EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2235-2245 ISSN 1307-5543 – ejpam.com Published by New York Business Global ε- Lukasiewicz Fuzzy UP (BCC)-Subalgebras of UP (BCC)-Algebras Aiyared Iampan1,∗, Ramasamy Subasini2, Neelamegarajan Rajesh3 1 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2 Department of Mathematics, Pollachi Institute of Engineering and Technology, Pollachi-642205, Tamilnadu, India 3 Department of Mathematics, Rajah Serfoji Government College (affiliated to Bharathidasan University), Thanjavur-613005, Tamilnadu, India Abstract. The idea of Lukasiewicz t-norm is used to construct the concept of ε- Lukasiewicz fuzzy sets based on a given fuzzy set. The ε- Lukasiewicz fuzzy sets are applied to UP (BCC)-algebras. Moreover, the notion of ε- Lukasiewicz fuzzy UP (BCC)-subalgebras is introduced, and its various properties are investigated. Three subsets, so-called ∈-set, q-set, and O-set, are constructed, and the conditions under which they can be UP (BCC)-subalgebras are explored. 2020 Mathematics Subject Classifications: 03G25; 08A72 Key Words and Phrases: UP (BCC)-algebra, ε- Lukasiewicz fuzzy set, ε- Lukasiewicz fuzzy UP (BCC)-subalgebra, ∈-set, q-set, O-set. 1. Introduction Zadeh [12] first proposed the idea of fuzzy sets. The theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. After introducing the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. The integration between fuzzy sets and some uncertainty approaches, such as soft sets and rough sets, has been discussed in [1–3]. The new tech- nology allows very complex inferences about variations on a theme to be anticipated and fixed in a program. Lukasiewicz logic, which is the logic of the Lukasiewicz t-norm, is a non-classical and many-valued logic. It was originally defined in the early 20th century by Lukasiewicz as a three-valued logic. Iampan [7] introduced a new algebraic structure called UP-algebra. Somjanta et al. [11] and Guntasow et al. [5] applied fuzzy set theory in UP-algebras. Dokkhamdang et al. [4] introduced the notion of fuzzy UP-subalgebras ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5311 Email addresses: aiyared.ia@up.ac.th (A. Iampan), subasinimaths@gmail.com (R. Subasini), nrajesh topology@yahoo.co.in (N. Rajesh) https://www.ejpam.com 2235 © 2024 EJPAM All rights reserved. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2236 with thresholds of UP-algebras. The concepts of UP-algebras (see [7]) and BCC-algebras (see [9]) are the same concept, as shown by Jun et al. [8] in 2022. In this publication and following investigations, our research team will refer to it as BCC rather than UP out of respect for Komori, who first characterized it in 1984. In this paper, using the idea of Lukasiewicz t-norm, we construct the concept of ε- Lukasiewicz fuzzy sets based on a given fuzzy set and apply it to BCC-algebras. We define the concepts of ε- Lukasiewicz fuzzy BCC-subalgebras and investigate several properties. We provide conditions for an ε- Lukasiewicz fuzzy set to be an ε- Lukasiewicz fuzzy BCC- subalgebra. We discuss the characterizations of ε- Lukasiewicz fuzzy BCC-subalgebras. We construct three kinds of subsets, so-called ∈-set, q-set, and O-set, and we find the conditions under which they can be BCC-subalgebras. 2. Preliminaries The concept of BCC-algebras (see [9]) can be redefined without the condition (2.6) as follows: An algebra X = (X, ∗, 0) of type (2, 0) is called a BCC-algebra (see [6]) if it satisfies the following conditions: (∀x, y, z ∈ X)((y ∗ z) ∗ ((x ∗ y) ∗ (x ∗ z)) = 0) (2.1) (∀x ∈ X)(0 ∗ x = x) (2.2) (∀x ∈ X)(x ∗ 0 = 0) (2.3) (∀x, y ∈ X)(x ∗ y = 0 = y ∗ x ⇒ x = y) (2.4) After this, we assign X instead of a BCC-algebra (X, ∗, 0) until otherwise specified. We define a binary relation ≤ on X as follows: (∀x, y ∈ X)(x ≤ y ⇔ x ∗ y = 0) (2.5) In X, the following assertions are valid (see [7]). (∀x ∈ X)(x ≤ x) (2.6) (∀x, y, z ∈ X)(x ≤ y, y ≤ z ⇒ x ≤ z) (2.7) (∀x, y, z ∈ X)(x ≤ y ⇒ z ∗ x ≤ z ∗ y) (2.8) (∀x, y, z ∈ X)(x ≤ y ⇒ y ∗ z ≤ x ∗ z) (2.9) (∀x, y, z ∈ X)(x ≤ y ∗ x, in particular, y ∗ z ≤ x ∗ (y ∗ z)) (2.10) (∀x, y ∈ X)(y ∗ x ≤ x ⇔ x = y ∗ x) (2.11) (∀x, y ∈ X)(x ≤ y ∗ y) (2.12) (∀a, x, y, z ∈ X)(x ∗ (y ∗ z) ≤ x ∗ ((a ∗ y) ∗ (a ∗ z))) (2.13) (∀a, x, y, z ∈ X)(((a ∗ x) ∗ (a ∗ y)) ∗ z ≤ (x ∗ y) ∗ z) (2.14) (∀x, y, z ∈ X)((x ∗ y) ∗ z ≤ y ∗ z) (2.15) A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2237 (∀x, y, z ∈ X)(x ≤ y ⇒ x ≤ z ∗ y) (2.16) (∀x, y, z ∈ X)((x ∗ y) ∗ z ≤ x ∗ (y ∗ z)) (2.17) (∀a, x, y, z ∈ X)((x ∗ y) ∗ z ≤ y ∗ (a ∗ z)) (2.18) Definition 1. [7] A nonempty subset S of X is called a BCC-subalgebra of X if it satisfies the following properties: (∀x, y ∈ S)(x ∗ y ∈ S) (2.19) A fuzzy set [12] in a nonempty set X is defined to be a function µ : X → [0, 1], where [0, 1] is the unit closed interval of real numbers. Definition 2. [11] A fuzzy set µ in X is called a fuzzy BCC-subalgebra of X if it satisfies the following property: (∀x, y ∈ X)(µ(x ∗ y) ≥ min{µ(x), µ(y)}). (2.20) A fuzzy set µ in a set X of the form µ(x) = { t ∈ (0, 1] if x = a 0 if x ̸= a, is said to be a fuzzy point with support a and value t and is denoted by [a/t]. For a fuzzy set µ in a set X, we say that a fuzzy point [a/t] is (1) contained in µ, denoted by [a/t] ∈ µ, (see [10]) if µ(a) ≥ t, (2) quasi-coincident with µ, denoted by [a/t]qµ, (see [10]) if µ(a) + t > 1. Proposition 1. If µ is a fuzzy set in a set X and ε ∈ (0, 1), then its ε- Lukasiewicz fuzzy set Lε µ satisfies the following property: (1) (∀x, y ∈ X)(µ(x) ≥ µ(y) ⇒ Lε µ(x) ≥ Lε µ(y)) (2) (∀x ∈ X)([x/ε]qµ ⇒ Lε µ(x) = µ(x) + ε− 1) (3) (∀x ∈ X,∀δ ∈ (0, 1))(ε ≥ δ ⇒ Lε µ(x) ≥ Lδ µ(x)) 3. ε- Lukasiewicz fuzzy BCC-subalgebra of a BCC-algebra In this section, we will recall the definition of ε- Lukasiewicz fuzzy sets and introduce a new concept called ε- Lukasiewicz fuzzy BCC-subalgebras. Definition 3. Let µ be a fuzzy set in a set X and let ε ∈ [0, 1]. A function Lε µ : X → [0, 1]; x 7→ max{0, µ(x) + ε− 1} is called an ε- Lukasiewicz fuzzy set of µ in X. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2238 Definition 4. Let µ be a fuzzy set in X. Then its ε- Lukasiewicz fuzzy set Lε µ in X is called an ε- Lukasiewicz fuzzy BCC-subalgebra of X if it satisfies the following property: (∀x, y ∈ X,∀ta, tb ∈ (0, 1])([x/ta] ∈ Lε µ, [y/tb] ∈ Lε µ ⇒ [(x ∗ y)/min{ta, tb}] ∈ Lε µ) (3.1) Theorem 1. If µ is a fuzzy BCC-subalgebra of X, then its ε- Lukasiewicz fuzzy set Lε µ in X is an ε- Lukasiewicz fuzzy BCC-subalgebra of X. Proof. Assume that µ is a fuzzy BCC-subalgebra of X. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that [x/ta] ∈ Lε µ and [y/tb] ∈ Lε µ. Then Lε µ(x) ≥ ta and Lε µ(y) ≥ tb. Thus Lε µ(x ∗ y) = max{0, µ(x ∗ y) + ε− 1} ≥ max{0,min{µ(x), µ(y)} + ε− 1} = max{0,min{µ(x) + ε− 1, µ(y) + ε− 1}} = min{max{0, µ(x) + ε− 1},max{0, µ(y) + ε− 1}} = min{Lε µ(x), Lε µ(y)} ≥ min{ta, tb}. Hence, [(x∗y)/min{ta, tb}] ∈ Lε µ. Therefore, Lε µ is an ε- Lukasiewicz fuzzy BCC-subalgebra of X. The following example shows that the converse of Theorem 1 may not be true. Example 1. Let X = {0, 1, 2, 3, 4} with the following Cayley table: ∗ 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 3 0 2 0 1 0 0 4 3 0 1 2 0 4 4 0 4 2 3 0 Then X is a BCC-algebra. Define a fuzzy set µ as follows: µ(x) =  1.0 if x = 0 0.4 if x = 1 0.2 if x = 2 0.3 if x = 3 0.6 if x = 4. Given ε = 0.9, the ε- Lukasiewicz fuzzy set Lε µ of µ in X is given as follows: Lε µ(x) =  0.9 if x = 0 0.3 if x = 1 0.1 if x = 2 0.2 if x = 3. 0.5 if x = 4. Then Lε µ is an ε- Lukasiewicz fuzzy BCC-subalgebra of X. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2239 Theorem 2. Let µ be a fuzzy set in X. Then its ε- Lukasiewicz fuzzy set Lε µ in X is an ε- Lukasiewicz fuzzy BCC-subalgebra of X if and only if it satisfies the following property: (∀x, y ∈ X)(Lε µ(x ∗ y) ≥ min{Lε µ(x), Lε µ(y)}) (3.2) Proof. Suppose Lε µ is an ε- Lukasiewicz fuzzy BCC-subalgebra of X. Let x, y ∈ X. Then [x/Lε µ(x)] ∈ Lε µ and [y/Lε µ(y)] ∈ Lε µ. Thus, [(x ∗ y)/min{Lε µ(x), Lε µ(y)}] ∈ Lε µ by (3.1), which implies that Lε µ(x ∗ y) ≥ min{Lε µ(x), Lε µ(y)}. Conversely, suppose that Lε µ satisfies the condition (3.2). Let x, y ∈ X and ta, tb ∈ (0, 1] be such that [x/ta] ∈ Lε µ and [y/tb] ∈ Lε µ. Then Lε µ(x) ≥ ta and Lε µ(y) ≥ tb, which implies from (3.2) that Lε µ(x∗y) ≥ min{Lε µ(x), Lε µ(y)} ≥ min{ta, tb}. Thus, [(x∗y)/min{ta, tb}] ∈ Lε µ. Hence, Lε µ is an ε- Lukasiewicz fuzzy BCC-subalgebra of X. Proposition 2. If µ is a fuzzy BCC-subalgebra of X, then its ε- Lukasiewicz fuzzy set Lε µ satisfies the following property: (∀x ∈ X)(Lε µ(0) ≥ Lε µ(x)) (3.3) Proof. If µ is a fuzzy BCC-subalgebra of X, then µ(0) = µ(x∗x) ≥ min{µ(x), µ(x)} = µ(x) for all x ∈ X. It follows from Proposition 1 (1) that Lε µ(0) ≥ Lε µ(x) for all x ∈ X. The following example shows that the converse of Proposition 2 is not true in general. Example 2. [5] Let X = {0, 1, 2, 3} with the following Cayley table: ∗ 0 1 2 3 0 0 1 2 3 1 0 0 1 2 2 0 0 0 1 3 0 0 0 0 Then X is a BCC-algebra. Define a fuzzy set µ as follows: µ : X → [0, 1];x 7→  1 if x = 0 0 if x = 1 1 if x = 2 1 if x = 3 Given ε = 0.9, the ε- Lukasiewicz fuzzy set Lε µ of µ in X is given as follows: Lε µ : X → [0, 1];x 7→  0.9 if x = 0 0 if x = 1 0.9 if x = 2 0.9 if x = 3 Then Lε µ(0) ≥= Lε µ(x) for all x ∈ X but µ is not a fuzzy BCC-subalgebra of X because µ(2 ∗ 3) = µ(1) = 0 ≱ 1 = min{µ(2), µ(3)}. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2240 Proposition 3. If µ is a fuzzy BCC-subalgebra of X, then its ε- Lukasiewicz fuzzy set Lε µ satisfies the following property: (∀x, y ∈ X)(Lε µ(y) = Lε µ(0) ⇔ Lε µ(x ∗ y) ≥ Lε µ(x)) (3.4) Proof. Assume that Lε µ(y) = Lε µ(0) for all y ∈ X. Then Lε µ(x∗y) ≥ min{Lε µ(x), Lε µ(y)} = min{Lε µ(x), Lε µ(0)} = Lε µ(x) for all x, y ∈ X by the combination of Theorem 1 and Propo- sition 2. Conversely, suppose that Lε µ(x ∗ y) ≥ Lε µ(x) for all x, y ∈ X. Using (2.2) induces Lε µ(y) = Lε µ(0 ∗ y) ≥ Lε µ(0). The combination of this and Proposition 2 leads to Lε µ(y) = Lε µ(0) for all y ∈ X. Proposition 4. If µ is a fuzzy BCC-subalgebra of X, then its ε- Lukasiewicz fuzzy set Lε µ satisfies the following property: (∀x, y ∈ X,∀ta, tb ∈ (0, 1]) ( [x/ta] ∈ Lε µ, [y/tb] ∈ Lε µ ⇒ [(x ∗ (0 ∗ y))/min{ta, tb}] ∈ Lε µ ) (3.5) Proof. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that [x/ta] ∈ Lε µ and [y/tb] ∈ Lε µ. Then Lε µ(x) ≥ ta and Lε µ(y) ≥ tb. Thus Lε µ(x ∗ (0 ∗ y)) = max{0, µ(x ∗ (0 ∗ y)) + ε− 1} ≥ max{0,min{µ(x), µ(0 ∗ y)} + ε− 1} ≥ max{0,min{µ(x),min{µ(0), µ(y)}} + ε− 1} = max{0,min{µ(x), µ(y)} + ε− 1} = max{0,min{µ(x) + ε− 1, µ(y) + ε− 1}} = min{max{0, µ(x) + ε− 1},max{0, µ(y) + ε− 1}} = min{Lε µ(x), Lε µ(y)} ≥ min{ta, tb}. Hence, [(x ∗ (0 ∗ y))/min{ta, tb}] ∈ Lε µ. We provide conditions for an ε- Lukasiewicz fuzzy set to be an ε- Lukasiewicz fuzzy BCC-subalgebra. Theorem 3. Let µ be a fuzzy set in X. If its ε- Lukasiewicz fuzzy set Lε µ satisfies the following property: [y/tb] ∈ Lε µ, [z/tc] ∈ Lε µ ⇒ [(x ∗ y)/min{tb, tc}] ∈ Lε µ (3.6) for all tb, tc ∈ (0, 1] and x, y, z ∈ X with z ≤ x, then Lε µ is an ε- Lukasiewicz fuzzy BCC- subalgebra of X. Proof. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that [x/ta] ∈ Lε µ and [y/tb] ∈ Lε µ. Since x ≤ x for all x ∈ X, it follows from (3.6) that [(x ∗ y)/min{ta, tb}] ∈ Lε µ. Hence, Lε µ is an ε- Lukasiewicz fuzzy BCC-subalgebra of X. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2241 Proposition 5. Let µ be a fuzzy set in X. Then every ε- Lukasiewicz fuzzy BCC-subalgebra Lε µ of X satisfies the following property: (∀x, y ∈ X,∀ta, tb ∈ (0, 1])([x/ta] ∈ Lε µ, [y/tb] ∈ Lε µ ⇒ [(x ∗ (0 ∗ y))/min{tb, tc}] ∈ Lε µ) (3.7) Proof. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that [x/ta] ∈ Lε µ and [y/tb] ∈ Lε µ. Then Lε µ(x) ≥ ta and Lε µ(y) ≥ tb. It follows from Theorem 2 and Proposition 2 that Lε µ(x ∗ (0 ∗ y)) ≥ min{Lε µ(x), Lε µ(0 ∗ y)} ≥ min{Lε µ(x),min{Lε µ(0), Lε µ(y)}} = min{Lε µ(x), Lε µ(y)} ≥ min{ta, tb}. Hence, [(x ∗ (0 ∗ y))/min{ta, tb}] ∈ Lε µ. Let µ be a fuzzy set in X. For an ε- Lukasiewicz fuzzy set Lε µ of µ in X and t ∈ (0, 1], consider the sets (Lε µ, t)∈ = {x ∈ X : [x/t] ∈ Lε µ}, (Lε µ, t)q = {x ∈ X : [x/t]qLε µ}, which are called the ∈-set and q-set, respectively, of Lε µ (with value t). We explore the conditions under which the ∈-set and q-set of ε- Lukasiewicz fuzzy sets can be BCC-subalgebras. Theorem 4. Let Lε µ be an ε- Lukasiewicz fuzzy set of a fuzzy set µ in X. Then the ∈-set (Lε µ, t)∈ of Lε µ with value t ∈ (0.5, 1] is a BCC-subalgebra of X if and only if the following assertion is valid: (∀x, y ∈ X)(min{Lε µ(x), Lε µ(y)} ≤ max{Lε µ(x ∗ y), 0.5}) (3.8) Proof. Assume that the ∈-set (Lε µ, t)∈ of Lε µ with value t ∈ (0.5, 1] is a BCC-subalgebra of X. If the condition (3.8) is not valid, then there exist a, b ∈ X such that min{Lε µ(a), Lε µ(b)} > max{Lε µ(a ∗ b), 0.5}. If we take s = min{Lε µ(a), Lε µ(b)}, then s ∈ (0.5, 1] and [a/s], [b/s] ∈ Lε µ, that is, a, b ∈ (Lε µ, s)∈. Since (Lε µ, s)∈ is a BCC-subalgebra of X, we have a ∗ b ∈ (Lε µ, s)∈. But [(a ∗ b)/s] /∈ Lε µ implies a ∗ b /∈ (Lε µ, s)∈, a contradiction. Thus, min{Lε µ(x), Lε µ(y)} ≤ max{Lε µ(x ∗ y), 0.5} for all x, y ∈ X. Conversely, suppose that Lε µ satisfies the condition (3.8). Let t ∈ (0.5, 1] and x, y ∈ X be such that x ∈ (Lε µ, t)∈ and y ∈ (Lε µ, t)∈. Then Lε µ(x) ≥ t and Lε µ(y) ≥ t, which imply from (3.8) that 0.5 < t ≤ min{Lε µ(x), Lε µ(y)} ≤ max{Lε µ(x∗y), 0.5}. Thus, [(x∗y)/t] ∈ Lε µ, that is, x ∗ y ∈ (Lε µ, t)∈. So, (Lε µ, t)∈ is a BCC-subalgebra of X for t ∈ (0.5, 1]. Theorem 5. Let Lε µ be an ε- Lukasiewicz fuzzy set of a fuzzy set µ in X. If µ is a fuzzy BCC-subalgebra of X, then the nonempty q-set (Lε µ, t)q of Lε µ with value t ∈ (0, 1] is a BCC-subalgebra of X. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2242 Proof. Let t ∈ (0, 1] and x, y ∈ (Lε µ, t)q. Then [x/t]qLε µ and [y/t]qLε µ, that is, Lε µ(x) + t > 1 and Lε µ(y) + t > 1. It follows from Theorems 1 and 2 that Lε µ(x ∗ y) + t ≥ min{Lε µ(x), Lε µ(y)} + t = min{Lε µ(x) + t, Lε µ(y) + t} > 1. Thus, [(x ∗ y)/t]qLε µ. So, x ∗ y ∈ (Lε µ, t)q. Hence, (Lε µ, t)q is a BCC-subalgebra of X. The following example shows that the converse of Theorem 5 is not true in general. Example 3. From the BCC-algebra X in Example 2, define a fuzzy set µ as follows: µ : X → [0, 1];x 7→  0.1 if x = 0 0 if x = 1 0.1 if x = 2 0.1 if x = 3 Given ε = 0.2, the ε- Lukasiewicz fuzzy set Lε µ of µ in X is given as follows: Lε µ : X → [0, 1];x 7→  0 if x = 0 0 if x = 1 0 if x = 2 0 if x = 3 Then the q-set (Lε µ, t)q of Lε µ with value t ∈ (0, 1] is empty but µ is not a fuzzy BCC- subalgebra of X because µ(2 ∗ 3) = µ(1) = 0 ≱ 0.1 = min{µ(2), µ(3)}. Theorem 6. Let µ be a fuzzy set in X. For an ε- Lukasiewicz fuzzy set Lε µ of µ in X, if the q-set (Lε µ, t)q is a BCC-subalgebra of X, then Lε µ satisfies the following property: (∀x, y ∈ X,∀ta, tb ∈ (0, 0.5]) ( [x/ta]qLε µ, [y/tb]qL ε µ ⇒ [(x ∗ y)/max{ta, tb}] ∈ Lε µ ) (3.9) Proof. Let x, y ∈ X and ta, tb ∈ (0, 0.5] be such that [x/ta]qLε µ and [y/tb]qL ε µ. Then x ∈ (Lε µ, ta)q ⊆ (Lε µ,max{ta, tb})q and y ∈ (Lε µ, tb)q ⊆ (Lε µ,max{ta, tb})q. Thus, x ∗ y ∈ (Lε µ,max{ta, tb})q. Since max{ta, tb} ≤ 0.5, we have Lε µ(x ∗ y) > 1 − max{ta, tb} ≥ max{ta, tb}. Hence, [(x ∗ y)/max{ta, tb}] ∈ Lε µ. Let µ be a fuzzy set in X. For an ε- Lukasiewicz fuzzy set Lε µ of µ in X, consider the set: O(Lε µ) = {x ∈ X : Lε µ(x) > 0}, which is called the O-set of Lε µ. It is observed that O(Lε µ) = {x ∈ X : µ(x) + ε− 1 > 0}. Theorem 7. Let Lε µ be an ε- Lukasiewicz fuzzy set of a fuzzy set µ in X. If µ is a fuzzy BCC-subalgebra of X, then the nonempty O-set O(Lε µ) of Lε µ is a BCC-subalgebra of X. Proof. Let x, y ∈ O(Lε µ). Then µ(x) + ε − 1 > 0 and µ(y) + ε − 1 > 0. If µ is a fuzzy BCC-subalgebra of X, then Lε µ is an ε- Lukasiewicz fuzzy BCC-subalgebra of X by Theorem 1. It follows from Theorem 2 that Lε µ(x ∗ y) ≥ min{Lε µ(x), Lε µ(y)} = min{µ(x) + ε − 1, µ(y) + ε − 1} > 0. Thus, x ∗ y ∈ O(Lε µ). Hence, O(Lε µ) is a BCC- subalgebra of X. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (3) (2024), 2235-2245 2243 Theorem 8. Let µ be a fuzzy set in X. If an ε- Lukasiewicz fuzzy set Lε µ of µ in X satisfies the following property: [x/ta] ∈ Lε µ, [y/tb] ∈ Lε µ ⇒ [(x ∗ y)/max{ta, tb}]qLε µ (3.10) for all x, y ∈ X and ta, tb ∈ (0, 1], then the nonempty O-set O(Lε µ) of Lε µ is a BCC- subalgebra of X. Proof. Assume that Lε µ satisfies the condition (3.10) for all x, y ∈ X and ta, tb ∈ (0, 1]. Let x, y ∈ O(Lε µ). Then µ(x) + ε− 1 > 0 and µ(y) + ε− 1 > 0. Since [x/Lε µ(x)] ∈ Lε µ and [y/Lε µ(y)] ∈ Lε µ, it follows from (3.10) that [(x ∗ y)/max{Lε µ(x ∗ (y ∗ z)), Lε µ(y)}]qLε µ. (3.11) If x ∗ y /∈ O(Lε µ), then Lε µ(x ∗ y) = 0. Thus, Lε µ(x ∗ y) + max{Lε µ(x), Lε µ(y)} = max{Lε µ(x), Lε µ(y)} = max{max{0, µ(x) + ε− 1},max{0, µ(y) + ε− 1}} = max{µ(x) + ε− 1, µ(y) + ε− 1} = max{µ(x), µ(y)} + ε− 1 ≤ 1 + ε− 1 = ε ≤ 1, which shows that (3.11) is not valid. This is a contradiction. Hence, x ∗ y ∈ O(Lε µ). Therefore, O(Lε µ) is a BCC-subalgebra of X. Theorem 9. Let µ be a fuzzy set in X. If an ε- Lukasiewicz fuzzy set Lε µ of µ in X satisfies the condition (3.9) for all x, y ∈ X and ta, tb ∈ (0, 1], then the nonempty O-set O(Lε µ) of Lε µ is a BCC-subalgebra of X. Proof. Let x, y ∈ O(Lε µ). Then µ(x) + ε − 1 > 0 and µ(y) + ε − 1 > 0. Thus, Lε µ(x) + 1 = max{0, µ(x) + ε− 1} + 1 = µ(x) + ε− 1 + 1 = µ(x) + ε > 1 and Lε µ(y) + 1 = max{0, µ(y) + ε−1}+ 1 = µ(y) + ε−1 + 1 = µ(y) + ε > 1, that is, [x/1]qLε µ and [y/1]qLε µ. It follows from (3.9) that [(x ∗ y)/1] = [(x ∗ y)/max{1, 1}] ∈ Lε µ. (3.12) If x∗ y /∈ O(Lε µ), then Lε µ(x∗ y) = 0 < 1 and so (3.12) is not valid. This is a contradiction. Thus, x ∗ y ∈ O(Lε µ). Hence, O(Lε µ) is a BCC-subalgebra of X. REFERENCES 2244 4. Conclusions The concept of ε- Lukasiewicz fuzzy sets using Lukasiewicz t-norm was introduced by Jun [8]. In this paper, the ε- Lukasiewicz fuzzy set has been applied to BCC-subalgebras in BCC-algebras, introducing the concept of ε- Lukasiewicz fuzzy BCC-subalgebras, and examining several properties. 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