆𝜉𝑖 ∗′(𝑠𝛼𝑥) = 𝑖𝑛𝑓𝑖∈𝐼𝑖𝑛𝑓𝑠∈𝑆𝜉𝑖(𝑠𝛼𝑥) = 𝑖𝑛𝑓𝑠∈𝑆𝑖𝑛𝑓𝑖∈𝐼𝜉𝑖(𝑠𝛼𝑥) ∴𝜉∗′ is the fuzzy ideal extension of Γ-semigroup. 4. Suppose ξ:S→[0,1] the fuzzy subset of semigroup and then (i) (𝜉𝑖 ) ∗ = ( 𝜉∗)𝑖 ∀ 𝑖 ∈ 𝑁 (𝑖𝑖) (𝜉(𝑖)) ∗ = ( 𝜉∗)𝑖 Proof: (i) if 𝑖 = 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 645 https://internationalpubls.com Then (𝜉1)∗ = (𝜉∗)1 → 𝜉∗ = 𝜉∗ ∴ 𝑖 = 1 the results are true We assume that it is true for 𝑖 > 1 Let 𝑥 𝜖 (𝜉𝑖+1 ) ∗ Then 𝜉𝑖+1(𝑥) = 𝜉𝑖+1 (0) = 𝜉(0) But 𝜉𝑖+1 (𝑥) = 𝑉{𝜉𝑖 (𝑦)𝜉(𝑧) /𝑦, 𝑧 ∈ 𝑆 𝑥 = 𝑦𝑧 } 𝜉(0) Thus 𝜉𝑖 (𝑦) = 𝜉(0) = 𝜉(𝑧) for some 𝑦, 𝑧 ∈ 𝑆 ∵ 𝜉𝑖 (𝑦) ≤ 𝜉𝑖 (0) = 𝜉(0) 𝜉𝑖 𝑖s finite-valued ∀ 𝑖 ≥ 1 Thus 𝑥 = 𝑦𝑧 for some 𝑦 ∈ (𝜉𝑖 ) ∗ = (𝜉∗ )𝑖 and 𝑧 ∈ 𝜉∗ Hence 𝑥 ∈ (𝜉∗ )𝑖 𝜉∗ = (𝜉∗ )𝑖+1 Thus (𝜉(𝑖+1)) ∗ ⊆ (𝜉∗ )(𝑖+1) Now let 𝑥 = ∑ 𝑥𝑗1 , 𝑥𝑗2 … 𝑥𝑗1 𝑟 (𝑗=1) ∈ (𝜉∗ )𝑖+1 Where 𝑥𝑗𝑘 ∈ 𝜉∗∀ 𝑘 = 1, 2, … 𝑖 + 1 𝑗 = 1,2, … 𝑟 Then 𝜉𝑖+1 (𝑥) ≥ ⋀ 𝜉𝑖+1(𝑥𝑗1 𝑥𝑗2 … 𝑥𝑗+1)𝑟 𝑗 = 1 ≥ ⋀ (⋀ 𝜉(𝑥𝑗𝑘)𝑖+1 𝑘 = 1 ) 𝑟 𝑗 = 1 = 𝜉(0) Thus 𝜉𝑖+1 (𝑥) = 𝜉(0) = 𝜉𝑖+1(0) Hence 𝑥 ∈ (𝜉𝑖+1) ∗ Thus (𝜉∗)𝑖+1 ⊆ (𝜉𝑖+1) ∗ Hence (〖𝜉𝑖+1)〗∗ = (𝜉∗)𝑖+1 Clearly, the result is true for 𝑖 = 1 Assume that it is true for 𝑖 ≥ 1 Let 𝑥 ∈ (𝜉𝑖+1) ∗ Then 𝜉𝑖+1(𝑥) = 𝜉𝑖+1(0) = 𝜉(0) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 646 https://internationalpubls.com But 𝜉𝑖+1(𝑥) = 𝑉{⋀ 𝜉𝑖(𝑦𝑘) ∧ 𝜉(𝑧𝑘))𝑛 𝑘=1 / 𝑦𝑘, 𝑧𝑘 ∈ 𝑆, 1 ≤ 𝑘 ≤ 𝑛, 𝑛 ∈ 𝑁;∑ 𝑦𝑘𝑧𝑘 = 𝑥 } = 𝑛 𝑘=1 𝜉(0) ∵ 𝜉𝑖 is finite-valued ∀ 𝑖 ≥ 1 This implies that 𝜉𝑖(𝑦𝑘) = 𝜉(0) = 𝜉(𝑧𝑘) ∀𝑘, 1 ≤ 𝑘 ≤ 𝑛 Thus 𝑧𝑘 ∈ 𝜉𝑘 ∀ 𝑘, 1 ≤ 𝑘 ≤ 𝑛 Also 𝜉(0) ≥ 𝜉(𝑖)(0) ≥ 𝜉(𝑖)(𝑦𝑘) = 𝜉(0) So 𝜉(𝑖)(0) = 𝜉(𝑖)(𝑦𝑘) i.e. 𝑦𝑘 ∈ (𝜉𝑖) ∗ ∀𝑘, 𝑖 ≤ 𝑘 ≤ 𝑛 hence 𝑥 = ∑ 𝑦𝑘𝑧𝑘 𝑛 𝑘 = 1 ∈ (𝜉∗)(𝑖+1)𝑤ℎ𝑒𝑟𝑒 𝑥𝑗𝑘 ∈ 𝜉∗∀ 𝑘 = 1,2, … 𝜉(𝑖+1)(𝑥) ≥ ⋀ 𝜉(𝑖+1)(𝑥𝑗1 𝑥𝑗2 … 𝑥𝑗+1)𝑟 𝑗 = 1 ≥ ⋀ ⋀ 𝜉(𝑥𝑗𝑘)𝑖+1 𝑘 = 1 𝑟 𝑗 = 1 = 𝜉(0) Thus 𝜉(𝑖+1)(𝑥) = 𝜉(0) = 𝜉(𝑖+1)(0) Hence 𝑥 ∈ (𝜉(𝑖+1)) ∗ Thus (𝜉∗)𝑖+1 ⊆ (𝜉(𝑖+1)) ∗ Hence (𝜉(𝑖+1)) ∗ = (𝜉∗)𝑖+1 Definition 6.1: Let 𝜉 ∈ 𝐹𝑆 define 𝜉𝑛 and 𝜉𝑛 as follows when 𝑥 ∈ 𝑁, 𝑛 > 1; 𝜉𝐼 = 𝜉 𝑎𝑛𝑑 𝜉𝑛 = 𝜉1 ° 𝜉𝑛−1 𝜉(𝐼) = 𝜉 𝜉(𝑛) = 𝜉(1)𝜉(𝑛−1) Definition 6.2: Let 𝜉1, 𝜉2 𝐹𝑆. Define 𝜉1𝜉2 ∈ 𝐹𝑆 ∀ 𝑥 ∈ 𝑆 (𝜉1𝜉2)(𝑥) = 𝑉{⋀ (𝜉(𝑦𝑖) ∧ 𝜉(𝑧𝑖))| 𝑦𝑖𝑧𝑖 ∈ 𝑆𝑛 𝑖=1 and 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ∈ 𝑁, ∑ 𝑦𝑖𝑧𝑖 = 𝑥𝑛 𝑖=1 } 5. If S is commutative then 𝜉1𝜉2 = 𝜉2𝜉1 (𝜉1𝜉2 )(𝑥) = ∨ {⋀ (𝜉1(𝑦𝑖) ∧ 𝜉2(𝑧𝑖)|𝑦𝑖, 𝑧𝑖 ∈ 𝑆𝑛 𝑖 = 1 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ∈ 𝑁, ∑ 〖𝑦𝑖𝑧𝑖 = 𝑥}𝑛 𝑖 = 1 = ∨ {⋀ (𝜉2(𝑧𝑖) ∧ 𝜉1(𝑦𝑖) |𝑛 𝑖 = 1 𝑦𝑖 , 𝑧𝑖 ∈ 𝑆 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ∈ 𝑁, ∑ 〖𝑧𝑖𝑦𝑖 = 𝑥}𝑛 𝑖 = 1 = (𝜉2𝜉1)(𝑥) 𝜉1𝜉2 = 𝜉2𝜉1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 647 https://internationalpubls.com 𝜉∗ = {𝑥 ∈ 𝑅|𝜉(𝑥) = 𝜉(0)} 6. Let 𝜉 ∈ 𝐹𝑆 and 𝑘 ∈ 𝑁, 𝑖𝑓 𝑥1, 𝑥2, … , 𝑥𝑘 ∈ 𝑆 Then 1. 𝜉∗ 𝑘(𝑥1, 𝑥2, … , 𝑥𝑘) ≥ ⋀ 𝜉∗(𝑥𝑖)𝑘 𝑖 = 1 2. 𝜉∗ (𝑘)(𝑥1, 𝑥2, … , 𝑥𝑘) ≥ ⋀ 𝜉∗(𝑥𝑖)𝑘 𝑖 = 1 Proof: 1. Clearly, the result is true for 𝑘 = 1 Assume that it is true for 𝑘 > 1 Now 𝜉∗ 𝑘+1(𝑥1, 𝑥2, … , 𝑥𝑘+1) = (𝜉∗ 𝑘 ∘ 𝜉∗)(𝑥1, 𝑥2, … , 𝑥𝑘+1) ≥ 𝜉∗ 𝑘(𝑥1, 𝑥2, … , 𝑥𝑘) ∧ 𝜉∗(𝑥𝑘+1) ≥ 𝜉∗ 𝑘(0,0, … ,0) ∧ 𝜉∗(𝑥𝑘+1) 𝜉∗ 𝑘(𝑥1, 𝑥2, … , 𝑥𝑘) ≥ ⋀ 𝜉∗(𝑥𝑖)𝑘 𝑖 = 1 2. this proof follows the same as lemma 1. 6. Discussion A fuzzy ternary gamma Semigroup is a mathematical structure that combines concepts from fuzzy set theory, ternary algebra, and gamma Semigroup. We discussed that fuzzy ternary gamma semigroups can be used to model fuzzy inference Systems. Fuzzy ternary gamma semigroups can be applied to computer science, particularly in studying fuzzy automata and fuzzy languages. Fuzzy ternary gamma semigroup may have applications in cryptography, especially in developing fuzzy cryptographic protocols. New problems are explored in studying fuzzy ternary gamma semigroups, a relatively new research area. ❖ Clarify Key Terms: While "star zeta" and "gamma semigroup" are central concepts, a brief clarification of their role in fuzzy semigroups could help readers unfamiliar with the topic. ❖ Strengthen Logical Progression: Ensure the discussion flows from definitions to properties, followed by implications and the collaborative nature of the work. References [1] Razaq, A., & Alhamzi, G. (2023). On Pythagorean fuzzy ideals of a classical ring. AIMS Math, 8(2), 4280-4303. [2] Clifford A.H and G. B. Preston, The Algebraic Theory of Semigroups, Vol. I, Mathematical Surveys No. 7, Amer. Math. Soc., Providence, 1961. [3] Clifford A.H and G. B. Preston, The Algebraic Theory of Semigroups, Vol. II, Mathematical Surveys No. 7, Amer. Math Soc., Providence, 1967. [4] J.M Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995. [5] J.M Howie An Introduction to Semigroup theory, academic press, London, 1976. IJIRT 154793 623. [6] Vasantha G, T SriLakshmi; “Important role of idempotent and regular classes in the distinguished semigroup theory” in VOLUME 15 – ISSUE VIII AUGUST 2022 – JAC: A JOURNAL OF COMPOSITION THEORY(JCT). https://jctjournal.com/volume-15-issue-viii-august-2022;ISSN NO.:0731-6755. [7] Vasantha G, T Sri Lakshmi; Background History of Semigroup Theory in Algebra in “The International Journal of Innovative Research in Technology”. IJIRT | Volume 8 Issue 12 | ISSN: 2349- 6002;htt://ijirt.org/master/publishedpaper/IJIRT154793_PAPER.pdf. [8] Ponizovskii, J.S.. "On a type of matrix semigroup.." Semigroup Forum 44.1 (1992): 125 128. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 648 https://internationalpubls.com Hollings, C. (2009). The Early Development of the Algebraic Theory of Semigroups. Archive for History of Exact Sciences, 63, 497-536 [9] ZADEH. L. (1965). FUZZY SETS, INFORMATION, AND CONTROL, 338 -3353. [10] Rosenfeld, A. (1971). Fuzzy groups. Journal of Mathematical Analysis and Applications, 35, 512–517. [11] Imtiaz, A., Alolaiyan, H., Shuaib, U., Razaq, A., & Liu, J. B. (2024). Applications of conjunctive complex fuzzy subgroups to Sylow theory. AIMS Mathematics, 9(1), 38-54. [12] Al-Masarwah, A., & Alqahtani, M. (2023). Operational algebraic properties and subsemigroups of semigroups given k-folded N-structures. AIMS Math, 8(9), 22081-22096. [13] Preethi, D., Vimala, J., & Rajareega, S. (2020). A systematic study in the applications of fuzzy hyperlattice. AIMS Mathematics, 6(2), 1695-1705. [14] Ali, A., Mateen, M. H., Xin, Q., Alsuraiheed, T., & Alhamzi, G. (2024). $(\epsilon,\delta) $-complex anti fuzzy subgroups and their applications. AIMS Mathematics, 9(5), 11580-11595. [15] Ali, A., Ameer, E., Aiadi, S. S., Tariq, M., Arshad, M., Mlaiki, N., & Shatanawi, W. (2022). New extension to fuzzy dynamic system and fuzzy fixed point results with an application. AIMS Math, 8, 1208-1229. [16] Ullah, A., Ibrahim, M., & Saeed, T. (2022). Fuzzy cosets in AG-groups. AIMS Mathematics, 7(3), 3321-3344. [17] Sezer, A. S. (2014). A new approach to LA-semigroup theory via the soft sets. Journal of Intelligent & Fuzzy Systems, 26(5), 2483-2495. [18] Feng, X., Tang, J., Davvaz, B., & Luo, Y. (2017). A novel study on fuzzy ideals and fuzzy filters of ordered*- semigroups. Journal of Intelligent & Fuzzy Systems, 33(1), 423-431. [19] Khan, F. M., Bibi, N., Xin, X. L., & Alam, A. (2022). Rough Fermatean fuzzy ideals in semigroups. Journal of Intelligent & Fuzzy Systems, 42(6), 5741-5752. [20] Yiarayong, P. (2021). On 2-absorbing bipolar fuzzy ideals over LA-semigroups. Journal of Intelligent & Fuzzy Systems, 41(2), 3173-3181. [21] Zhang, X., Wu, X., Mao, X., Smarandache, F., & Park, C. (2019). On neutrosophic extended triplet groups (loops) and Abel-Grassmann’s groupoids (AG-groupoids). Journal of Intelligent & Fuzzy Systems, 37(4), 5743-5753. [22] Budimirović, B., Budimirović, V., Šešelja, B., & Tepavčević, A. (2014). Fuzzy identities with application to fuzzy semigroups. Information Sciences, 266, 148-159. [23] Yiarayong, P. (2022). On bipolar-valued fuzzy quasi-semiprime ideals of LA-semigroups. Afrika Matematika, 33(3), 81. [24] Jun, Y. B., Song, S. Z., & Muhiuddin, G. (2016). Hesitant fuzzy semigroups with a frontier. Journal of Intelligent & Fuzzy Systems, 30(3), 1613-1618. [25] Habib, S., Muhammad Khan, F., & Yufeng, N. (2019). A new concept of possibility fuzzy soft ordered semigroups via its applications. Journal of Intelligent & Fuzzy Systems, 36(4), 3685-3696. [26] Muhiuddin, G., Mahboob, A., Khan, N. M., & Al-Kadi, D. (2021). New types of fuzzy (m, n)-ideals in ordered semigroups. Journal of Intelligent & Fuzzy Systems, 41(6), 6561-6574. [27] Khan, A., Sarmin, N. H., Davvaz, B., & Khan, F. M. (2012). New types of fuzzy bi-ideals in ordered semigroups. Neural Computing and Applications, 21(Suppl 1), 295-305. [28] Li, C., Xu, B., & Huang, H. (2020). Bipolar fuzzy abundant semigroups with applications. Journal of Intelligent & Fuzzy Systems, 39(1), 167-176. [29] Rehman, N., & Shabir, M. (2014). Some characterizations of ternary semigroups by the properties of their $\left (\in _ {\gamma},\in _ {\gamma}\vee q_ {\delta}\right) $-fuzzy ideals. Journal of Intelligent & Fuzzy Systems, 26(5), 2107-2117. [30] Li, C., Xu, B., & Huang, H. (2021). A new characterisation of fuzzy ideals of semigroups and its applications. Automatika, 62(3-4), 407-414. [31] Anis, Saima, Madad Khan, and Young Bae Jun. "Hybrid ideals in semigroups." Cogent Mathematics 4.1 (2017): 1352117. [32] Mursaleen, M., Srivastava, H. M., & Sharma, S. K. (2016). Generalised statistically convergent sequences of fuzzy numbers. Journal of Intelligent & Fuzzy Systems, 30, 1511–1518. [33] Torra, V. (2010). Hesitant fuzzy sets. International Journal of Computational Intelligence Systems, 25, 529–539 [34] Torra, V., & Narukawa, Y. (2009). On hesitant fuzzy sets and decisions. In The 18th IEEE International Conference on Fuzzy Systems (1378–1382). Jeju Island Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 649 https://internationalpubls.com [35] Hedayati, 2012 intuitionistic (S, T)-fuzzy (1, 2)-ideals of semigroups with interval-valued membership functions HInternational Journal of Fuzzy Systems (2012) 14(1) [36] Singer, 2007Linearly ordered semigroups for fuzzy set theory DAnnals of Mathematics and Artificial Intelligence (2007) 49(1-4). [37] Murthy, A., Deen, M., & Fang, Q. (2015). LED Array Design with Image-Based Planar Verification. International Journal of Electronics Communication and Computer Engineering, 6(5), 565-571. [38] Kumar, V., Gupta, H., & Sharma, S. (2014). Neutralising the Impact of Account or Service Traffic Hijacking in Cloud Computing. International Journal of Electronics Communication and Computer Engineering, 5(1), 206-209.