EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3209-3222 ISSN 1307-5543 – ejpam.com Published by New York Business Global ε- Lukasiewicz Fuzzy UP (BCC)-Ideals: a New Frontier in 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. This paper presents the development of ε- Lukasiewicz fuzzy sets using the Lukasiewicz t-norm derived from a given fuzzy set. These ε- Lukasiewicz fuzzy sets are subsequently applied to UP (BCC)-algebras. In addition, the paper introduces the concept of ε- Lukasiewicz fuzzy UP (BCC)-ideals and examines their various properties. Three specific subsets, termed the ∈-set, q- set, and O-set, are constructed, with an exploration of the conditions under which these subsets qualify as UP (BCC)-ideals. 2020 Mathematics Subject Classifications: 03G25, 08A72 Key Words and Phrases: UP (BCC)-algebra, ε- Lukasiewicz fuzzy set, ε- Lukasiewicz fuzzy UP (BCC)-ideal, ∈-set, q-set, O-set 1. Introduction Zadeh [21] originally proposed the concept of fuzzy sets. Fuzzy set theory finds numer- ous real-world applications, and many researchers have extensively explored its principles. Following the introduction of fuzzy sets, various studies have focused on their generaliza- tions. The intersection of fuzzy sets with other uncertainty models, such as soft and rough sets, has been explored in [1–3]. Modern technology enables sophisticated inferences and problem-solving capabilities, particularly in handling theme variations through program- ming. Lukasiewicz logic, governed by the Lukasiewicz t-norm, represents a non-classical, multi-valued logic initially formulated in the early 20th century with three truth values. One significant extension is the ε- Lukasiewicz fuzzy set, derived from the Lukasiewicz logic, a non-classical, many-valued logic. The ε- Lukasiewicz fuzzy set is based on the Lukasiewicz ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5450 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 3209 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3210 t-norm and t-conorm, which define fuzzy operations such as intersection, union, and com- plement. The parameter ε is introduced to provide additional flexibility and control over the set’s fuzziness level. Fundamental concepts of the ε- Lukasiewicz fuzzy set can be found in [5, 6, 14]. ε- Lukasiewicz fuzzy sets are highly applicable across various fields by effectively mod- elling uncertainty and imprecision. In decision-making systems, they enable the incorpo- ration of ambiguous information, allowing for more nuanced evaluations of alternatives. In medical diagnosis, these fuzzy sets help assess unclear symptoms and test results, lead- ing to personalized treatment plans. In financial analysis, they enhance risk assessment by capturing market uncertainties, while in artificial intelligence, they improve knowledge representation and reasoning, enabling AI systems to make human-like decisions. Over- all, ε- Lukasiewicz fuzzy sets offer valuable insights and practical solutions in uncertain environments across multiple disciplines. Iampan [9] introduced UP-algebras as a novel algebraic structure. Somjanta et al. [20] and Guntasow et al. [7] applied fuzzy set theory within the framework of UP-algebras. Dokkhamdang et al. [4] introduced the concept of fuzzy UP-subalgebras with thresholds in UP-algebras. Poungsumpao et al. [16] studied fuzzy UP-subalgebras and fuzzy UP-ideals of UP-algebras in terms of upper t-(strong) level subsets and lower t-(strong) level sub- sets. Senapati et al. [18] pioneered the concept of cubic sets within UP-subalgebras and UP-ideals in the framework of UP-algebras. Their research delved into the intricate rela- tionships between cubic UP-subalgebras and cubic UP-ideals, revealing new insights into their structural connections. Senapati et al. [19] explored the concept of interval-valued intuitionistic fuzzy sets, applying it to both UP-subalgebras and UP-ideals in UP-algebras. Their work examined the homomorphic images and inverse images of these interval-valued intuitionistic fuzzy UP-subalgebras and UP-ideals, providing deeper insights into their structural behaviour. Jana et al. [11] introduced the concept of quasi-coincidence between an intuitionistic fuzzy point and an intuitionistic fuzzy set. They further developed and ex- plored the notions of (∈,∈ ∨q)-intuitionistic fuzzy BCI-subalgebras within the framework of BCI-algebras, offering new perspectives on their structure and properties. UP-algebras (see [9]) and BCC-algebras (see [15]) are identified as the same concept, as demonstrated by Jun et al. [13] in 2022. For consistency with Komori’s initial characterization in 1984, our research team will adopt the term BCC rather than UP in subsequent investigations and publications. In this paper, we utilize the Lukasiewicz t-norm to introduce the concept of ε- Lukasiewicz fuzzy sets derived from a given fuzzy set, applying this framework to BCC-algebras. We define ε- Lukasiewicz fuzzy BCC-ideals and explore their properties. Conditions are established for an ε- Lukasiewicz fuzzy set to qualify as an ε- Lukasiewicz fuzzy BCC- subalgebra, and we characterize these structures. Additionally, we introduce three specific subsets—referred to as ∈-set, q-set, and O-set—and determine the conditions under which they can function as BCC-ideals. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3211 2. Preliminaries The concept of BCC-algebras (referenced in [15]) can be reformulated without the condition (2.6) as follows: An algebra X = (X, ◦, 0) of type (2, 0) is called a BCC-algebra (see [8]) 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 = 0 ⇒ x = y) (2.4) Following this, we will denote X as a BCC-algebra (X, ◦, 0) unless stated otherwise. 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 [9]). (∀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) (∀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. [9] A nonempty subset S of X is called (1) a BCC-subalgebra of X if it satisfies the following property: (∀x, y ∈ S)(x ◦ y ∈ S) (2.19) (2) a BCC-ideal of X if it satisfies the following properties: 0 ∈ S (2.20) (∀x, y, z ∈ X)(x ◦ (y ◦ z), y ∈ S ⇒ x ◦ z ∈ S) (2.21) A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3212 A fuzzy set [21] 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. [20] A fuzzy set µ in X is said to be (1) a fuzzy BCC-subalgebra of X if it satisfies the following property: (∀x, y ∈ X)(µ(x ◦ y) ≥ min{µ(x), µ(y)}) (2.22) (2) a fuzzy BCC-ideal of X if it satisfies the following properties: (∀x ∈ X)(µ(0) ≥ µ(x)) (2.23) (∀x, y, z ∈ X)(µ(x ◦ z) ≥ min{µ(x ◦ (y ◦ z)), µ(y)}) (2.24) A fuzzy set µ in a set X of the form µ(x) = { t ∈ (0, 1] if x = a 0 otherwise, 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 [17]) if µ(a) ≥ t, (2) quasi-coincident with µ, denoted by [a/t]qµ, (see [17]) 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-ideals of BCC-algebras In this section, we revisit the concept of ε- Lukasiewicz fuzzy sets and introduce an innovative idea: ε- Lukasiewicz fuzzy BCC-ideals. 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. Definition 4. [10] 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) A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3213 Definition 5. 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-ideal of X if it satisfies the following properties: (∀x ∈ X,∀ta ∈ (0, 1])([x/ta] ∈ Lε µ ⇒ [0/ta] ∈ Lε µ) (3.2) (∀x, y, z ∈ X,∀ta, tb ∈ (0, 1]) ( [(x ◦ (y ◦ z))/ta] ∈ Lε µ, [y/tb] ∈ Lε µ ⇒ [(x ◦ z)/min{ta, tb}] ∈ Lε µ ) (3.3) Example 1. [7] Let X = {0, 1, 2, 3} with the following Cayley table: ◦ 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 1 0 3 3 0 1 2 0 Then X is a BCC-algebra. Define a fuzzy set µ as follows: µ : X → [0, 1];x 7→  0.6 if x = 0 0.4 if x = 1 0.3 if x = 2 0.2 if x = 3 Given ε = 0.85, the ε- Lukasiewicz fuzzy set Lε µ of µ in X is given as follows: Lε µ : X → [0, 1];x 7→  0.45 if x = 0 0.25 if x = 1 0.15 if x = 2 0.05 if x = 3 Then Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X. Theorem 1. Let µ be a fuzzy set in X. Then its ε- Lukasiewicz fuzzy set Lε µ in X is an ε- Lukasiewicz fuzzy BCC-ideal of X if and only if it satisfies the following properties: (∀x ∈ X)(Lε µ(0) ≥ Lε µ(x)) (3.4) (∀x, y, z ∈ X)(Lε µ(x ◦ z) ≥ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)}) (3.5) Proof. Assume that Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X. Let x ∈ X. Since [x/Lε µ(x)] ∈ Lε µ, we have [0/Lε µ(x)] ∈ Lε µ by (3.2), and so Lε µ(0) ≥ Lε µ(x). Note that [(x◦(y◦z))/Lε µ(x◦(y◦z))] ∈ Lε µ, [y/L ε µ(y)] ∈ Lε µ for all x, y, z ∈ X. It follows from (3.3) that [Lε µ(x◦z)/min{Lε µ(x◦(y◦z)), Lε µ(y)}] ∈ Lε µ, that is, Lε µ(x◦z) ≥ min{Lε µ(x◦(y◦z)), Lε µ(y)} for all x, y, z ∈ X. Conversely, let Lε µ be an ε- Lukasiewicz fuzzy set satisfying the conditions (3.4) and (3.5). If [x/t] ∈ Lε µ for all x ∈ X and t ∈ (0, 1], then Lε µ(0) ≥ Lε µ(x) ≥ t for all x ∈ X by (3.4). Hence, [0/t] ∈ Lε µ. Let x, y, z ∈ X and ta, tb ∈ (0, 1] be such that [(x◦(y◦z))/ta] ∈ Lε µ and [y/tb] ∈ Lε µ. Then Lε µ(x ◦ (y ◦ z)) ≥ ta and Lε µ(y) ≥ tb. It follows from (3.5) that Lε µ(x ◦ z) ≥ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} ≥ min{ta, tb}. Hence, [((x ◦ z)/min{ta, tb} ∈ Lε µ. Therefore, Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3214 Proposition 2. If Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X, then (∀x, y ∈ X)(y ≤ x ⇒ Lε µ(y) ≤ Lε µ(x)). (3.6) Proof. Let Lε µ be an ε- Lukasiewicz fuzzy BCC-ideal of X. Let x, y ∈ X be such that y ≤ x. Then Lε µ(x) = Lε µ(0 ◦ x) = max{0, µ(0 ◦ x) + ε− 1} ≥ max{0,min{µ(0 ◦ (y ◦ x)), µ(y)} + ε− 1} = max{0,min{µ(y ◦ x) + ε− 1, µ(y) + ε− 1}} = min{max{0, µ(0) + ε− 1},max{0, µ(y) + ε− 1}} = min{Lε µ(0), Lε µ(y)} = Lε µ(y). Proposition 3. If Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X, then (∀w, x, y, z ∈ X)(x ≤ w ◦ (y ◦ z) ⇒ Lε µ(x ◦ z) ≥ min{Lε µ(w), Lε µ(y)}). (3.7) Proof. Let Lε µ be an ε- Lukasiewicz fuzzy BCC-ideal of X. Let w, x, y, z ∈ X be such that x ≤ w ◦ (y ◦ z). Then Lε µ(x ◦ z) = max{0, µ(x ◦ z) + ε− 1} ≥ max{0,min{µ(x ◦ (y ◦ z)), µ(y)} + ε− 1} ≥ max{0,min{min{µ(x ◦ (w ◦ (y ◦ z))), µ(w)}, µ(y)} + ε− 1} = max{0,min{min{µ(0), µ(w)}, µ(y)} + ε− 1} = max{0,min{µ(w), µ(y)} + ε− 1} = max{0,min{µ(w) + ε− 1, µ(y) + ε− 1}} = min{max{0, µ(w) + ε− 1},max{0, µ(y) + ε− 1}} = min{Lε µ(w), Lε µ(y)}. Proposition 4. If Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X, then (∀x, y, z ∈ X)(x ≤ y ◦ z ⇒ Lε µ(x ◦ z) ≥ Lε µ(y)). (3.8) Proof. Let Lε µ be an ε- Lukasiewicz fuzzy BCC-ideal of X. Let x, y, z ∈ X be such that x ≤ y ◦ z in X. By Proposition 3, put w = 0. By (3.7), we have x ≤ 0 ◦ (y ◦ z). Hence, Lε µ(x ◦ z) ≥ min{Lε µ(0), Lε µ(y)} = Lε µ(y). Proposition 5. Every ε- Lukasiewicz fuzzy BCC-ideal of X is an ε- Lukasiewicz fuzzy BCC-subalgebra of X. Proof. Let Lε µ ba an ε- Lukasiewicz fuzzy BCC-ideal of X and let x, y ∈ X. By (2.10), we have x ≤ y ◦ x. It follows from (3.6) that Lε µ(y ◦ x) ≥ Lε µ(x) ≥ min{Lε µ(y), Lε µ(x)}. The following example shows that the converse of Proposition 5 is not generally true. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3215 Example 2. [7] 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 0.6 if x = 1 0.4 if x = 2 0.1 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 0.5 if x = 1 0.3 if x = 2 0 if x = 3 Then Lε µ is an ε- Lukasiewicz fuzzy BCC-subalgebra of X but not an ε- Lukasiewicz fuzzy BCC-ideal of X because Lε µ(0 ◦ 3) = Lε µ(3) = 0 ≱ 0.3 = min{0.3, 0.5} = min{Lε µ(0 ◦ (1 ◦ 3)), Lε µ(1)} by Theorem 1. Theorem 2. If µ is a fuzzy BCC-ideal of X, then its ε- Lukasiewicz fuzzy set Lε µ in X is an ε- Lukasiewicz fuzzy BCC-ideal of X. Proof. Assume that µ is a fuzzy BCC-ideal of X. Let x ∈ X and ta ∈ (0, 1] be such that [x/ta] ∈ Lε µ. Then Lε µ(x) ≥ ta. Thus Lε µ(0) = max{0, µ(0) + ε− 1} ≥ max{0, µ(x) + ε− 1} = Lε µ(x) ≥ ta. Then [0/ta] ∈ Lε µ. Let x, y, z ∈ X and ta, tb ∈ (0, 1] be such that [x ◦ (y ◦ z)/ta] ∈ Lε µ and [y/tb] ∈ Lε µ. Then Lε µ(x ◦ (y ◦ z)) ≥ ta and Lε µ(y) ≥ tb. Thus Lε µ(x ◦ z) = max{0, µ(x ◦ z) + ε− 1} ≥ max{0,min{µ(x ◦ (y ◦ z)), µ(y)} + ε− 1} = max{0,min{µ(x ◦ (y ◦ z)) + ε− 1, µ(y) + ε− 1}} = min{max{µ(x ◦ (y ◦ z)) + ε− 1},max{µ(y) + ε− 1}} = min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} ≥ min{ta, tb}. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3216 Then [(x ◦ z)/min{ta, tb}] ∈ Lε µ. Hence, Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X. 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 a BCC-ideal. Theorem 3. 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-ideal of X if and only if the following assertions are valid: (∀x ∈ X)(max{Lε µ(0), 0.5} ≥ Lε µ(x)) (3.9) (∀x, y, z ∈ X)(max{Lε µ(x ◦ z), 0.5} ≥ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)}) (3.10) Proof. Assume that the ∈-set (Lε µ, t)∈ of Lε µ with value t ∈ (0.5, 1] is a BCC-ideal of X. If the condition (3.9) is not valid, then there exists a ∈ X such that max{Lε µ(0), 0.5} < Lε µ(a). Thus Lε µ(a) ∈ (0.5, 1] and Lε µ(a) > Lε µ(0). If we take t = Lε µ(a), then [a/t] ∈ Lε µ, that is, a ∈ (Lε µ, s)∈ and 0 /∈ (Lε µ, t)∈. This is a contradiction and so Lε µ(x) ≤ max{Lε µ(0), 0.5} for all x ∈ X. Now, if the condition (3.10) is not valid, then there exist a, b, c ∈ X such that max{Lε µ(a ◦ c), 0.5} < min{Lε µ(a ◦ (b ◦ c)), Lε µ(b)}. If we take s = min{Lε µ(a ◦ (b ◦ c)), Lε µ(b)}, then s ∈ (0.5, 1] and [a ◦ (b ◦ c)/s], [b/s] ∈ Lε µ, that is, a ◦ (b ◦ c), b ∈ (Lε µ, s)∈. Since (Lε µ, s)∈ is a BCC-ideal of X, we have a ◦ c ∈ (Lε µ, s)∈. But [(a ◦ c)/s]∈Lε µ implies a ◦ c /∈ (Lε µ, s)∈, a contradiction. Thus, max{Lε µ(x ◦ z), 0.5} ≥ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} for all x, y, z ∈ X. Conversely, suppose that Lε µ satisfies the conditions (3.9) and (3.10). For every t ∈ (0.5, 1], we have 0.5 < t ≤ Lε µ(x) ≤ max{Lε µ(0), 0.5} for all x ∈ (Lε µ, t)∈ by (3.9). Then 0 ∈ (Lε µ, t)∈. Let t ∈ (0.5, 1] and x, y, z ∈ X be such that x ◦ (y ◦ z) ∈ (Lε µ, t)∈ and y ∈ (Lε µ, t)∈. Then Lε µ(x ◦ (y ◦ z)) ≥ t and Lε µ(y) ≥ t, which imply from (3.10) that 0.5 < t ≤ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} ≤ max{Lε µ(x ◦ z), 0.5}. Thus, [(x ◦ z)/t] ∈ Lε µ, that is, x ◦ z ∈ (Lε µ, t)∈. Hence, (Lε µ, t)∈ is a BCC-ideal of X for t ∈ (0.5, 1]. In Theorem 3, if t /∈ (0.5, 1], that is, there exists at least one t ≤ 0.5, then Theorem 3 is incorrect, as shown in the following example. Example 3. Let X = {0, 1, 2, 3} with the following Cayley table: ◦ 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 1 0 0 3 0 1 2 0 A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3217 Then X is a BCC-algebra. Define a fuzzy set µ as follows: µ : X → [0, 1];x 7→  0.72 if x = 0 0.68 if x = 1 0.61 if x = 2 0.57 if x = 3 Given ε = 0.42, the ε- Lukasiewicz fuzzy set Lε µ of µ in X is given as follows: Lε µ : X → [0, 1];x 7→  0.14 if x = 0 0.10 if x = 1 0.03 if x = 2 0 if x = 3 Then (Lε µ, 0.25)∈ = ∅ is not a BCC-ideal of X. Theorem 4. Let Lε µ be an ε- Lukasiewicz fuzzy set of a fuzzy set µ in X. If µ is a fuzzy BCC-ideal of X, then the q-set (Lε µ, t)q of Lε µ with value t ∈ (0, 1] is a BCC-ideal of X. Proof. Assume that µ is a fuzzy BCC-ideal of X and let t ∈ (0, 1]. If 0 /∈ (Lε µ, t)q, then [0/t]qLε µ, that is, Lε µ(0) + t ≤ 1. Since Lε µ(0) ≥ Lε µ(x) for x ∈ (Lε µ, t)q, it follows that Lε µ(x) ≤ Lε µ(0) ≤ 1 − t. Hence, [x/t]qLε µ, and so x /∈ (Lε µ, t)q. This contradiction is thus 0 ∈ (Lε µ, t)q. Let t ∈ (0, 1] and x, y, z ∈ (Lε µ, t)q be such that x ◦ (y ◦ z) ∈ (Lε µ, t)q and y ∈ (Lε µ, t)q. Then [x ◦ (y ◦ z)/t]qL ε µ and [y/t]qL ε µ, that is, Lε µ(x ◦ (y ◦ z)) + t > 1 and Lε µ(y) + t > 1. It follows from Theorem 1 and Theorem 2 that Lε µ(x ◦ z) + t ≥ min{Lε µ(x◦(y◦z)), Lε µ(y)}+t = min{Lε µ(x◦(y◦z))+t, Lε µ(y)+t} > 1. Thus, [(x◦z)/t]qLε µ. So x ◦ z ∈ (Lε µ, t)q. Hence, (Lε µ, t)q is a BCC-ideal of X. Theorem 5. 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-ideal of X, then Lε µ satisfies the following properties: (∀ta ∈ (0, 0.5])(0 ∈ (Lε µ, ta)∈) (3.11) (∀x, y, z ∈ X,∀ta, tb ∈ (0, 0.5]) ( [x ◦ (y ◦ z)/ta]qLε µ, [y/tb]qL ε µ ⇒ (x ◦ z) ∈ (Lε µ,max{ta, tb})∈ ) (3.12) Proof. Let x, y, z ∈ X and ta, tb ∈ (0, 0.5]. If 0 /∈ (Lε µ, ta)∈, then [0/ta]∈Lε µ and so Lε µ(0) < ta ≤ 1 − ta since ta ≤ 0.5. Hence, [0/ta]qLε µ. This contradiction is hence 0 ∈ (Lε µ, ta)∈. Let x, y, z ∈ X be such that x ◦ (y ◦ z) ∈ (Lε µ, t)q and y ∈ (Lε µ, t)q. Then [x ◦ (y ◦ z)/t]qLε µ and [y/t]qLε µ, that is, Lε µ(x ◦ (y ◦ z)) > 1 − t and Lε µ(y) > 1 − t. It follows that Lε µ(x ◦ z) ≥ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} > 1 − t. Thus, [x ◦ z/t]qLε µ and so x ◦ z ∈ (Lε µ, t)q. Hence, (Lε µ, t)q is a BCC-ideal of X. Theorem 6. If an ε- Lukasiewicz fuzzy set Lε µ in X satisfies the following properties: (∀x ∈ X,∀t ∈ (0.5, 1])([x/t]qLε µ ⇒ [0/t] ∈ (Lε µ)) (3.13) A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3218 (∀x, y, z ∈ X,∀ta, tb ∈ (0.5, 1]) ( [x ◦ (y ◦ z)/ta]qLε µ, [y/tb]qL ε µ ⇒ [(x ◦ z)/max{ta, tb}] ∈ Lε µ ) (3.14) then the nonempty ∈-set (Lε µ,max{ta, tb})∈ of Lε µ is a BCC-ideal of X for all ta, tb ∈ (0.5, 1]. Proof. Let ta, tb ∈ (0.5, 1] and assume that the ∈-set (Lε µ,max{ta, tb})∈ of Lε µ is nonempty. Then there exists x ∈ (Lε µ,max{ta, tb})∈, and so Lε µ(x) ≥ max{ta, tb} > 1 − max{ta, tb}, that is, [x/max{ta, tb}]qLε µ. Hence, [0/max{ta, tb}] ∈ Lε µ by (3.13), and thus 0 ∈ (Lε µ,max{ta, tb})∈. Let x, y, z ∈ X be such that x ◦ (y ◦ z) ∈ (Lε µ,max{ta, tb})∈ and y ∈ (Lε µ,max{ta, tb})∈. Then Lε µ(x ◦ (y ◦ z)) ≥ max{ta, tb} > 1 − max{ta, tb} and Lε µ(y) ≥ max{ta, tb} > 1 − max{ta, tb}, that is, [(x ◦ (y ◦ z))/max{ta, tb}]qLε µ and [y/max{ta, tb}]qLε µ. It follows from (3.14) that [(x ◦ z)/max{ta, tb}] ∈ Lε µ. Hence, x ◦ z ∈ (Lε µ,max{ta, tb})∈, and therefore (Lε µ,max{ta, tb})∈ is a BCC-ideal of X for all ta, tb ∈ (0.5, 1]. Theorem 7. If an ε- Lukasiewicz fuzzy set Lε µ in X satisfies the conditions (3.13) and (∀x, y, z ∈ X,∀ta, tb ∈ (0.5, 1]) ( [x ◦ (y ◦ z)/ta]qLε µ, [y/tb]qL ε µ ⇒ [(x ◦ z)/min{ta, tb}] ∈ Lε µ ) , (3.15) then the nonempty ∈-set (Lε µ,min{ta, tb})∈ of Lε µ is a BCC-ideal of X for all ta, tb ∈ (0.5, 1]. Proof. Let ta, tb ∈ (0.5, 1] and assume that the ∈-set (Lε µ,min{ta, tb})∈ of Lε µ is nonempty. Then there exists x ∈ (Lε µ,min{ta, tb})∈, and so Lε µ(x) ≥ min{ta, tb} > 1 − min{ta, tb}, that is, [x/min{ta, tb}]qLε µ. Hence, [0/min{ta, tb}] ∈ Lε µ by (3.13), and thus 0 ∈ (Lε µ,min{ta, tb})∈. Let x, y, z ∈ X be such that x◦(y◦z) ∈ (Lε µ,min{ta, tb})∈ and y ∈ (Lε µ,min{ta, tb})∈. Then Lε µ(x ◦ (y ◦ z)) ≥ min{ta, tb} > 1 − min{ta, tb} and Lε µ(y) ≥ min{ta, tb} > 1−min{ta, tb}, that is, [(x◦ (y ◦z))/min{ta, tb}]qLε µ and [y/min{ta, tb}]qLε µ. It follows from (3.15) that [(x ◦ z)/min{ta, tb}] ∈ Lε µ. Hence, x ◦ z ∈ (Lε µ,min{ta, tb})∈, and therefore (Lε µ,min{ta, tb})∈ is a BCC-ideal of X for all ta, tb ∈ (0.5, 1]. Theorem 8. If an ε- Lukasiewicz fuzzy set Lε µ in X satisfies the conditions (3.13) and (∀x ∈ X,∀t ∈ (0.5, 1])([x/t]qLε µ ⇒ [0/t] ∈ (Lε µ)) (3.16) (∀x, y, z ∈ X,∀ta, tb ∈ (0.5, 1]) ( [x ◦ (y ◦ z)/ta]qLε µ, [y/tb]qL ε µ ⇒ [(x ◦ z)/min{ta, tb}] ∈ Lε µ ) (3.17) then the nonempty ∈-set (Lε µ,max{ta, tb})∈ of Lε µ is a BCC-ideal of X for all ta, tb ∈ (0.5, 1]. Proof. Let y ∈ (Lε µ,max{ta, tb})∈ for ta, tb ∈ (0.5, 1]. Then Lε µ(y) ≥ max{ta, tb} > 1 − max{ta, tb}, and so [y/max{ta, tb}]qLε µ. Hence, [(x ◦ y)/max{ta, tb}] ∈ Lε µ for all x ∈ X by (3.16), which implies that x ◦ y ∈ (Lε µ,max{ta, tb})∈ for all x ∈ X. Let x, y ∈ A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3219 (Lε µ,max{ta, tb})∈ for ta, tb ∈ (0.5, 1]. Then Lε µ(x) ≥ max{ta, tb} > 1 − max{ta, tb} and Lε µ(y) ≥ max{ta, tb} > 1−max{ta, tb}, that is, [x/max{ta, tb}]qLε µ and [y/max{ta, tb}]qLε µ. It follows from (3.17) that [(x◦ (y ◦z)/max{ta, tb}] ∈ Lε µ for all z ∈ X. Hence, x◦ (y ◦z) ∈ (Lε µ,max{ta, tb})∈ for all z ∈ X. Therefore, (Lε µ,max{ta, tb})∈ of Lε µ is a BCC-ideal of X for all ta, tb ∈ (0.5, 1]. Lemma 1. Every ε- Lukasiewicz fuzzy ideal Lε µ of X satisfies the following property: (∀x, y, z ∈ X)(Lε µ(x ◦ z) ≥ max{Lε µ(x ◦ (y ◦ z)), Lε µ(y)}) Proof. Note that [(x ◦ (y ◦ z))/Lε µ(x ◦ (y ◦ z))] ∈ Lε µ and [y/Lε µ(y)] ∈ Lε µ for all x, y, z ∈ X. It follows that [(x ◦ z)/min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)}] ∈ Lε µ, that is, Lε µ(x ◦ z) ≥ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} for all x, y, z ∈ X. Theorem 9. If an ε- Lukasiewicz fuzzy set Lε µ in X satisfies the following properties: (∀x ∈ X,∀t ∈ (0.5, 1])([x/t]qLε µ ⇒ [0/t] ∈ (Lε µ)) (3.18) (∀x, y, z ∈ X,∀ta, tb ∈ (0.5, 1]) ( [x ◦ (y ◦ z)/ta] ∈ Lε µ, [y/tb] ∈ Lε µ ⇒ [(x ◦ z)/min{ta, tb}]qLε µ ) (3.19) then the nonempty q-set (Lε µ,min{ta, tb})q of Lε µ is a BCC-ideal of X for all ta, tb ∈ (0, 0.5]. Proof. Let ta, tb ∈ (0, 0.5]. If (Lε µ,min{ta, tb})q is nonempty, then there exists x ∈ (Lε µ,min{ta, tb})q. Hence, Lε µ(x) > 1 − min{ta, tb} ≥ min{ta, tb}, which shows that [x/min{ta, tb}] ∈ Lε µ. It follows from (3.18) that [0/min{ta, tb}]qLε µ. Thus, 0 ∈ (Lε µ,min{ta, tb})q. Let x, y, z ∈ X be such that x◦(y◦z) ∈ (Lε µ,min{ta, tb})q and y ∈ (Lε µ,min{ta, tb})q. Then Lε µ(x ◦ (y ◦ z)) > 1 − min{ta, tb} ≥ min{ta, tb} and Lε µ(y) > 1 − min{ta, tb} ≥ min{ta, tb}. Thus, [(x ◦ (y ◦ z))/min{ta, tb}] ∈ Lε µ and [y/min{ta, tb}] ∈ Lε µ. It follows from (3.19) that [(x ◦ z)/min{ta, tb}]qLε µ, that is, x ◦ z ∈ (Lε µ,min{ta, tb})q. Therefore, (Lε µ,min{ta, tb})q is a BCC-ideal of X. Theorem 10. If an ε- Lukasiewicz fuzzy set Lε µ in X satisfies the conditions (3.11) and (3.12), then the q-set (Lε µ, t)q of Lε µ is a BCC-ideal of X for all t ∈ (0.5, 1]. Proof. Assume that Lε µ satisfies the conditions (3.11) and (3.12). The condition (3.11) induces Lε µ(0) + t ≥ 2t > 1, that is, [0/t]qLε µ. Hence, 0 ∈ (Lε µ, t)q. Let x, y, z ∈ X be such that x◦ (y ◦z) ∈ (Lε µ, t)q and y ∈ (Lε µ, t)q. Then [(x◦ (y ◦z))/t]qLε µ and [y/t]qLε µ. It follows from (3.12) that x ◦ z ∈ (Lε µ,min{t, t})∈ = (Lε µ, t)∈. Hence, Lε µ(x ◦ z) ≥ t > 1 − t, that is, x ◦ z ∈ (Lε µ, t)q. Therefore, (Lε µ, t)q is a BCC-ideal of X for all t ∈ (0.5, 1]. Let µ be a fuzzy set in X. Consider an ε- Lukasiewicz fuzzy set Lε µ associated with µ in X. Define the set O(Lε µ) = {x ∈ X : Lε µ(x) > 0}, known as the O-set of Lε µ. It is noted that O(Lε µ) can be expressed as O(Lε µ) = {x ∈ X : µ(x) + ε− 1 > 0}. Theorem 11. Let Lε µ be an ε- Lukasiewicz fuzzy set of a fuzzy set µ in X. If µ is a fuzzy BCC-ideal of X, then the O-set O(Lε µ) of Lε µ is a BCC-ideal of X. A. Iampan, R. Subasini, N. Rajesh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3209-3222 3220 Proof. Assume that µ is a fuzzy BCC-ideal of X. Then Lε µ is an ε- Lukasiewicz fuzzy BCC-ideal of X by Theorem 2. It is clear that 0 ∈ Lε µ. Let x, y, z ∈ O(Lε µ) be such that µ(x ◦ (y ◦ z)) + ε − 1 > 0 and µ(y) + ε − 1 > 0. It follows from (3.5) that Lε µ(x ◦ z) ≥ min{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} = min{µ(x ◦ (y ◦ z)) + ε − 1, µ(y) + ε − 1} > 0. Thus, x ◦ z ∈ O(Lε µ). Hence, O(Lε µ) is a BCC-ideal of X. Theorem 12. Let µ be a fuzzy set in X. If an ε- Lukasiewicz fuzzy set Lε µ of µ in X satisfies the following properties: (∀x ∈ X,∀t ∈ (0, 1])([x/t]qLε µ ⇒ [0/t]qLε µ) (3.20) (∀x, y, z ∈ X,∀ta, tb ∈ (0.5, 1]) ( [x ◦ (y ◦ z)/ta] ∈ Lε µ, [y/tb] ∈ Lε µ ⇒ [(x ◦ z)/min{ta, tb}]qLε µ ) (3.21) then the O-set O(Lε µ) of Lε µ is a BCC-ideal of X. Proof. If y ∈ O(Lε µ), then µ(y) > 1−ε, that is, [y/(1−ε)] ∈ µ. Hence, [0/(1−ε)]qLε µ by (3.20), and thus Lε µ(0)+1−ε > 1. Thus, Lε µ(0) > ε > 0, which shows that 0 ∈ O(Lε µ). Let x, y, z ∈ X be such that x ◦ (y ◦ z) ∈ O(Lε µ) and y ∈ O(Lε µ). Then µ(x ◦ (y ◦ z)) + ε− 1 > 0 and µ(y) + ε− 1 > 0. Since [x ◦ (y ◦ z)/Lε µ(x ◦ (y ◦ z))] ∈ Lε µ and [y/Lε µ(y)] ∈ Lε µ. It follows from (3.21) that [(x ◦ z)/max{Lε µ(x ◦ (y ◦ z)), Lε µ(y)}]qLε µ. (3.22) If x ◦ z /∈ O(Lε µ), then Lε µ(x ◦ z) = 0. Thus, Lε µ(x ◦ z) + max{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} = max{Lε µ(x ◦ (y ◦ z)), Lε µ(y)} = max{max{0, µ(x ◦ (y ◦ z)) + ε− 1},max{0, µ(y) + ε− 1}} = max{µ(x ◦ (y ◦ z)) + ε− 1, µ(y) + ε− 1} = max{µ(x ◦ (y ◦ z)), µ(y)} + ε− 1 ≤ 1 + ε− 1 = ε ≤ 1, which shows that (3.22) is not valid. This is a contradiction. So x ◦ z ∈ O(Lε µ). Hence, O(Lε µ) is a BCC-ideal of X. Theorem 13. Let µ be a fuzzy set in X. If an ε- Lukasiewicz fuzzy set Lε µ of µ in X satisfies [0/ε]qµ and the following property: (∀x, y, z ∈ X) ( [x ◦ (y ◦ z)/ε]qµ, [y/ε]qµ ⇒ [(x ◦ z)/ε]qLε µ ) , (3.23) then the O-set O(Lε µ) of Lε µ is a BCC-ideal of X. REFERENCES 3221 Proof. If [0/ε]qµ, then µ(0)+ε > 1 and so Lε µ(0) = max{0, µ(0)+ε−1} = µ(0)+ε−1 > 0. Hence, 0 ∈ O(Lε µ). Let x, y, z ∈ X be such that x ◦ (y ◦ z) ∈ O(Lε µ) and y ∈ O(Lε µ). Then µ(x ◦ (y ◦ z)) + ε − 1 > 0 and µ(y) + ε − 1 > 0. Hence, Lε µ(x ◦ (y ◦ z)) + 1 = max{0, µ(x ◦ (y ◦ z)) + ε − 1} + 1 = µ(x ◦ (y ◦ z)) + ε − 1 + 1 = µ(x ◦ (y ◦ z)) + 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 ◦ (y ◦ z)/ε]qLε µ and [y/ε]qLε µ. It follows from (3.23) that [(x ◦ z)/ε] = [(x ◦ z)/ε] ∈ Lε µ, which shows that Lε µ(x ◦ z) ≥ ε > 0. Hence, x ◦ z ∈ O(Lε µ). Therefore, O(Lε µ) is a BCC-ideal of X. 4. Conclusions The concept of ε- Lukasiewicz fuzzy sets utilizing the Lukasiewicz t-norm was intro- duced by Jun [12]. This paper applies ε- Lukasiewicz fuzzy sets to BCC-ideals within BCC-algebras, introducing the concept of ε- Lukasiewicz fuzzy BCC-ideals and explor- ing their properties. The characterization of ε- Lukasiewicz fuzzy BCC-ideals is discussed, along with the relationship between fuzzy BCC-ideals and ε- Lukasiewicz fuzzy BCC-ideals. Conditions are provided under which ε- Lukasiewicz fuzzy sets qualify as ε- Lukasiewicz fuzzy BCC-ideals. 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