EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6327 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Quotient KS-semigroups Induced by Fuzzy KS-ideals Hulsen T. Sarapuddin1,∗, Jocelyn P. Vilela1 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Mathematical and Theoretical Physical Sciences-PRISM, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. In this paper, we introduce new types of fuzzy ideals on KS-semigroups, compare them with existing fuzzy KS-ideals and investigate their properties. We discuss the construction of quotient KS-semigroups induced by fuzzy KS-ideals, show that this quotient structure is a generalization of the existing quotient structure and prove properties of this generalized structure. Moreover, we investigate the quotient structure of product KS-semigroups induced by fuzzy KS- ideals. 2020 Mathematics Subject Classifications: 08A30, 08A72 Key Words and Phrases: Quotient KS-semigroups, congruence relation, fuzzy KS-ideals, fuzzy KS-p-ideals, fuzzy (commutative, implicative) KS-ideals 1. Introduction The class of BCK-algebras was introduced by Imai and Iseki [1] in 1966 that describes fragments of propositional calculus involving implication, known as BCK logic. It was also developed as a generalization of set difference in set theory. Since then, a great deal of literature has been produced in the theory of BCK-algebras. The initial paper on the class of semigroups emerged in 1905 as a concise work by Dickson. However, the true inception of the theory occurred in 1928 when Suschkewitsch [2] published a paper of paramount significance. In contemporary language, he demonstrated that within any finite semigroup, there exists a “kernel” (referred to as a simple ideal), and he comprehensively characterized the structure of finite simple semigroups. Semigroups provide a foundational framework for understanding how elements combine under certain operations, and their applications span across multiple branches of mathematics and various interdisciplinary fields such as Coding theory, Automata, etc. For more notions about semigroups and history, we refer to [3]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6327 Email addresses: hulsen.sarapuddin@g.msuiit.edu.ph (H. T. Sarapuddin), jocelyn.vilela@g.msuiit.edu.ph (J. P. Vilela) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 2 of 23 The notion of fuzzy sets was introduced by Zadeh [4] in 1965 to provide a mathematical framework for dealing with vagueness and imprecision. It departs from classical set theory by allowing elements to have degrees of membership in a set, rather than requiring absolute membership or non-membership. Fuzzy sets quickly garnered attention across various disciplines, including algebraic structures. The combination of the two concepts gave rise to fuzzy algebraic structures. The latter was established by Rosenfeld in 1971 when he applied the concept of fuzzy sets to groups. In 2006, Kim [5] introduced the class of KS-semigroups which is both a BCK-algebra and a semigroup. The quotient KS-semigroups via ideals was established and the isomor- phism theorems were proved in 2009 by Cawi and Vilela [6]. In 2007, Prince Williams and Husain [7] applied the concept of fuzzy sets to KS-semigroup, and referred to it as a fuzzy KS-semigroup. The notions of fuzzy KS-ideals and fuzzy KS-p-ideals were introduced and some of their properties were investigated. In 2011, Bautista and Vilela [8] introduced and investigated fuzzy topology on a KS-semigroup and fuzzy topological KS-semigroup. There are many research items in the literature characterizing the class of semigroups through its (fuzzy) sets, particularly, through generalization and fuzzification of ideals. For example, generalization of bi-antiideals in semigroups and their fuzzification were studied in [9] and (fuzzy) KS-H-ideals in KS-semigroups and their properties were studied in [10]. Moreover, the construction of quotient semigroups induced by fuzzy ideals and their properties were studied in [11]. In this paper, we introduce new types of fuzzy ideals on KS-semigroups, compare them with existing fuzzy KS-ideals and investigate their properties. We construct quotient KS-semigroups induced by fuzzy KS-ideals and investigate their properties. Furthermore, we investigate the quotient structure of product KS-semigroups induced by fuzzy KS-ideals. 2. Preliminaries In this section, we present some basic concepts and known results that are useful in this study. Definition 1. [5] An algebraic system (X, ∗, 0) is called a BCK-algebra if it satisfies the following conditions: for all x, y, z ∈ X, (i) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (iv) 0 ∗ x = 0, (ii) (x ∗ (x ∗ y)) ∗ y = 0, (v) x ∗ y = 0 and y ∗ x = 0 imply x = y. (iii) x ∗ x = 0, Definition 2. [5] If X is a BCK-algebra, then the relation x ≤ y if and only if x ∗ y = 0 is a partial order on X, which will be called the natural ordering on X. Remark 1. [5] A BCK-algebra X has the following properties for any x, y, z ∈ X: (i) x ∗ 0 = x, (iv) x ≤ y implies that x ∗ z ≤ y ∗ z and z ∗ y ≤ z ∗ x, H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 3 of 23 (ii) x ∗ y ≤ x, (v) (x ∗ z) ∗ (y ∗ z) ≤ x ∗ y. (iii) (x ∗ y) ∗ z = (x ∗ z) ∗ y, Proposition 1. [12] In a BCK-algebra X, the following hold: for all x, y, z ∈ X, (i) ((x ∗ z) ∗ z) ∗ (y ∗ z) ≤ (x ∗ y) ∗ z, (ii) (x ∗ z) ∗ (x ∗ (x ∗ z)) = (x ∗ z) ∗ z, (iii) (x ∗ (y ∗ (y ∗ x))) ∗ (y ∗ (x ∗ (y ∗ (y ∗ x)))) ≤ x ∗ y. Definition 3. [13] Let X be a non-empty set. The system (X, ·) is called a semigroup if “ · ” is an associative binary operation. For convenience, we write x · y by xy. Definition 4. [5] An algebraic system (X, ∗, ·, 0) is called a KS-semigroup if it satisfies the following conditions: (i) (X, ∗, 0) is a BCK-algebra (ii) (X, ·) is a semigroup (iii) The operation · is distributive over the operation ∗ on both sides, that is, for all x, y, z ∈ X, x · (y ∗ z) = (x · y) ∗ (x · z) and (x ∗ y) · z = (x · z) ∗ (y · z). Example 1. [5] Let X = {0, a, b, c}. Define ∗ and · by the following tables: ∗ 0 a b c 0 0 0 0 0 a a 0 a a b b b 0 b c c c c 0 · 0 a b c 0 0 0 0 0 a 0 a 0 0 b 0 0 b 0 c 0 c 0 0 Then X is a KS-semigroup. Definition 5. [5] A KS-semigroup X is said to be (i) commutative if x ∗ (x ∗ y) = y ∗ (y ∗ x) for all x, y ∈ X, (ii) positive implicative if (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z), or equivalently, x ∗ y = (x ∗ y) ∗ y for all x, y, z ∈ X, (iii) implicative if x∗ (y ∗x) = x for all x, y ∈ X, or if it is both commutative and positive implicative. Proposition 2. [5] A KS-semigroup X is commutative if and only if for all x, y ∈ X, x ≤ y implies x = y ∗ (y ∗ x). Definition 6. [5] A non-empty subset A of a semigroup (X, ·) is said to be left (resp. right) stable if xa ∈ A (resp. ax ∈ A) whenever x ∈ X and a ∈ A. A non-empty subset of X which is both left and right stable is called two-sided stable or simply stable. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 4 of 23 Definition 7. [5] A non-empty subset A of a KS-semigroup X is called a left (resp. right) ideal of X if (i) A is a left (resp. right) stable subset of (X, ·), (ii) for any x, y ∈ X, x ∗ y ∈ A and y ∈ A imply that x ∈ A. If A is both a left and a right ideal, then it is called a two-sided ideal or simply an ideal. Remark 2. [5] Let X be a KS-semigroup. Then (i) {0} and X are ideals of X. (ii) If A is a left (resp. right) ideal of X, then 0 ∈ A. Definition 8. [5] Let X be a KS-semigroup and let ∼ be a binary relation on X. Then (i) ∼ is said to be right (resp. left) compatible if whenever x ∼ y then x ∗ z ∼ y ∗ z (resp. z ∗ x ∼ z ∗ y) and xz ∼ yz (resp. zx ∼ zy) for all x, y, z ∈ X; (ii) ∼ is said to be compatible if x ∼ y and u ∼ v imply x ∗ u ∼ y ∗ v and xu ∼ yv for all x, y, u, v ∈ X; (iii) A compatible equivalence relation is called a congruence relation. Theorem 1. [5] Let X be a KS-semigroup. An equivalence relation ∼ on X is a congru- ence relation if and only if it is both left and right compatible. Let X be a non-empty set. A fuzzy set on X is a function µ : X → [0, 1]. The characteristic function of a set I is a fuzzy set denoted by χI . Definition 9. [7] A fuzzy set µ on a KS-semigroup X is called a left (resp. right) fuzzy KS-ideal of X if for all x, y, a ∈ X, (F1) µ(0) ≥ µ(x), (F2) µ(x) ≥ min{µ(x ∗ y), µ(y)}, (F3) µ(xa) ≥ min{µ(x), µ(a)} (resp. µ(ax) ≥ min{µ(x), µ(a)}). A fuzzy set µ on a KS-semigroup X is called a fuzzy KS-ideal of X if it is both a left and a right fuzzy KS-ideal of X. Definition 10. [7] Let X be a KS-semigroup. A fuzzy set µ on X is called a left (resp. right) fuzzy KS-p-ideal of X if it satisfies (F1), (F3) and (F4) : µ(x ∗ z) ≥ min{µ((x ∗ y) ∗ z), µ(y ∗ z)} for all x, y, z ∈ X. A fuzzy set µ on a KS-semigroup X is called a fuzzy KS-p-ideal of X if it is both a left and a right fuzzy KS-p-ideal of X. Theorem 2. [7] Let X be a KS-semigroup. Then any fuzzy KS-p-ideal of X is a fuzzy KS-ideal of X. However, the converse is not true. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 5 of 23 3. Some Fuzzy Ideals on KS-semigroup In this section, we investigate useful properties of fuzzy KS-ideals and give criterion for a fuzzy KS-ideal to be a fuzzy KS-p-ideal. Moreover, we introduce new types of fuzzy ideals on KS-semigroups, compare them with existing fuzzy KS-ideals and investigate their properties. The following lemma gives some useful properties of fuzzy KS-ideals. Lemma 1. Let µ be a fuzzy KS-ideal of a KS-semigroup X. (i) If x, y ∈ X such that x ≤ y, then µ(y) ≤ µ(x). (ii) If x, y, z ∈ X such that x ∗ y ≤ z, then µ(x) ≥ min{µ(y), µ(z)}. Proof. Let µ be a fuzzy KS-ideal of a KS-semigroup X. (i) Suppose x, y ∈ X such that x ≤ y. Then by (F2), µ(x) ≥ min{µ(x ∗ y), µ(y)} and x ∗ y = 0. Thus, µ(x) ≥ min{µ(x ∗ y), µ(y)} = min{µ(0), µ(y)} = µ(y). (ii) Suppose x, y, z ∈ X such that x ∗ y ≤ z. Then by (i), µ(z) ≤ µ(x ∗ y). By (F2), µ(x) ≥ min{µ(x ∗ y), µ(y)} ≥ min{µ(z), µ(y)}. In Theorem 2, a fuzzy KS-ideal may not be a fuzzy KS-p-ideal. The following theorem gives a criterion for a fuzzy KS-ideal to be a fuzzy KS-p-ideal. Theorem 3. A fuzzy set µ on a KS-semigroup X is a fuzzy KS-p-ideal if and only if it is a fuzzy KS-ideal satisfying µ(x ∗ y) = µ((x ∗ y) ∗ y) for all x, y ∈ X. Proof. Let X be a KS-semigroup and µ be a fuzzy set on X. Suppose µ is a fuzzy KS-p-ideal of X. Then by Theorem 2, µ is a fuzzy KS-ideal. By (F4), (F1) and Definition 1, for all x, y ∈ X, µ(x ∗ y) ≥ min{µ((x ∗ y) ∗ y), µ(y ∗ y)} = min{µ((x ∗ y) ∗ y), µ(0)} = µ((x ∗ y) ∗ y). Moreover, observe that by Remark 1(ii), (x ∗ y) ∗ y ≤ x ∗ y. Thus, by Lemma 1(i), µ((x ∗ y) ∗ y) ≥ µ(x ∗ y). Hence, µ(x ∗ y) = µ((x ∗ y) ∗ y) for all x, y ∈ X. Conversely, suppose µ is a fuzzy KS-ideal of X satisfying µ(x ∗ y) = µ((x ∗ y) ∗ y) for all x, y ∈ X. Note that by Proposition 1(i), ((x ∗ z) ∗ z) ∗ (y ∗ z) ≤ (x ∗ y) ∗ z. Thus, by Lemma 1(ii), we have µ(x ∗ z) = µ((x ∗ z) ∗ z) ≥ min{µ((x ∗ y) ∗ z), µ(y ∗ z)}. Therefore, µ is a fuzzy KS-p-ideal of X. Now, we introduce some fuzzy ideals on KS-semigroups, investigate their relationships with fuzzy KS-ideals and discuss some of their properties. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 6 of 23 Definition 11. Let X be a KS-semigroup. A fuzzy set µ on X is called a left (resp. right) fuzzy commutative KS-ideal of X if it satisfies (F1), (F3) and (F5) : µ(x ∗ (y ∗ (y ∗ x))) ≥ min{µ((x ∗ y) ∗ z), µ(z)} for all x, y, z ∈ X. A fuzzy set µ on a KS-semigroup X is called a fuzzy commutative KS-ideal of X if it is both a left and a right fuzzy commutative KS-ideal of X. Example 2. Let X = {0, a, b, c}. Define the operations ∗ and · by the following tables. ∗ 0 a b c 0 0 0 0 0 a a 0 0 a b b a 0 b c c c c 0 · 0 a b c 0 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c 0 a b c By routine calculations, we can see that X is a KS-semigroup. Let t0, t1, t2 ∈ [0, 1] such that t0 > t1 > t2. Define a fuzzy set µ on X by µ(0) = t0, µ(a) = t1 and µ(b) = µ(c) = t2. Then µ is a fuzzy commutative KS-ideal of X. The following theorem tells us that every fuzzy commutative KS-ideal is a fuzzy KS- ideal. Theorem 4. Let X be a KS-semigroup. Then any fuzzy commutative KS-ideal of X is a fuzzy KS-ideal of X. Proof. Let µ be a fuzzy commutative KS-ideal of X. Then by Remark 1(i) and Definition 1, for any x, y ∈ X, µ(x) = µ(x ∗ 0) = µ(x ∗ (0 ∗ 0)) = µ(x ∗ (0 ∗ (0 ∗ x))) ≥ min{µ((x ∗ 0) ∗ y), µ(y)} = min{µ(x ∗ y), µ(y)}. Therefore, µ is a fuzzy KS-ideal of X. A fuzzy KS-ideal of a KS-semigroup X may not be a fuzzy commutative KS-ideal of X as shown in the following example. Example 3. Let X = {0, 1, 2, 3, 4}. Define the operations ∗ and · by the following tables. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 0 0 3 3 3 3 0 0 4 4 4 4 3 0 · 0 1 2 3 4 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 3 4 0 1 2 3 4 H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 7 of 23 Then X is a KS-semigroup. Let t0, t1, t2 ∈ [0, 1] such that t0 > t1 > t2. Define a fuzzy set µ on X by µ(0) = t0, µ(1) = t1 and µ(2) = µ(3) = µ(4) = t2. By routine calculations, we can see that µ is a fuzzy KS-ideal of X. However, µ is not a fuzzy commutative KS-ideal of X since (F5) is not satisfied: µ(2 ∗ (3 ∗ (3 ∗ 2))) = µ(2) = t2 < t0 = µ(0) = min{µ((2 ∗ 3) ∗ 0), µ(0)}. The following theorem gives a criterion for a fuzzy KS-ideal to be a fuzzy commutative KS-ideal. Theorem 5. A fuzzy set µ on a KS-semigroup X is a fuzzy commutative KS-ideal if and only if it is a fuzzy KS-ideal satisfying µ(x ∗ (y ∗ (y ∗ x))) = µ(x ∗ y) for all x, y ∈ X. Proof. Let X be a KS-semigroup and µ be a fuzzy set on X. Suppose µ is a fuzzy commutative KS-ideal of X. Then by Theorem 4, µ is a fuzzy KS-ideal. By (F5), for all x, y ∈ X, µ(x ∗ (y ∗ (y ∗ x))) ≥ min{µ((x ∗ y) ∗ 0), µ(0)} = min{µ((x ∗ y)), µ(0)} = µ(x ∗ y). Moreover, note that by Remark 1(ii), y ∗ (y ∗ x) ≤ y. Thus, by Remark 1(iv), x ∗ y ≤ x ∗ (y ∗ (y ∗ x)). Hence, by Lemma 1(i), µ(x ∗ y) ≥ µ(x ∗ (y ∗ (y ∗ x))). Therefore, µ(x ∗ (y ∗ (y ∗ x))) = µ(x ∗ y). Conversely, suppose µ is a fuzzy KS-ideal of X satisfying µ(x ∗ (y ∗ (y ∗ x))) = µ(x ∗ y) for all x, y ∈ X. By (F2), µ(x ∗ (y ∗ (y ∗ x))) = µ(x ∗ y) ≥ min{µ((x ∗ y) ∗ z), µ(z)} for all x, y, z ∈ X. Therefore, µ is a fuzzy commutative KS-ideal of X. The following theorem provides a necessary condition for a fuzzy KS-ideal to be a fuzzy commutative KS-ideal. Theorem 6. Let X be a commutative KS-semigroup. Then every fuzzy KS-ideal is a fuzzy commutative KS-ideal. Proof. Let µ be a fuzzy KS-ideal of a commutative KS-semigroup X. It is sufficient to show that µ satisfies (F5). Let x, y, z ∈ X. Then by Remark 1 and sinceX is commutative, ((x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z)) ∗ z = ((x ∗ (y ∗ (y ∗ x))) ∗ z) ∗ ((x ∗ y) ∗ z) ≤ (x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ y) = (x ∗ (x ∗ y)) ∗ (y ∗ (y ∗ x)) = 0. By Definition 2, (x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z) ≤ z. Thus, by Lemma 1(ii), µ(x ∗ (y ∗ (y ∗ x))) ≥ min{µ((x ∗ y) ∗ z), µ(z)}. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 8 of 23 Hence, (F5) holds. Therefore, µ is a fuzzy commutative KS-ideal of X. Definition 12. Let X be a KS-semigroup. A fuzzy set µ on X is called a left (resp. right) fuzzy implicative KS-ideal of X if it satisfies (F1), (F3) and (F6) : µ(x) ≥ min{µ((x ∗ (y ∗ x)) ∗ z), µ(z)} for all x, y, z ∈ X. A fuzzy set µ on a KS-semigroup X is called a fuzzy implicative KS-ideal of X if it is both a left and a right fuzzy implicative KS-ideal of X. Example 4. Let X = {0, 1, 2, 3, 4}. Define the operations ∗ and · by the following tables. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 0 0 3 3 3 3 0 0 4 4 3 4 1 0 · 0 1 2 3 4 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 4 Then X is a KS-semigroup. Let s, t ∈ [0, 1] such that s < t. Define a fuzzy set µ on X by µ(0) = µ(1) = µ(2) = t and µ(3) = µ(4) = s. By routine calculations, µ is a fuzzy implicative KS-ideal of X. We now give a relationship between a fuzzy KS-ideal and a fuzzy implicative KS-ideal. Theorem 7. Let X be a KS-semigroup. Then any fuzzy implicative KS-ideal of X is a fuzzy KS-ideal of X. Proof. Let µ be a fuzzy implicative KS-ideal of a KS-semigroup X. Then for all x, y ∈ X, µ(x) ≥ min{µ((x ∗ (x ∗ x)) ∗ y), µ(y)} = min{µ((x ∗ 0) ∗ y), µ(y)} = min{µ(x ∗ y), µ(y)}. Therefore, µ is a fuzzy KS-ideal of X. The converse of Theorem 7 may not be true as shown in the following example. Example 5. Consider the KS-semigroup in Example 4. Define a fuzzy set µ on X by µ(0) = µ(2) = 0.8 and µ(1) = µ(3) = µ(4) = 0.5. By routine calculations, µ is a fuzzy KS-ideal of X. However, µ is not a fuzzy implicative KS-ideal of X since µ(1) = 0.5 < 0.8 = min{µ((1 ∗ (3 ∗ 1)) ∗ 2), µ(2)}. The following result provides a criterion for a fuzzy KS-ideal to be a fuzzy implicative KS-ideal. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 9 of 23 Theorem 8. A fuzzy set µ on a KS-semigroup X is a fuzzy implicative KS-ideal if and only if it is a fuzzy KS-ideal satisfying µ(x) = µ(x ∗ (y ∗ x)) for all x, y ∈ X. Proof. Let X be a KS-semigroup and µ be a fuzzy set on X. Suppose µ is a fuzzy implicative KS-ideal of X. Then by Theorem 7, µ is a fuzzy KS-ideal of X. By (F6), for all x, y ∈ X, µ(x) ≥ min{µ(x ∗ (y ∗ x) ∗ 0), µ(0)} = min{µ(x ∗ (y ∗ x)), µ(0)} = µ(x ∗ (y ∗ x)). Moreover, observe that x∗(y∗x) ≤ x by Remark 1(ii). Thus, by Lemma 1(i), µ(x∗(y∗x)) ≥ µ(x). Hence, µ(x) = µ(x ∗ (y ∗ x)). Conversely, suppose µ is a fuzzy KS-ideal of X satisfying µ(x) = µ(x ∗ (y ∗ x)) for all x, y ∈ X. By (F2), µ(x) = µ(x∗ (y ∗x)) ≥ min{µ((x∗ (y ∗x))∗ z), µ(z)} for all x, y, z ∈ X. Thus, µ is a fuzzy implicative KS-ideal of X. The following theorem gives a necessary condition for a fuzzy KS-ideal to be a fuzzy implicative KS-ideal. Theorem 9. If X is an implicative KS-semigroup, then every fuzzy KS-ideal of X is fuzzy implicative KS-ideal. Proof. Let X be an implicative KS-semigroup and µ a fuzzy KS-ideal of X. Then by Definition 5(ii), x = x ∗ (y ∗ x) for all x, y ∈ X. Since µ is a fuzzy KS-ideal of X, by (F2), µ(x) ≥ min{µ(x ∗ z), µ(z)} = min{µ((x ∗ (y ∗ x)) ∗ z), µ(z)} for all z ∈ X. Thus, (F6) holds. Hence, µ is a fuzzy implicative KS-ideal of X. Theorem 10. A fuzzy set µ on a KS-semigroup X is a fuzzy implicative KS-ideal if and only if it is both a fuzzy KS-p-ideal and fuzzy commutative KS-ideal. Proof. Let X be a KS-semigroup and µ be a fuzzy set on X. Suppose µ is a fuzzy implicative KS-ideal of X. By Theorem 8, Proposition 1 and Lemma 1(ii), for all x, y, z ∈ X, µ(x ∗ z) = µ((x ∗ z) ∗ (x ∗ (x ∗ z))) = µ((x ∗ z) ∗ z) ≥ min{µ((x ∗ y) ∗ z), µ(y ∗ z)}. Thus, (F4) holds. Hence, µ is a fuzzy KS-p-ideal. Moreover, by Proposition 1, Lemma 1(i) and Theorem 8, µ(x ∗ y) ≤ µ((x ∗ (y ∗ (y ∗ x))) ∗ (y ∗ (x ∗ (y ∗ (y ∗ x))))) = µ(x ∗ (y ∗ (y ∗ x))) for all x, y ∈ X. By Remark 1, y ∗ (y ∗ x) ≤ y implies that x ∗ y ≤ x ∗ (y ∗ (y ∗ x)). Thus, by Lemma 1(i), µ(x∗y) ≤ µ(x∗ (y ∗ (y ∗x))). Hence, µ(x∗y) = µ(x∗ (y ∗ (y ∗x))). Therefore, by Theorem 5, µ is a fuzzy commutative KS-ideal of X. Conversely, suppose µ is both fuzzy KS-p-ideal and fuzzy commutative KS-ideal of X. By Definition 1(i), (y ∗ (y ∗x)) ∗ (y ∗x) ≤ x ∗ (y ∗x) for all x, y ∈ X. Thus, by Lemma 1(i), µ(x∗(y∗x)) ≤ µ((y∗(y∗x))∗(y∗x)). By Theorem 3, µ((y∗(y∗x))∗(y∗x)) = µ(y∗(y∗x)). Thus, we have µ(x ∗ (y ∗ x)) ≤ µ(y ∗ (y ∗ x)). (1) H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 10 of 23 Now, by Remark 1, y ∗ x ≤ y implies that x ∗ y ≤ x ∗ (y ∗ x) for all x, y ∈ X. Thus, by Lemma 1(i), µ(x ∗ (y ∗ x)) ≤ µ(x ∗ y). Since µ is a fuzzy commutative KS-ideal of X, by Theorem 5, µ(x ∗ y) = µ(x ∗ (y ∗ (y ∗ x))). Hence, for all x, y ∈ X, µ(x ∗ (y ∗ x)) ≤ µ(x ∗ (y ∗ (y ∗ x))). (2) Note that by Definition 1(ii), x ∗ (x ∗ (y ∗ (y ∗ x))) ≤ y ∗ (y ∗ x). Thus, by Lemma 1(ii), µ(x) ≥ min{µ(x ∗ (y ∗ (y ∗ x))), µ(y ∗ (y ∗ x))}. Combining (1) and (2), we get µ(x) ≥ min{µ(x ∗ (y ∗ (y ∗ x))), µ(y ∗ (y ∗ x))} ≥ µ(x ∗ (y ∗ x)). Since x ∗ (y ∗ x) ≤ x, it follows that by Lemma 1(i), µ(x) ≤ µ(x ∗ (y ∗ x)). Thus, µ(x) = µ(x ∗ (y ∗ x)) for all x, y ∈ X. Therefore, by Theorem 8, µ is a fuzzy implicative KS-ideal of X. 4. Quotient KS-semigroups induced by Fuzzy KS-ideals Recall that in [5], if A is an ideal of a KS-semigroup X, then the relation “ ∼A ” on X defined by x ∼A y if and only if x ∗ y ∈ A and y ∗ x ∈ A is a congruence relation on X. Denote Ax as the equivalence class containing x ∈ X and X/A as the set of all equivalence classes of X with respect to “ ∼A ”, that is, Ax = {y ∈ X : x ∼A y} and X/A = {Ax : x ∈ X}. Furthermore, (X/A,⊛,⊙, A0) is a KS-semigroup under the binary operation Ax ⊛Ay = Ax∗y and Ax ⊙Ay = Axy for all Ax, Ay ∈ X/A. In this section, we present the construction of quotient KS-semigroups via fuzzy KS- ideals. Definition 13. Let µ be a non-zero fuzzy KS-ideal of a KS-semigroup X. Define a binary relation ∼µ on X by x ∼µ y if and only if µ(x ∗ y) > 0 and µ(y ∗ x) > 0. We will prove that ∼µ is a congruence relation on a KS-semigroup X. Proposition 3. ∼µ is an equivalence relation on a KS-semigroup X. Proof. Let µ be a non-zero fuzzy KS-ideal of a KS-semigroup X. Let x ∈ X. Since µ is non-zero, there exists z ∈ X such that µ(z) ̸= 0, that is, µ(z) > 0. Then by (F1), µ(0) ≥ µ(z) > 0. Thus by Definition 1, µ(x ∗ x) = µ(0) > 0. Hence, x ∼µ x, which means ∼µ is reflexive. Suppose x ∼µ y. Then µ(x ∗ y) > 0 and µ(y ∗ x) > 0, or equivalently, µ(y ∗ x) > 0 and µ(x ∗ y) > 0. Thus, y ∼µ x. Hence, ∼µ is symmetric. Now, suppose x ∼µ y and y ∼µ z. Then µ(x ∗ y) > 0, µ(y ∗ x) > 0, µ(y ∗ z) > 0 and µ(z ∗ y) > 0. Since X is a KS-semigroup, (x ∗ z) ∗ (x ∗ y) ≤ y ∗ z and (z ∗ x) ∗ (z ∗ y) ≤ y ∗ x for all x, y, z ∈ X. Thus, by Lemma 1(ii), we have µ(x ∗ z) ≥ min{µ(x ∗ y), µ(y ∗ z)} > 0 and µ(z ∗ x) ≥ min{µ(y ∗ x), µ(z ∗ y)} > 0. Thus, x ∼µ z. Hence, ∼µ is transitive. Therefore, ∼µ is an equivalence relation on X. The following example shows that if a non-zero fuzzy KS-ideal µ of a KS-semigroup X does not satisfy condition (C): µ(xy) ≥ µ(x) and µ(xy) ≥ µ(y) for all x, y ∈ X, then ∼µ is not compatible. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 11 of 23 Example 6. Consider the KS-semigroup X = {0, a, b, c} in Example 1. Define a fuzzy set µ in X by µ(0) = µ(a) = µ(b) = 0.8 and µ(c) = 0. Then µ is a fuzzy KS-ideal of X. Note that µ(ca) = µ(c) = min{µ(c), µ(a)} < µ(a). Thus, µ does not satisfy condition (C). Observe that µ(a ∗ b) = µ(a) = 0.8 > 0 and µ(b ∗ a) = µ(b) = 0.8 > 0. Thus, a ∼µ b. However, ∼µ is not compatible since µ(ca ∗ cb) = µ(c ∗ 0) = µ(c) = 0, that is, ca ≁µ cb. Proposition 4. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfying condition (C). Then ∼µ is both left and right compatible. Proof. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfying condition (C). Let x, y, z ∈ X such that x ∼µ y. Then µ(x∗y) > 0 and µ(y ∗x) > 0. Since (z ∗ x) ∗ (z ∗ y) ≤ y ∗ x and (z ∗ y) ∗ (z ∗ x) ≤ x ∗ y, we have µ((z ∗ x) ∗ (z ∗ y)) ≥ µ(y ∗ x) > 0 and µ((z ∗ y) ∗ (z ∗ x)) ≥ µ(x ∗ y) > 0 by Lemma 1(i). Thus, z ∗ x ∼µ z ∗ y. Now, since zx ∗ zy = z(x ∗ y), we have µ(zx ∗ zy) = µ(z(x ∗ y)). Thus by condition (C), µ(zx ∗ zy) = µ(z(x ∗ y)) ≥ µ(x ∗ y) > 0. Similarly, µ(zy ∗ zx) > 0. Hence, zx ∼µ zy. By Definition 8(i), ∼µ is left compatible. Now, since (x ∗ z) ∗ (y ∗ z) ≤ x ∗ y and (y ∗ z) ∗ (x ∗ z) ≤ y ∗ x, we have µ((x ∗ z) ∗ (y ∗ z)) ≥ µ(x ∗ y) > 0 and µ((y ∗ z) ∗ (x ∗ z)) > µ(y ∗ x) > 0 by Lemma 1(i). Thus, x ∗ z ∼µ y ∗ z. Moreover, since xz ∗ yz = (x ∗ y)z, we have µ(xz ∗ yz) = µ((x ∗ y)z). Thus by condition (C), µ(xz ∗ yz) = µ((x ∗ y)z) ≥ µ(x ∗ y) > 0. Similarly, µ(yz ∗xz) > 0. Hence, xz ∼µ yz. By Definition 8(i), ∼µ is right compatible. The following corollary follows from Propositions 3 and 4 and Theorem 1. Corollary 1. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfying condition (C). Then ∼µ is a congruence relation on X. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfying condition (C). Denote µx as the equivalence class containing x ∈ X and X/µ as the set of all equivalence classes of X with respect to “ ∼µ ”, that is, µx = {y ∈ X : y ∼µ x} and X/µ = {µx : x ∈ X}. Remark 3. µx = µy if and only if x ∼µ y. Theorem 11. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfying condition (C). Then (X/µ,⊛,⊙, µ0) is a KS-semigroup under the binary operations µx ⊛ µy = µx∗y and µx ⊙ µy = µxy for all µx, µy ∈ X/µ. Proof. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfy- ing condition (C). By Corollary 1, the operations ⊛ and ⊙ are well-defined. For every µx, µy, µz ∈ X/µ, H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 12 of 23 (i) ((µx ⊛ µy)⊛ (µx ⊛ µz))⊛ (µz ⊛ µy) = µ((x∗y)∗(x∗z))∗(z∗y) = µ0; (ii) (µx ⊛ (µx ⊛ µy))⊛ µy = µ(x∗(x∗y))∗y = µ0; (iii) µx ⊛ µx = µx∗x = µ0; (iv) µ0 ⊛ µx = µ0∗x = µ0. (v) Suppose µx ⊛ µy = µ0 and µy ⊛ µx = µ0. Then µx∗y = µy∗x = µ0. This implies that x ∗ y ∼µ 0 and y ∗ x ∼µ 0. Thus, µ(x ∗ y) = µ((x ∗ y) ∗ 0) > 0 and µ(y ∗ x) = µ((y ∗ x) ∗ 0) > 0, that is, x ∼µ y. Hence, µx = µy. Therefore, (X/µ,⊛, µ0) is a BCK-algebra. Moreover, (µx ⊙ µy)⊙ µz = µxy ⊙ µz = µ(xy)z = µx(yz) = µx ⊙ µyz = µx ⊙ (µy ⊙ µz). Thus, (X/µ,⊙) is a semigroup. Now, observe that µx ⊙ (µy ⊛ µz) = µx ⊙ µy∗z = µx(y∗z) = µxy∗xz = µxy ⊛ µxz = (µx ⊙ µy)⊛ (µx ⊙ µz) and (µx ⊛ µy)⊙ µz = µx∗y ⊙ µz = µ(x∗y)z = µxz∗yz = µxz ⊛ µyz = (µx ⊙ µz)⊛ (µy ⊙ µz). Therefore, (X/µ,⊛,⊙, µ0) is a KS-semigroup. The algebraic system (X/µ,⊛,⊙, µ0) is called the quotient KS-semigroup induced by a fuzzy KS-ideal µ. The following theorem tells us that the condition (C) is necessary to extend the notion of ideal of KS-semigroup. Theorem 12. Let I be a nonempty subset of a KS-semigroup X. Then χI is a fuzzy KS-ideal of X satisfying condition (C) if and only if I is an ideal of X. Proof. Let I be a nonempty subset of a KS-semigroup X. Suppose χI is a fuzzy KS-ideal of X satisfying condition (C). Let x ∈ X and a ∈ I. Then by condition (C), χI(xa) ≥ χI(a) = 1. Thus, xa ∈ I. Similarly, ax ∈ I. Hence, I is stable. Now, let x, y ∈ X such that x ∗ y ∈ I and y ∈ I. Then χI(x ∗ y) = χI(y) = 1. Since χI is a fuzzy KS-ideal in X, χI(x) ≥ min{χI(x ∗ y), χI(y)} = 1. Thus, χI(x) = 1, that is, x ∈ I. Hence, I is an ideal of X. Conversely, suppose I is an ideal of X. Then by Remark 2(ii), 0 ∈ I. Thus, χI(0) = 1 ≥ χI(x) for all x ∈ X. Now, let x, y ∈ X. We consider the following cases. Case 1: x ∗ y ∈ I and y ∈ I. Since I is an ideal of X, x ∈ I. Thus, χI(x) = 1 ≥ min{χI(x ∗ y), χI(y)}. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 13 of 23 Case 2: x ∗ y ∈ I and y /∈ I. Then χI(y) = 0. Thus, χI(x) ≥ 0 = min{χI(x ∗ y), χI(y)}. Case 3: x ∗ y /∈ I and y ∈ I. Then χI(x ∗ y) = 0. Thus, χI(x) ≥ 0 = min{χI(x ∗ y), χI(y)}. Case 4: x ∗ y /∈ I and y /∈ I. Then χI(x ∗ y) = χI(y) = 0. Thus, χI(x) ≥ 0 = min{χI(x ∗ y), χI(y)}. In either case, χI(x) ≥ min{χI(x∗y), χI(y)}. Now, let x, a ∈ X. We consider the following cases. Case 1: x, a ∈ I. Since I is an ideal of X, xa, ax ∈ I. Thus, χI(xa) = 1 ≥ min{χI(x), χI(a)} and χI(ax) = 1 ≥ min{χI(x), χI(a)}. Case 2: x /∈ I or a /∈ I. If xa ∈ I, then χI(xa) = 1. Thus, χI(xa) = 1 ≥ min{χI(x), χI(a)}. If xa /∈ I, then x, a /∈ I. Thus, χI(xa) = 0 and χI(x) = 0 = χI(a). Hence, χI(xa) = 0 ≥ 0 = min{χI(x), χI(a)}. Similarly, χI(ax) ≥ min{χI(x), χI(a)}. Thus, χI is a fuzzy KS-ideal of X . Moreover, by observing Case 1 and Case 2 above, χI satisfies condition (C). The following proposition tells us that the equivalence class containing any element of a KS-semigroup X are equal with respect to ∼I and ∼χI for any ideal I of X. Proposition 5. Let I be an ideal of a KS-semigroup X. Then x ∼I y if and only if x ∼χI y. Proof. Let I be an ideal of a KS-semigroup X. Then for all x, y ∈ X, x ∼I y ⇔ x ∗ y ∈ I and y ∗ x ∈ I ⇔ χI(x ∗ y) = 1 and χI(y ∗ x) = 1 ⇔ χI(x ∗ y) > 0 and χI(y ∗ x) > 0 ⇔ x ∼χI y. Let I be an ideal of a KS-semigroup X. Then for all x ∈ X, Ix = (χI)x. Hence, X/I = X/χI . Therefore, if µ is a non-zero fuzzy KS-ideal of X satisfying condition (C), the quotient KS-semigroup X/µ is a generalization of the quotient KS-semigroup X/I. Theorem 13. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfying condition (C). If J is an ideal of X, then J/µ = {µj : j ∈ J} is an ideal of X/µ. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 14 of 23 Proof. Let µ be a non-zero fuzzy KS-ideal in a KS-semigroup X satisfying condition (C) and J an ideal of X. Since J ⊆ X, it follows that J/µ ⊆ X/µ. Now, let µa ∈ J/µ and µx ∈ X/µ. Then a ∈ J and x ∈ X. Since J is an ideal of X, ax ∈ J . Thus, µaµx = µax ∈ J/µ. Similarly, µxµa ∈ J/µ. For any µx, µy ∈ X/µ, suppose µx∗y = µx ∗ µy ∈ J/µ and µy ∈ J/µ. Then x ∗ y ∈ J and y ∈ J . Since J is an ideal of X, x ∈ J . Thus, µx ∈ J/µ. Therefore, J/µ is an ideal of X/µ. Theorem 14. Let X be a KS-semigroup and µ a non-zero fuzzy KS-ideal of X satisfying condition (C). If J∗ is an ideal of X/µ, then there exists an ideal J = ⋃ {x ∈ X : µx ∈ J∗} of X such that J/µ = J∗. Definition 14. [5] Let X and Y be KS-semigroups and f : X → Y be a mapping. Then f is called a KS-semigroup homomorphism (briefly, homomorphism) if for all x, y ∈ X, f(x ∗ y) = f(x) ∗ f(y) and f(xy) = f(x)f(y). If f is a one-to-one homomorphism, f is said to be a monomorphism. If f is an onto homomorphism, f is called an epimorphism. If f is a bijective homomorphism, f is called an isomorphism. In this case, X and Y are said to be isomorphic (written X ∼= Y ). Theorem 15. Let µ be a non-zero fuzzy KS-ideal of a KS-semigroup X satisfying condi- tion (C). If J is an ideal of X, then X/µ J/µ ∼= X/J . Proof. Let µ be a non-zero fuzzy KS-ideal of a KS-semigroupX satisfying condition (C) and J an ideal ofX. By Theorem 13, J/µ is an ideal ofX/µ, X/µ J/µ = {(J/µ)µx : µx ∈ X/µ} and X/J = {Jx : x ∈ X}. Consider a map φ : X/µ J/µ → X/J defined by φ((J/µ)µx) = Jx. Let (J/µ)µx , (J/µ)µy ∈ X/µ J/µ such that (J/µ)µx = (J/µ)µy . Then µx ∼J/µ µy. Thus, µx∗y = µx ∗ µy ∈ J/µ, µy∗x = µy ∗ µx ∈ J/µ. This means that x ∗ y ∈ J , y ∗ x ∈ J , that is, x ∼J y and Jx = Jy. Thus, φ((J/µ)µx) = Jx = Jy = φ((J/µ)µy). Hence, φ is well-defined. Now, let (J/µ)µx , (J/µ)µy ∈ X/µ J/µ . Then φ((J/µ)µx ∗ (J/µ)µy) = φ((J/µ)µx∗µy) = φ((J/µ)µx∗y) = Jx∗y = Jx ∗ Jy = φ((J/µ)µx) ∗ φ((J/µ)µy) and φ((J/µ)µx(J/µ)µy) = φ((J/µ)µxµy) H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 15 of 23 = φ((J/µ)µxy) = Jxy = JxJy = φ((J/µ)µx)φ((J/µ)µy). Thus, φ is a KS-semigroup homomorphism. For each Jx ∈ X/J , there exists (J/µ)µx such that φ((J/µ)µx) = Jx. Thus, φ is onto. Now, suppose φ((J/µ)µx) = φ((J/µ)µy). Then Jx = Jy. Thus, x ∼J y. Let µα ∈ (J/µ)µx . Then µα ∼J/µ µx. It follows that µα∗x = µα∗µx ∈ J/µ and µx∗α = µx∗µα ∈ J/µ. Thus, α∗x ∈ J and x∗α ∈ J , that is, α ∼J x. Since ∼J is an equivalence relation, α ∼J y. Thus, µα ∈ (J/µ)µy . Hence, (J/µ)µx ⊆ (J/µ)µy . Similarly, (J/µ)µy ⊆ (J/µ)µx . Thus, (J/µ)µx = (J/µ)µy . Hence, φ is one-to-one. Therefore, φ is an isomorphism, that is, X/µ J/µ ∼= X/J . The following are the properties of the generalized quotient via fuzzy KS-ideals with respect to fuzzy KS-p-ideal, fuzzy commutative KS-ideal and fuzzy implicative KS-ideal. Theorem 16. Let µ be a non-zero fuzzy commutative KS-ideal of a KS-semigroup X satisfying condition (C). Then X/µ is a commutative KS-semigroup. Proof. Let µ be a non-zero fuzzy commutative KS-ideal of a KS-semigroupX satisfying condition (C). For all x, y ∈ X, denote u = y ∗ (y ∗ x). Then u ∗ x = (y ∗ (y ∗ x)) ∗ x = (y∗x)∗(y∗x) = 0. Thus, by Theorem 5, µ(u∗(x∗(x∗u))) = µ(u∗x) = µ(0) > 0. Moreover, since (x ∗ (x ∗ u)) ∗ u = (x ∗ u) ∗ (x ∗ u) = 0, it follows that µ((x ∗ (x ∗ u)) ∗ u) = µ(0) > 0. Thus, y ∗ (y ∗ x) ∼µ x ∗ (x ∗ (y ∗ (y ∗ x))). Hence, µy ⊛ (µy ⊛ µx) = µx ⊛ (µx ⊛ (µy ⊛ (µy ⊛ µx))). Therefore, by Proposition 2, X/µ is a commutative KS-semigroup. Theorem 17. Let µ be a non-zero fuzzy KS-p-ideal of a KS-semigroup X satisfying con- dition (C). Then X/µ is a positive implicative KS-semigroup. Proof. Let µ be a non-zero fuzzy KS-p-ideal of a KS-semigroup X satisfying condition (C). Let x, y ∈ X and denote u = x ∗ ((x ∗ y) ∗ y). Then (u ∗ y) ∗ y = 0. Thus, µ((u∗y)∗y) = µ(0) > 0 since µ is non-constant. By Theorem 3, µ(u∗y) = µ((u∗y)∗y) > 0. Thus, µ((x ∗ y) ∗ ((x ∗ y) ∗ y)) = µ(u ∗ y) > 0. Moreover, since ((x ∗ y) ∗ y) ∗ (x ∗ y) = 0, µ(((x ∗ y) ∗ y) ∗ (x ∗ y)) = µ(0) > 0. This implies that x ∗ y ∼µ (x ∗ y) ∗ y. Hence, µx ⊛ µy = µx∗y = µ(x∗y)∗y = (µx ⊛ µy)⊛ µy. Therefore, X/µ is a positive implicative KS-semigroup. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 16 of 23 Theorem 18. Let µ be a non-zero fuzzy implicative KS-ideal of a KS-semigroup X sat- isfying condition (C). Then X/µ is a implicative KS-semigroup. Proof. Let µ be a non-zero fuzzy implicative KS-ideal of a KS-semigroup X satisfying condition (C). Then by Theorem 10, µ is both a non-zero fuzzy commutative KS-ideal and fuzzy KS-p-ideal satisfying condition (C). Thus, by Theorems 16 and 17, X/µ is both commutative and positive implicative KS-semigroup. Therefore, by Definition 5(iii), X/µ is implicative. 5. Homomorphic Properties of the Generalized Quotient via fuzzy KS-ideals In this section, we investigate some results on homomorphism of KS-semigroup. Lemma 2. If µ is a non-zero fuzzy KS-ideal of a KS-semigroup X satisfying condition (C), then the set Xµ = {x ∈ X : µ(x) > 0} is an ideal of X. Proof. Let µ be a non-zero fuzzy KS-ideal of a KS-semigroup X satisfying condition (C) and let Xµ = {x ∈ X : µ(x) > 0}. Let x ∈ X and a ∈ Xµ. Then µ(a) > 0. Thus, µ(xa) ≥ µ(a) > 0 and µ(ax) ≥ µ(a) > 0. This implies that xa, ax ∈ Xµ. Thus, Xµ is stable. Now, suppose x ∗ y ∈ Xµ and y ∈ Xµ. Then µ(x ∗ y) = µ(y) > 0. Since µ is a fuzzy KS-ideal of X, µ(x) ≥ min{µ(x ∗ y), µ(y)} > 0. Thus, x ∈ Xµ. Therefore, Xµ is an ideal of X. Definition 15. [5] Let f : X → Y be a homomorphism of KS-semigroup. Then the kernel of f is denoted and defined as ker f = {x ∈ X : f(x) = 0}. Theorem 19. Let µ be a non-zero fuzzy KS-ideal of a KS-semigroup X satisfying con- dition (C). Then the mapping γ : X → X/µ given by γ(x) = µx is an epimorphism with ker γ = Xµ. Recall that in [7], if f : X → Y is a mapping of KS-semigroup and ν is a fuzzy set on Y , then the fuzzy set νf = ν ◦ f on X is said to be the preimage of ν under f . If f is a homomorphism and ν is a fuzzy KS-ideal of Y , then νf is a fuzzy KS-ideal of X. Furthermore, if f is an epimorphism and νf is a fuzzy KS-ideal of X, then ν is a fuzzy KS-ideal of Y . Lemma 3. Let f : X → Y be a KS-semigroup homomorphism. If µ is a non-zero fuzzy KS-ideal of Y satisfying condition (C), then µf is a non-zero fuzzy KS-ideal of X satisfying condition (C). Proof. Let f : X → Y be a KS-semigroup homomorphism and µ a non-zero fuzzy KS- ideal of Y satisfying condition (C). Then µf is a fuzzy KS-ideal of X. Now, let x, y ∈ X. Then, µf (xy) = µ(f(xy)) = µ(f(x)f(y)) ≥ µ(f(x)) = µf (x) H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 17 of 23 and µf (xy) = µ(f(xy)) = µ(f(x)f(y)) ≥ µ(f(y)) = µf (y). Theorem 20. Let f : X → Y be an epimorphism and µ be a non-zero fuzzy KS-ideal of Y satisfying condition (C). Then X/µf ∼= Y/µ. Proof. Let f : X → Y be an epimorphism and µ be a non-zero fuzzy KS-ideal of Y satisfying condition (C). Then by Lemma 3, µf is a non-zero fuzzy KS-ideal of X satisfying condition (C). By Theorem 11, X/µf and Y/µ are KS-semigroups. Consider a mapping φ : X/µf → Y/µ defined by φ(µfx) = µf(x) for each x ∈ X. Let µfx, µ f y ∈ X/µf such that µfx = µfy . Then x ∼µf y, that is, µf (x ∗ y) = µf (0) and µf (y ∗ x) = µf (0). Since f is a homomorphism, µ(f(x) ∗ f(y)) = µ(f(x ∗ y)) = µf (x ∗ y) = µf (0) and µ(f(y)∗f(x)) = µ(f(y∗x)) = µf (y∗x) = µf (0). Thus, f(x) ∼µ f(y), that is, µf(x) = µf(y). Hence, φ is well-defined. Now, let µfx, µ f y ∈ X/µf . Then φ(µfx ∗ µfy) = φ(µfx∗y) = µf(x∗y) = µf(x)∗f(y) = µf(x) ∗ µf(y) = φ(µfx) ∗ φ(µfy) and φ(µfxµ f y) = φ(µfxy) = µf(xy) = µf(x)f(y) = µf(x)µf(y) = φ(µfx)φ(µ f y). Thus, by Definition 14(i), φ is a homomorphism. Now, since f is an epimorphism, for each y ∈ Y , there exists x ∈ X such that y = f(x). Thus, φ(µfx) = µf(x) = µy. Hence φ is onto. Suppose φ(µfx) = φ(µfy) for µ f x, µ f y ∈ X/µf . Then µf(x) = µf(y) ⇒ f(x) ∼µ f(y) ⇒ µ(f(x) ∗ f(y)) > 0 and µ(f(y) ∗ f(x)) > 0 ⇒ µ(f(x ∗ y)) > 0 and µ(f(y ∗ x)) > 0 ⇒ µf (x ∗ y) > 0 and µf (y ∗ x) > 0 ⇒ x ∼µf y ⇒ µfx = µfy . Thus, φ is one-to-one. Hence, φ is an isomorphism. Therefore, X/µf ∼= Y/µ. 6. Quotient Structure of Product KS-semigroup via fuzzy KS-ideals In this section, we investigate the quotient structure of product KS-semigroup induced by fuzzy KS-ideals. H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 18 of 23 Definition 16. Let (X, ∗X , ·X , 0X) and (Y, ∗Y , ·Y , 0Y ) be KS-semigroups. Define the op- erations ∗ and · in X × Y by (x1, y1) ∗ (x2, y2) = (x1 ∗X x2, y1 ∗Y y2) and (x1, y1) · (x2, y2) = (x1 ·X x2, y1 ·Y y2). Then (X × Y, ∗, ·, 0) is also KS-semigroup, where 0 = (0X , 0Y ) and it is called product KS-semigroup. Definition 17. [4] Let µ and ν be fuzzy sets on the sets X and Y , respectively. The Cartesian product µ × ν : X × Y → [0, 1] is defined by µ × ν(x, y) = min{µ(x), ν(y)} for all (x, y) ∈ X × Y . In [7], if µ and ν are fuzzy KS-ideals of a KS-semigroup X, then µ× ν is also a fuzzy KS-ideal in X ×X. The following result is a general case. Proposition 6. Let µ and ν be fuzzy KS-ideals of KS-semigroups X and Y , respectively. Then µ× ν is also a fuzzy KS-ideal of X × Y . Proof. Let µ and ν be fuzzy KS-ideals of KS-semigroups X and Y , respectively. (i) For any (x, y) ∈ X × Y , µ× ν(0X , 0Y ) = min{µ(0X), ν(0Y )} ≥ min{µ(x), µ(y)} = µ× ν(x, y). (ii) Let (x1, x2), (y1, y2) ∈ X × Y . Then µ× ν(x1, x2) = min{µ(x1), ν(x2)} ≥ min{min{µ(x1 ∗X y1), µ(y1)},min{ν(x2 ∗Y y2), ν(y2)}} = min{min{µ(x1 ∗X y1), ν(x2 ∗Y y2)},min{µ(y1), ν(y2)}} = min{µ× ν(x1 ∗X y1, x2 ∗Y y2), µ× ν(y1, y2)} = min{µ× ν((x1, x2) ∗ (y1, y2)), µ× ν(y1, y2)}. (iii) Let (x1, x2), (y1, y2) ∈ X × Y . Then µ× ν((x1, x2)(y1, y2)) = µ× ν(x1 ·X y1, x2 ·Y y2) = min{µ(x1 ·X y1), ν(x2 ·Y y2)} ≥ min{min{µ(x1), µ(y1)},min{ν(x2), ν(y2)}} = min{min{µ(x1), ν(x2)},min{µ(y1), ν(y2)}} = min{µ× ν(x1, x2), µ× ν(y1, y2)}. Therefore, µ× ν is a fuzzy KS-ideal of X × Y . H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 19 of 23 Corollary 2. Let µ and ν be non-zero fuzzy KS-ideals of KS-semigroups X and Y , re- spectively, both satisfying condition (C). Then µ×ν is a non-zero fuzzy KS-ideal of X×Y satisfying condition (C). Proof. Let µ and ν be non-zero fuzzy KS-ideals of KS-semigroups X and Y , respec- tively, both satisfying condition (C). Then there exist x ∈ X and y ∈ Y such that µ(x) ̸= 0 and ν(y) ̸= 0. Thus, µ × ν(x, y) = min{µ(x), ν(y)} ≠ 0. Thus, µ × ν is non-zero. By Proposition 6, µ×ν is a fuzzy KS-ideal of X×Y . Moreover, since µ and ν satisfy condition (C), for all (x1, x2), (y1, y2) ∈ X × Y , µ× ν((x1, x2)(y1, y2)) = µ× ν(x1 ·X y1, x2 ·Y y2) = min{µ(x1 ·X y1), ν(x2 ·Y y2)} ≥ min{µ(x1), ν(x2)} = µ× ν(x1, x2) and µ× ν((x1, x2)(y1, y2)) = µ× ν(x1 ·X y1, x2 ·Y y2) = min{µ(x1 ·X y1), ν(x2 ·Y y2)} ≥ min{µ(y1), ν(y2)} = µ× ν(y1, y2). Therefore, µ× ν satisfies condition (C). The following corollary follows from Theorems 16, 17 and 18. Corollary 3. Let µ and ν be fuzzy KS-ideals of KS-semigroups X and Y , respectively. (i) If µ×ν is a non-zero fuzzy commutative KS-ideal of X×Y satisfying condition (C), then X × Y/µ× ν is a commutative KS-semigroup. (ii) If µ × ν is a non-zero fuzzy KS-p-ideal of X × Y satisfying condition (C), then X × Y/µ× ν is a positive implicative KS-semigroup. (iii) If µ× ν is a non-zero fuzzy implicative KS-ideal of X × Y satisfying condition (C), then X × Y/µ× ν is an implicative KS-semigroup. Recall that in [5], if f : X → Y is an epimorphism of KS-semigroup, thenX/ ker f ∼= Y . Theorem 21. Let µ and ν be non-zero fuzzy KS-ideals of KS-semigroups X and Y , respectively, both satisfying condition (C). Then X × Y µ× ν ∼= X/µ× Y/ν. Proof. Let µ and ν be non-zero fuzzy KS-ideals of KS-semigroups X and Y , re- spectively, both satisfying condition (C). Define a map ψ : X × Y → X/µ × Y/ν by H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 20 of 23 ψ(x, y) = (µx, νy). Let (x1, y1), (x2, y2) ∈ X × Y such that (x1, y1) = (x2, y2). Then x1 = x2 and y1 = y2. Thus, ψ(x1, y1) = (µx1 , νy1) = (µx2 , νy2) = ψ(x2, y2). Thus, ψ is well-defined. Now, let (x1, y1), (x2, y2) ∈ X × Y . Then ψ((x1, y1) ∗ (x2, y2)) = ψ(x1 ∗X x2, y1 ∗Y y2) = (µx1∗Xx2 , νy1∗Y y2) = (µx1 ∗X µx2 , νy1 ∗Y νy2) = (µx1 , νy1) ∗ (µx2 , νy2) = ψ(x1, y1) ∗ ψ(x2, y2) and ψ((x1, y1)(x2, y2)) = ψ(x1 ·X x2, y1 ·Y y2) = (µx1·Xx2 , νy1·Y y2) = (µx1 ·X µx2 , νy1 ·Y νy2) = (µx1 , νy1)(µx2 , νy2) = ψ(x1, y1)ψ(x2, y2). Thus, ψ is a homomorphism. Moreover, for each (µx, νy) ∈ X/µ × Y/ν, there exists (x, y) ∈ X × Y such that ψ(x, y) = (µx, νy). Thus, ψ is an epimorphism. By First Isomorphism Theorem, we have X × Y kerψ ∼= X/µ× Y/ν. Let K = kerψ. Then X × Y kerψ = X × Y K = {K(x,y) : (x, y) ∈ X × Y }. Similarly, X × Y µ× ν = {µ× ν(x,y) : (x, y) ∈ X × Y }. Claim: K(x,y) ∼= µ× ν(x,y). For (α, β) ∈ K(x,y), (α, β) ∈ K(x,y) ⇔ (α, β) ∼K (x, y) ⇔ (α, β) ∗ (x, y) ∈ K, (x, y) ∗ (α, β) ∈ K ⇔ ψ((α, β) ∗ (x, y)) = (µ0, ν0) = ψ((x, y) ∗ (α, β)), where µ0 and ν0 are fixed elements in X/µ and Y/ν, respectively ⇔ ψ(α ∗ x, β ∗ y) = (µ0, ν0) = ψ(x ∗ α, y ∗ β) ⇔ (µα∗x, νβ∗y) = (µ0, ν0) = (µx∗α, νy∗β) H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 21 of 23 ⇔ µα∗x = µx∗α = µ0, νβ∗y = νy∗β = ν0 ⇔ µα = µx, νβ = νy. Also, for (α, β) ∈ µ× ν(x,y), (α, β) ∈ µ× ν(x,y) ⇔ (α, β) ∼µ×ν (x, y) ⇔ µ× ν((α, β) ∗ (x, y)) > 0 and µ× ν((x, y) ∗ (α, β)) > 0 ⇔ µ× ν(α ∗ x, β ∗ y) > 0 and µ× ν(x ∗ α, y ∗ β) > 0 ⇔ min{µ(α ∗ x), ν(β ∗ y)} > 0 and min{µ(x ∗ α), ν(y ∗ β)} > 0 ⇔ µ(α ∗ x) > 0, ν(β ∗ y) > 0 and µ(x ∗ α) > 0, ν(y ∗ β) > 0 ⇔ µ(α ∗ x) > 0 and µ(x ∗ α) > 0, ν(β ∗ y) > 0 and ν(y ∗ β) > 0 ⇔ α ∼µ x, β ∼ν y ⇔ µα = µβ, νβ = νy. Thus, (α, β) ∈ K(x,y) ⇔ (α, β) ∈ µ× ν(x,y). This proves the claim. Hence, X × Y µ× ν ∼= X × Y kerψ ∼= X/µ× Y/ν. Lemma 4. Let J and J be ideals of KS-semigroups X and Y , respectively. Then J × I is an ideal of X × Y . Moreover, X × Y/J × I ∼= X/J × Y/I. Proof. Let J and J be ideals of KS-semigroups X and Y , respectively. Let (x, y) ∈ X × Y and (a, b) ∈ J × I. Since J and I are ideals, (x, y) · (a, b) = (x ·X a, y ·Y b) ∈ J × I. Similarly, (a, b) · (x, y) ∈ J × I. Now, let (x1, y1), (x2, y2) ∈ X × Y such that (x1 ∗X x2, y1 ∗Y y2) = (x1, y1) ∗ (x2, y2) ∈ J × I and (x2, y2) ∈ J × I. Since J and I are ideals, (x1, y1) ∈ J × I. Thus, J × I is an ideal of X × Y . Define φ : X×Y → X/J×Y/I by φ(x, y) = (Jx, Iy). Let (x1, y1), (x2, y2) ∈ X×Y such that (x1, y1) = (x2, y2). Then x1 = x2 and y1 = y2 so that x1 ∼J x2 and y1 ∼I y2. Thus, Jx1 = Jx2 and Iy1 = Iy2 , that is, (Jx1 , Iy1) = (Jx2 , Iy2). This shows that φ is well-defined. Now, let (x1, y1), (x2, y2) ∈ X × Y . Then φ((x1, y1) ∗ (x2, y2)) = φ(x1 ∗X x2, y1 ∗Y y2) = (Jx1∗Xx2 , Iy1∗Y y2) = (Jx1 ∗X/J Jx2 , Iy1 ∗Y/I Iy2) = (Jx1 , Iy1) ∗ (Jx2 , Iy2) = φ(x1, y1) ∗ φ(x2, y2) and φ((x1, y1) · (x2, y2)) = φ(x1 ·X x2, y1 ·Y y2) H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 22 of 23 = (Jx1·Xx2 , Iy1·Y y2) = (Jx1 ·X/J Jx2 , Iy1 ·Y/I Iy2) = (Jx1 , Iy1) · (Jx2 , Iy2) = φ(x1, y1) · φ(x2, y2). Thus, φ is a homomorphism. For any (Jx, Iy) ∈ X/J × Y/I, there exists (x, y) ∈ X × Y such that φ(x, y) = (Jx, Iy). This means that φ is an epimorphism. Moreover, kerφ = {(x, y) ∈ X × Y : φ(x, y) = (J0, I0)} = {(x, y) ∈ X × Y : (Jx, Iy) = (J0, I0)} = {(x, y) ∈ X × Y : Jx = J0 and Iy = I0} = {(x, y) ∈ X × Y : x ∼J 0 and y ∼I 0} = {(x, y) ∈ X × Y : x = x ∗ 0 ∈ J and y = y ∗ 0 ∈ I} = J × I. By First Isomorphism Theorem, X × Y/J × I = X × Y/ kerφ ∼= X/J × Y/I. Theorem 22. Let µ and ν be non-zero fuzzy KS-ideals of KS-semigroups X and Y , respectively, both satisfying condition (C). If J and I are ideals of X and Y , respectively, then X × Y/µ× ν J × I/µ× ν ∼= X/J × Y/I. Proof. Let µ and ν be non-zero fuzzy KS-ideals of KS-semigroups X and Y , respec- tively, both satisfying condition (C). Let J , I be ideals of X and Y , respectively. Then by Lemma 4, J × I is an ideal of X × Y . By Proposition 6, µ × ν is a fuzzy KS-ideal in X × Y , so is in J × I. Thus, by Theorem 21, X × Y µ× ν ∼= X/µ× Y/ν and J × I µ× ν ∼= J/µ× I/ν. Hence, by Theorem 15 and Lemma 4, X × Y/µ× ν J × I/µ× ν ∼= X/µ× Y/ν J/µ× I/ν ∼= X/µ J/µ × Y/ν I/ν ∼= X/J × Y/I. 7. Conclusion In this paper, we studied new types of fuzzy ideals, namely, fuzzy commutative KS- ideals and fuzzy implicative KS-ideals. We compared them with existing fuzzy KS-ideals H. Sarapuddin, J. Vilela / Eur. J. Pure Appl. Math, 18 (3) (2025), 6327 23 of 23 and investigated some of their properties. We defined binary relation on a KS-semigroup using fuzzy KS-ideals with certain condition and showed that it is a congruence relation on a KS-semigroup. Using this congruence relation, we constructed a quotient KS-semigroup induced by fuzzy KS-ideals and showed that it is a generalization of the quotient KS- semigroup via ideals. 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