EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1417-1428 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Connections of Strongest Fuzzy Γ-Ideals on Ternary Γ-Semigroups Warud Nakkhasen1,∗, Onnalin Yangnok1, Kewarin Chaidet1, Wichayaporn Jantanan2 1 Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham 44150, Thailand 2 Department of Mathematics, Faculty of Science, Buriram Rajabhat University, Buriram 31000, Thailand Abstract. The fuzzy relation Rµ on µ, where µ is a fuzzy set of a set X, is called a strongest fuzzy relation on X if Rµ(x, y) = min{µ(x), µ(y)}, for all x, y ∈ X. The notion of strongest fuzzy relations will be applied in our investigation on ternary Γ-semigroups. In order to achieve this, we will define the concepts of strongest fuzzy ternary Γ-subsemigroups, strongest fuzzy Γ-ideals (resp. left, right, and lateral), and strongest fuzzy bi-Γ-ideals on ternary Γ-semigroups. Then, we study the connections and characterizations of these concepts in ternary Γ-semigroups. 2020 Mathematics Subject Classifications: 20M75, 08A72 Key Words and Phrases: Strongest fuzzy relation, Strongest fuzzy Γ-ideal, Γ-Ideal, Ternary Γ-semigroup 1. Introduction The notion of ternary Γ-semigroups was introduced by Madhusudhana Rao et al. [6] in 2015. The ternary Γ-semigroups were generalized the concepts of semigroups, Γ- semigroups and ternary semigroups. They characterized and examined about several some elements of ternary Γ-semigroups. Then Vasantha and Madhusudhana Rao [8] developed and characterized the terms completely semiprime ternary Γ-ideal and semiprime ternary Γ-ideal in ternary Γ-semigroups. After that, Vasantha et al. [11] introduced the concepts of trio L-trio TΓ-ideals, La-trio TΓ-ideals, R-trio TΓ-ideals, and trio TΓ-ideals in trio ternary Γ-semigroups. Afterwards, Ali et al. [1] introduced and discussed some properties of po-bi quasi-Γ-ideals, po-bi-Γ-ideals, and generalized po-bi quasi-Γ-ideals in po-bi-ternary Γ-semigroups. For other research related to ternary Γ-semigroups, additional studies can be done in general (e.g., [7, 9, 10]). ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5309 Email addresses: warud.n@msu.ac.th (W. Nakkhasen), 63010213011@msu.ac.th (O. Yangnok), 63010213018@msu.ac.th (K. Chaidet), wichayaporn.jan@bru.ac.th (W. Jantanan) https://www.ejpam.com 1417 © 2024 EJPAM All rights reserved. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1418 Fuzzy subsets or fuzzy sets are defined by Zadeh [13] as a function from a nonempty set X to the unit interval [0, 1]. This idea is a mathematical extension of the classical sets in mathematics. Then, in 1985, Bhattacharya and Mukherjee [3] proved that a strongest fuzzy relation µσ on a group S is a fuzzy subgroup if and only if σ is a fuzzy subgroup. This concept of strongest fuzzy relations has been studied continuously. Mostafa et al. [5] presented some properties of KU-ideals in terms of strongest fuzzy relations in KU- algebras. Subsequently, the concept of strongest fuzzy relations in the Cartesian product of B-algebras was investigated by Yamini and Kailasavalli [12] in 2014. Following that, Bhargavi et al. [2] gave and analyzed the concept of the Cartesian product of fuzzy sets in ternary Γ-semigroups. In addition, they characterized different types of fuzzy Γ-ideals in terms of their Cartesian product of ternary Γ-semigroups. Recently, Derseh et al. [4] considered some properties of strongest intuitionistic fuzzy PMS-relations on PMS-algebras in 2023. The purpose of this article is applying the fuzzy relation to define the concepts of strongest fuzzy Γ-subsemigroups, strongest fuzzy (resp. left, right, lateral) Γ-ideals, and strongest fuzzy bi-Γ-ideals of ternary Γ-semigroups. Later on, we consider the connections of strongest fuzzy Γ-subsemigroups, strongest fuzzy (resp. left, right, lateral) Γ-ideals, and strongest fuzzy bi-Γ-ideals on ternary Γ-semigroups. 2. Preliminaries In this section, we will review important basic concepts for use in the next section. A fuzzy set [13] µ of a nonempty set X is a mapping form the set X into [0, 1]. The fuzzy relation [3] R on a nonempty set X is a fuzzy set R : X ×X → [0, 1]. Let R be any fuzzy relation on a nonempty set X, and µ be a fuzzy set of X. Then R is said to be a fuzzy relation on µ [3] if R(x, y) ≤ min{µ(x), µ(y)}, for all x, y ∈ X. Definition 1. [3] Let µ be a fuzzy set of a nonempty X, and Rµ be a fuzzy relation on µ. Then Rµ is called a strongest fuzzy relation on X if Rµ(x, y) = min{µ(x), µ(y)}, for all x, y ∈ X. For any strongest fuzzy relation Rµ on a nonempty set X, and for each t ∈ [0, 1], we denote by (Rµ)t the level subset of Rµ where (Rµ)t := {(x, y) | Rµ(x, y) ≥ t} (see [3]). Let X be a nonempty set, and µ be a fuzzy set of X. For any subset A of X, the characteristic function χA µ of A is a strongest fuzzy relation on X defined by for every x, y ∈ X, χA µ (x, y) = { 1 if x, y ∈ A, 0 otherwise. Definition 2. (cf. [6]) Let T and Γ be two nonempty sets. A ternary Γ-semigroup is an algebraic structure (T,Γ, [ ]) if there exist a mapping [ ] : T × Γ× T × Γ× T → T , written as (a, α, b, β, c) → [aαbβc] satisfying the associative law: [[aαbβc]γdδe] = [aα[bβcγd]δe] = [aαbβ[cγdδe]], W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1419 for all a, b, c, d, e ∈ T and α, β, γ, δ ∈ Γ. For the sake of simplicity, we will write aαbβc instead of [aαbβc], for each a, b, c ∈ T and α, β ∈ Γ. Let A,B and C be any nonempty subsets of a ternary Γ-semigroup T . We denote the set AΓBΓC := {aαbβc | a ∈ A, b ∈ B, c ∈ C,α, β ∈ Γ}. We now review the concepts of various kinds of Γ-ideals and fuzzy Γ-ideals in ternary Γ-semigroups that appeared in [2] in the following ways. Definition 3. [2] Let A be any nonempty subset of a ternary Γ-semigroup T . Then: (i) A is called a ternary Γ-subsemigroup of T if AΓAΓA ⊆ A; (ii) A is called a left (resp. right, lateral) Γ-ideal of T if TΓTΓA ⊆ A (resp. AΓTΓT ⊆ A, TΓAΓT ⊆ A); (iii) A is called a Γ-ideal of T if it is a left, a right, and a lateral Γ-ideal of T ; (iv) a ternary Γ-subsemigroup A of T is called a bi-Γ-ideal of T if TΓAΓTΓAΓT ⊆ A. Definition 4. [2] Let µ be any fuzzy set of a ternary Γ-semigroup T . Then: (i) µ is called a fuzzy ternary Γ-subsemigroup of T if µ(aαbβc) ≥ min{µ(a), µ(b), µ(c)}, for all a, b, c ∈ T and α, β ∈ Γ; (ii) µ is called a fuzzy left (resp. right, lateral) Γ-ideal of T if µ(aαbβc) ≥ µ(c) (resp. µ(aαbβc) ≥ µ(a), µ(aαbβc) ≥ µ(b)), for all a, b, c ∈ T and α, β ∈ Γ; (iii) µ is called a fuzzy Γ-ideal of T if it is a fuzzy left Γ-ideal, a fuzzy right Γ-ideal, and a fuzzy lateral Γ-ideal of T ; (iv) a fuzzy ternary Γ-subsemigroup µ of T is called a fuzzy bi-Γ-ideal of T if µ(aαbβcγdδe) ≥ min{µ(a), µ(c), µ(e)}, for all a, b, c, d, e ∈ T and α, β, γ, δ ∈ Γ. Let S and T be ternary Γ-semigroups with respect to the same set Γ. The mapping · : (S × T )× Γ× (S × T )× Γ× (S × T ) → S × T is defined by (s1, t1)α(s2, t2)β(s3, t3) = (s1αs2βs3, t1αt2βt3), for all (s1, t1), (s2, t2), (s3, t3) ∈ S × T and α, β ∈ Γ. Then S × T forms a ternary Γ- semigroup (see [2]). 3. Strongest fuzzy Γ-ideals on ternary Γ-semigroups In this section, we introduce the concepts of strongest fuzzy ternary Γ-subsemigroups, strongest fuzzy (resp. left, right, and lateral) Γ-ideals, and strongest fuzzy bi-Γ-ideals on ternary Γ-semigroups. Then we study the relationships and characterizations of these concepts in ternary Γ-semigroups. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1420 Definition 5. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then Rµ is called a strongest fuzzy ternary Γ-subsemigroup on T if Rµ(a1αb1βc1, a2αb2βc2) ≥ min{Rµ(a1, a2), Rµ(b1, b2), Rµ(c1, c2)}, for all a1, a2, b1, b2, c1, c2 ∈ T and α, β ∈ Γ. Definition 6. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then Rµ is called: (i) a strongest fuzzy left Γ-ideal on T if Rµ(a1αb1βc1, a2αb2βc2) ≥ Rµ(c1, c2), for all a1, a2, b1, b2, c1, c2 ∈ T and α, β ∈ Γ; (ii) a strongest fuzzy right Γ-ideal on T if Rµ(a1αb1βc1, a2αb2βc2) ≥ Rµ(a1, a2), for all a1, a2, b1, b2, c1, c2 ∈ T and α, β ∈ Γ; (iii) a strongest fuzzy lateral Γ-ideal on T if Rµ(a1αb1βc1, a2αb2βc2) ≥ Rµ(b1, b2), for all a1, a2, b1, b2, c1, c2 ∈ T and α, β ∈ Γ; (iv) a strongest fuzzy Γ-ideal on T if it is a strongest fuzzy left Γ-ideal, a strongest fuzzy right Γ-ideal, and a strongest fuzzy lateral Γ-ideal on T . Definition 7. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy ternary Γ-subsemigroup on T . Then Rµ is said to be a strongest fuzzy bi-Γ-ideal on T if Rµ(a1αb1βc1γd1δe1, a2αb2βc2γd2δe2) ≥ min{Rµ(a1, a2), Rµ(c1, c2), Rµ(e1, e2)}, for all a1, a2, b1, b2, c1, c2, d1, d2, e1, e2 ∈ T and α, β, γ, δ ∈ Γ. By Definition 7, it is clear that every strongest fuzzy bi-Γ-ideal on a ternary Γ- semigroup is also a strongest fuzzy ternary Γ-subsemigroup, but the converse is not always true, as the following example. Example 1. Let T = {a, b, c} and Γ = T . Define the operation · on T by xαyβz = (x ∗ y) ∗ z, for all x, y, z ∈ T and α, β ∈ Γ where the binary operation ∗ on T is defined by the following table: ∗ a b c a a a a b a b b c a c c Then, T is a ternary Γ-semigroup [6]. Next, we define a fuzzy set µ of T by µ(a) = 0.2, µ(b) = 0.5, and µ(c) = 0.9. Following a careful analysis, we have Rµ is a strongest fuzzy ternary Γ-subsemigroup on T , but it is not a strongest fuzzy bi-Γ-ideal on T , since Rµ(cαaβcγaδc, cαaβcγaδc) = 0.2 < 0.9 = min{Rµ(c, c)Rµ(c, c), Rµ(c, c)}, for all α, β, γ, δ ∈ Γ. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1421 Proposition 1. Let T be a ternary Γ-semigroup. Then: (i) every strongest fuzzy left Γ-ideal on T is also a strongest fuzzy bi-Γ-ideal; (ii) every strongest fuzzy right Γ-ideal on T is also a strongest fuzzy bi-Γ-ideal; (iii) every strongest fuzzy lateral Γ-ideal on T is also a strongest fuzzy bi-Γ-ideal; (iv) every strongest fuzzy Γ-ideal on T is also a strongest fuzzy bi-Γ-ideal. Proof. (i) Let Rµ be a strongest fuzzy left Γ-ideal on T . It is not difficult to verify that Rµ is a strongest fuzzy ternary Γ-subsemigroup on T . For any a1, a2, b1, b2, c1, c2, d1, d2, e1, e2 ∈ T , and any α, β, γ, δ ∈ Γ, we have (a1αb1)β(c1γd1)δe1 = x1βy1δe1 and (a2αb2)β(c2γd2)δe2 = x2βy2δe2, for some x1 = a1αb1, y1 = c1γd1, x2 = a2αb2, and y2 = c2γd2. It follows that Rµ(a1αb1βc1γd1δe1, a2αb2βc2γd2δe2) = Rµ(x1βy1δe1, x2βy2δe2) ≥ Rµ(e1, e2) ≥ min{Rµ(a1, a2), Rµ(c1, c2), Rµ(e1, e2)}. Hence, Rµ is a strongest fuzzy bi-Γ-ideal on T . The proofs of (ii) and (iii) are similar to the proof of (i). (iv) It is obvious. The converses of statements in Proposition 1 don’t have to be true as shown in the following example. Example 2. Let T = {a, b, c} and Γ = T . Define the mapping · on T in Example 1. Now, we define a fuzzy set µ of T by µ(a) = 0.7, µ(b) = 0.7 and µ(c) = 0.2. After a thorough examination, we obtain Rµ is a strongest fuzzy bi-Γ-ideal on T . Never- theless, Rµ is not a strongest fuzzy left Γ-ideal on T , since Rµ(cαcβb, cαcβb) = 0.2 < 0.7 = Rµ(b, b), for all α, β ∈ Γ. Furthermore, it is not a strongest fuzzy lateral Γ-ideal on T either, because Rµ(cαbβc, cαbβc) = 0.2 < 0.7 = Rµ(b, b), for all α, β ∈ Γ. Next, we present the characterizations of strongest fuzzy ternary Γ-subsemigroups, strongest fuzzy (resp. left, right, lateral) Γ-ideals, and strongest fuzzy bi-Γ-ideals on ternary Γ-semigroups. Theorem 1. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then, µ is a fuzzy ternary Γ-subsemigroup of T if and only if Rµ is a strongest fuzzy ternary Γ-subsemigroup on T . W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1422 Proof. Assume that µ is a fuzzy ternary Γ-subsemigroup of T . Let a1, a2,b1, b2,c1, c2 ∈ T and α, β ∈ Γ. Then, we have Rµ(a1αb1βc1, a2αb2βc2) = min{µ(a1αb1βc1), µ(a2αb2βc2)} ≥ min{min{µ(a1), µ(b1), µ(c1)},min{µ(a2), µ(b2), µ(c2)}} = min{min{µ(a1), µ(a2)},min{µ(b1), µ(b2)},min{µ(c1), µ(c2)}} = min{Rµ(a1, a2), Rµ(b1, b2), Rµ(c1, c2)}. Thus, Rµ is a strongest fuzzy ternary Γ-subsemigroup on T . Conversely, assume that Rµ is a strongest fuzzy ternary Γ-subsemigroup on T . Let a, b, c ∈ T and α, β ∈ Γ. Then, we have µ(aαbβc) = min{µ(aαbβc), µ(aαbβc)} = Rµ(aαbβc, aαbβc) ≥ min{Rµ(a, a), Rµ(b, b), Rµ(c, c)} = min{min{µ(a), µ(a)},min{µ(b), µ(b)},min{µ(c), µ(c)}} = min{µ(a), µ(b), µ(c)}. Hence, µ is a fuzzy ternary Γ-subsemigroup of T . Theorem 2. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then the following statements hold: (i) µ is a fuzzy left Γ-ideal of T if and only if Rµ is a strongest fuzzy left Γ-ideal on T ; (ii) µ is a fuzzy right Γ-ideal of T if and only if Rµ is a strongest fuzzy right Γ-ideal on T ; (iii) µ is a fuzzy lateral Γ-ideal of T if and only if Rµ is a strongest fuzzy lateral Γ-ideal on T ; (iv) µ is a fuzzy Γ-ideal of T if and only if Rµ is a strongest fuzzy Γ-ideal on T . Proof. (i) Assume that µ is a fuzzy left Γ-ideal of T . Let a1, a2, b1, b2, c1, c2 ∈ T and α, β ∈ Γ. Thus, we have Rµ(a1αb1βc1, a2αb2βc2) = min{µ(a1αb1βc1), µ(a2αb2βc2)} ≥ min{µ(c1), µ(c2)} = Rµ(c1, c2). This implies that Rµ is a strongest fuzzy left Γ-ideal on T . Conversely, assume that Rµ is a strongest fuzzy left Γ-ideal on T . Let a, b, c ∈ T and α, β ∈ Γ. Then, we have µ(aαbβc) = min{µ(aαbβc), µ(aαbβc)} W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1423 = Rµ(aαbβc, aαbβc) ≥ Rµ(c, c) = min{µ(c), µ(c)} = µ(c). We obtain that µ is a fuzzy left Γ-ideal of T . For the proofs of (ii) and (iii), we can prove similarly. (iv) It follows by the conditions of (i), (ii), and (iii). Theorem 3. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then, µ is a fuzzy bi-Γ-ideal of T if and only if Rµ is a strongest fuzzy bi-Γ-ideal on T . Proof. Assume that µ is a fuzzy bi-Γ-ideal of T . Then µ is a fuzzy ternary Γ- subsemigroup of T . By Theorem 1, we get Rµ is a strongest fuzzy ternary Γ-subsemigroup on T . Let a1, a2, b1, b2, c1, c2, d1, d2, e1, e2 ∈ T and α, β, γ, δ ∈ Γ. Thus, we have Rµ(a1αb1βc1γd1δe1, a2αb2βc2γd2δe2) = min{µ(a1αb1βc1γd1δe1), µ(a2αb2βc2γd2δe2)} ≥ min{min{µ(a1), µ(c1), µ(e1)},min{µ(a2), µ(c2), µ(e2)}}} = min{min{µ(a1), µ(a2)},min{µ(c1), µ(c2)},min{µ(e1), µ(e2)}} = min{Rµ(a1, a2), Rµ(c1, c2), Rµ(e1, e2)}. Hence, Rµ is a strongest fuzzy bi-Γ-ideal on T . Conversely, assume that Rµ is a strongest fuzzy bi-Γ-ideal on T . Again, by Theorem 1, we have µ is a fuzzy ternary Γ-subsemigroup of T . Now, let a, b, c, d, e ∈ T and α, β, γ, δ ∈ Γ. It follows that µ(aαbβcγdδe) = min{µ(aαbβcγdδe), µ(aαbβcγdδe)} = Rµ(aαbβcγdδe, aαbβcγdδe) ≥ min{Rµ(a, a), Rµ(c, c), Rµ(e, e)} = min{µ(a), µ(c), µ(e)}. Therefore, µ is a fuzzy bi-Γ-ideal of T . In the following, we will write T ×T instead of a ternary Γ-semigroup T ×T , where T is a ternary Γ-semigroup. Theorem 4. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then, Rµ is a strongest fuzzy ternary Γ-subsemigroup on T if and only if for every t ∈ [0, 1], (Rµ)t is a ternary Γ-subsemigroup of T × T if it is nonempty. Proof. Assume that Rµ is a strongest fuzzy ternary Γ-subsemigroup on T . Let t ∈ [0, 1] be such that (Rµ)t ̸= ∅, and let (a1, a2), (b1, b2), (c1, c2) ∈ (Rµ)t and α, β ∈ Γ. Then Rµ(a1, a2) ≥ t, Rµ(b1, b2) ≥ t, and Rµ(c1, c2) ≥ t. It turns out that W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1424 Rµ(a1αb1βc1, a2αb2βc2) ≥ min{Rµ(a1, a2), Rµ(b1, b2), Rµ(c1, c2)} ≥ t. This means that (a1, a2)α(b1, b2)β(c1, c2) = (a1αb1βc1, a2αb2βc2) ∈ (Rµ)t. So, (Rµ)tΓ(Rµ)tΓ(Rµ)t ⊆ (Rµ)t. Hence, (Rµ)t is a ternary Γ-subsemigroup of T × T . Conversely, for any t ∈ [0, 1], (Rµ)t ̸= ∅ is a ternary Γ-subsemigroup of T × T . Let a1, a2, b1, b2, c1, c2 ∈ T and α, β ∈ Γ. Choose Rµ(a1, a2) = t1, Rµ(b1, b2) = t2, and Rµ(c1, c2) = t3, for some t1, t2, t3 ∈ [0, 1]. Let t = min{t1, t2, t3}. Then, we have (a1, a2), (b1, b2), (c1, c2) ∈ (Rµ)t. By the hypothesis, we get (a1, a2)α(b1, b2)β(c1, c2) ∈ (Rµ)tΓ(Rµ)tΓ(Rµ)t ⊆ (Rµ)t. Thus, (a1αb1βc1, a2αb2βc2) = (a1, a2)α(b1, b2)β(c1, c2) ∈ (Rµ)t. This implies that Rµ(a1αb1βc1, a2αb2βc2) ≥ t = min{t1, t2, t3} = min{Rµ(a1, a2), Rµ(b1, b2), Rµ(c1, c2)}. Therefore, Rµ is a strongest fuzzy ternary Γ-subsemigroup on T . Theorem 5. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then the following statements hold: (i) Rµ is a strongest fuzzy left Γ-ideal on T if and only if for any t ∈ [0, 1], (Rµ)t is a left Γ-ideal of T × T if it is nonempty; (ii) Rµ is a strongest fuzzy right Γ-ideal on T if and only if for any t ∈ [0, 1], (Rµ)t is a right Γ-ideal of T × T if it is nonempty; (iii) Rµ is a strongest fuzzy lateral Γ-ideal on T if and only if for any t ∈ [0, 1], (Rµ)t is a lateral Γ-ideal of T × T if it is nonempty; (iv) Rµ is a strongest fuzzy Γ-ideal on T if and only if for any t ∈ [0, 1], (Rµ)t is a Γ-ideal of T × T if it is nonempty. Proof. (i) Assume that Rµ is a strongest fuzzy left Γ-ideal on T . Let (a1, a2), (b1, b2) ∈ T × T and (c1, c2) ∈ (Rµ)t, and α, β ∈ Γ. Then Rµ(a1αb1βc1, a2αb2βc2) ≥ Rµ(c1, c2) ≥ t. We obtain that (a1, a2)α(b1, b2)β(c1, c2) = (a1αb1βc1, a2αb2βc2) ∈ (Rµ)t; that is, (T × T )Γ(T × T )Γ(Rµ)t ⊆ (Rµ)t. This shows that (Rµ)t is a left Γ-ideal of T × T . W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1425 Conversely, assume that for any t ∈ [0, 1], (Rµ)t ̸= ∅ is a left Γ-ideal of T × T . Let a1, a2, b1, b2, c1, c2 ∈ T and α, β ∈ Γ. Take Rµ(c1, c2) = t, for some t ∈ [0, 1]. It follows that (c1, c2) ∈ (Rµ)t, and then (Rµ)t ̸= ∅. By the given assumption, we have (a1αb1βc1, a2αb2βc2) = (a1, a2)α(b1, b2)β(c1, c2) ∈ (T × T )Γ(T × T )Γ(Rµ)t ⊆ (Rµ)t. This implies that Rµ(a1αb1βc1, a2αb2βc2) ≥ t = Rµ(c1, c2). Therefore, Rµ is a strongest fuzzy left Γ-ideal on T . The proofs of (ii) and (iii) can proved in a similar way. (iv) It obtains from (i), (ii), and (iii). Theorem 6. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and Rµ be a strongest fuzzy relation on T . Then, Rµ is a strongest fuzzy bi-Γ-ideal on T if and only if for each t ∈ [0, 1], (Rµ)t is a bi-Γ-ideal of T × T when it is nonempty. Proof. Assume that Rµ is a strongest fuzzy bi-Γ-ideal on T . Let t ∈ [0, 1] be such that (Rµ)t ̸= ∅. Let (a1, a2), (c1, c2), (e1, e2) ∈ (Rµ)t and (b1, b2), (d1, d2) ∈ T × T , and let α, β, γ, δ ∈ Γ. Thus, we have Rµ(a1αc1βe1, a2αc2βe2) ≥ min{Rµ(a1, a2), Rµ(c1, c2), Rµ(e1, e2)} ≥ t and Rµ(a1αb1βc1γd1δe1, a2αb2βc2γd2δe2) ≥ min{Rµ(a1, a2), Rµ(b1, b2), Rµ(e1, e2)} ≥ t. Also, (a1, a2)α(c1, c2)β(e1, e2) = (a1αc1βe1, a2αc2βe2) ∈ (Rµ)t and (a1, a2)α(b1, b2)β(c1, c2)γ(d1, d2)δ(e1, e2) = (a1αb1βc1γd1δe1, a2αb2βc2γd2δe2) ∈ (Rµ)t, respectively. This shows that (Rµ)tΓ(Rµ)tΓ(Rµ)t ⊆ (Rµ)t and (Rµ)tΓ(T × T )Γ(Rµ)tΓ(T × T )Γ(Rµ)t ⊆ (Rµ)t. Therefore, (Rµ)t is a bi-Γ-ideal of T × T . Conversely, let a1, a2, b1, b2, c1, c2, d1, d2, e1, e2 ∈ T and α, β, γ, δ ∈ Γ. ChooseRµ(a1, a2) = t1, Rµ(c1, c2) = t2, and Rµ(e1, e2) = t3, for some t1, t2, t3 ∈ [0, 1]. Let t = min{t1, t2, t3}. It turns out that (a1, a2), (c1, c2), (e1, e2) ∈ (Rµ)t. By assumption, we have (Rµ)t is a bi-Γ-ideal of T × T . So, we obtain (a1αc1βe1, a2αc2βe2) = (a1, a2)α(c1, c2)β(e1, e2) ∈ (Rµ)t and (a1αb1βc1γd1δe1, a2αb2βc2γd2δe2) = (a1, a2)α(b1, b2)β(c1, c2)γ(d1, d2)δ(e1, e2) ∈ (Rµ)t. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1417-1428 1426 It means that Rµ(a1αc1βe1, a2αc2βe2) ≥ t = min{t1, t2, t3} = min{Rµ(a1, a2), Rµ(c1, c2), Rµ(e1, e2)} and Rµ(a1αb1βc1γd1δe1, a2αb2βc2γd2δe2) ≥ t = min{t1, t2, t3} = min{Rµ(a1, a2), Rµ(c1, c2), Rµ(e1, e2)}. Consequently, Rµ is a strongest fuzzy bi-Γ-ideal on T . Example 3. By Example 1, we obtain Rµ is a strongest fuzzy bi-Γ-ideal on a ternary Γ- semigroup T . It turns out that the set of all level subsets of Rµ are (Rµ)0.7 = {(a, a), (a, b), (b, a), (b, b)} and (Rµ)0.2 = T × T . By Theorem 6, we have (Rµ)0.7 and (Rµ)0.2 are bi-Γ- ideals of a ternary Γ-semigroup T × T . This is the process of finding some bi-Γ-ideals of a ternary Γ-semigroup T × T using Theorem 6 such as the sets {(a, a), (a, b), (b, a), (b, b)} and T × T . Let X be a nonempty set, and µ be a fuzzy set of X. We observe that all level subsets of the strongest fuzzy relation χA µ on X only include that the sets A and X, for each subset A of X. Therefore, we obtain the following results by Theorem 4, Theorem 5, and Theorem 6, respectively. Corollary 1. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and A be a nonempty subset of T . Then, χA µ is a strongest fuzzy ternary Γ-subsemigroup on T if and only if A is a ternary Γ-subsemigroup of T . Corollary 2. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and A be a nonempty subset of T . Then the following conditions hold: (i) χA µ is a strongest fuzzy left Γ-ideal on T if and only if A is a left Γ-ideal of T ; (ii) χA µ is a strongest fuzzy right Γ-ideal on T if and only if A is a right Γ-ideal of T ; (iii) χA µ is a strongest fuzzy lateral Γ-ideal on T if and only if A is a lateral Γ-ideal of T ; (iv) χA µ is a strongest fuzzy Γ-ideal on T if and only if A is a Γ-ideal of T . Corollary 3. Let T be a ternary Γ-semigroup, µ be a fuzzy set of T , and A be a nonempty subset of T . Then, χA µ is a strongest fuzzy bi-Γ-ideal on T if and only if A is a bi-Γ-ideal of T . REFERENCES 1427 4. Conclusions The concept of fuzzy relation was applied to define the notions of strongest fuzzy ternary Γ-subsemigroups, strongest fuzzy (resp. left, right, lateral) Γ-ideals, and strongest fuzzy bi-Γ-ideals on ternary Γ-semigroups. Following this, we investigated the connections of these concepts that every strongest fuzzy (resp. left, right, lateral) Γ-ideal is also a strongest fuzzy bi-Γ-ideal, while every strongest fuzzy bi-Γ-ideal is also a strongest fuzzy ternary Γ-subsemigroup on a ternary Γ-semigroup. In addition, as Example 1 and Example 2 indicate, the converses of the above mentioned relationships are not true. After that, we studied the links between different types of fuzzy Γ-ideals of ternary Γ-semigroups and their respective types of strongest fuzzy Γ-ideals on ternary Γ-semigroups, which occurred in Theorem 1, Theorem 2, and Theorem 3. Finally, the characterizations of strongest fuzzy ternary Γ-subsemigroups, strongest fuzzy (resp. left, right, lateral) Γ-ideals, and strongest fuzzy bi-Γ-ideals on ternary Γ-semigroups by the various types if their level subsets in ternary Γ-semigroups are presented in Theorem 4, Theorem 5, and Theorem 6. 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