EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1307-1320 ISSN 1307-5543 – ejpam.com Published by New York Business Global Ideals in BE-algebras based on Lukasiewicz fuzzy set Sun Shin Ahn1,∗, Eun Hwan Roh2, Young Bae Jun3 1 Department of Mathematics Education, Dongguk University, Seoul 04620, Korea 2 Department of Mathematics Education, Chinju National University of Education, Jinju 52673, Korea 3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea Abstract. For the purpose of applying the concept of Lukasiewicz fuzzy set to ideals in BE- algebras, Lukasiewicz fuzzy ideal is introduced, and its properties are studied. The relationship between fuzzy ideal and Lukasiewicz fuzzy ideal is discussed. Conditions for the Lukasiewicz fuzzy set to be a Lukasiewicz fuzzy ideal are provided, and characterizations of Lukasiewicz fuzzy ideal are displayed. Conditions in which three subsets, called ∈-set, q-set and O-set, are ideals are explored. 2020 Mathematics Subject Classifications: 2020 Mathematics Subject Classification. 03G25, 06F35, 08A72. Key Words and Phrases: Fuzzy ideal, Lukasiewicz fuzzy ideal, ∈-set, q-set, O-set. 1. Introduction In 1966, Y. Imai, K. Iséki and S. Tanaka introduced BCK-algebra and BCI-algebra as algebraic structures of universal algebra which describe fragments of propositional calculus related to implications known as BCK and BCI-logic. Various generalizations were then attempted, and BCC-algebra, BCH-algebra, BE-algebra, BH-algebra, and d-algebra etc. appeared. In 2008, S. S. Ahn and K. S. So studied ideal theory in BE-algebras (see [1]), and its fuzzy set theory is studied by Y. B. Jun, K. J. Lee and S. Z. Song (see [9]). 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 Jan Lukasiewicz as a three-valued logic. Using the idea of Lukasiewicz t-norm, Y. B. Jun [3] constructed the concept of Lukasiewicz fuzzy sets based on a given fuzzy set and applied it to BCK- algebras and BCI-algebras. S. S. Ahn et al. [8], and A. Rezaei and A. Borumand Saeid [7] studied fuzzy BE-algebras. G. Dymek and A. Walendziak [2] developed the theory of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4467 Email addresses: sunshine@dongguk.edu (S. S. Ahn), ehroh9988@gmail.com (E. H. Roh), skywine@gmail.com (Y. B. Jun) https://www.ejpam.com 1307 © 2022 EJPAM All rights reserved. S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1308 fuzzy filters in BE-algebras. Y. B. Jun and S. S. Ahn applied the Lukasiewicz fuzzy set to BE-filters and subalgebras (see [4]). The purpose of this paper is to apply the Lukasiewicz fuzzy set to ideals in BE- algebras. We introduce the notion of Lukasiewicz fuzzy ideal, and investigate several properties. We discuss the characterization of Lukasiewicz fuzzy ideal. We consider the relationship between fuzzy ideal and Lukasiewicz fuzzy ideal. We provide conditions for the Lukasiewicz fuzzy set to be a Lukasiewicz fuzzy ideal. We explore the conditions under which three subsets, called ∈-set, q-set and O-set, will become ideals. 2. Preliminaries This section lists the known default content that will be used later. Definition 1 ([5]). A BE-algebra is defined to be a set X together with a binary operation “ ∗ ” and a special element “1” satisfying the conditions: (BE1) (∀a ∈ X) (a ∗ a = 1), (BE2) (∀a ∈ X) (a ∗ 1 = 1), (BE3) (∀a ∈ X) (1 ∗ a = a), (BE4) (∀a, y, c ∈ X) (a ∗ (y ∗ c) = y ∗ (a ∗ c)). In the following, the BE-algebra is expressed as (X, 1)∗. A relation “ ≤ ” in (X, 1)∗ is defined as follows: (∀x, b ∈ X)(x ≤ b ⇔ x ∗ b = 1). (1) Definition 2. A subset K of X is called an ideal of (X, 1)∗ (see [1]) if it satisfies: (∀a, b ∈ X) (b ∈ K ⇒ a ∗ b ∈ K) , (2) (∀x, y, a ∈ X) (x, y ∈ K ⇒ (x ∗ (y ∗ a)) ∗ a ∈ K) . (3) Lemma 1 ([9]). A subset K of X is an ideal of (X, 1)∗ if and only if it satisfies: 1 ∈ K, (4) (∀a, b, c ∈ X)(a ∗ (b ∗ c) ∈ K, b ∈ K ⇒ a ∗ c ∈ K). (5) Definition 3. A fuzzy set ψ in X is called a fuzzy ideal of (X, 1)∗ (see [9]) if it satisfies: (∀x, b ∈ X) (ψ(x ∗ b) ≥ ψ(b)) , (6) (∀x, b, c ∈ X) (ψ((b ∗ (c ∗ x)) ∗ x) ≥ min{ψ(b), ψ(c)}) . (7) A fuzzy set ψ in a set X of the form ψ(b) := { t ∈ (0, 1] if b = a, 0 if b ̸= a, is said to be a fuzzy point with support a and value t and is denoted by ⟨a/t⟩. For a fuzzy set ψ in a set X, we say that a fuzzy point ⟨a/t⟩ is S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1309 (i) contained in ψ, denoted by ⟨a/t⟩ ∈ ψ, ([6]) if ψ(a) ≥ t. (ii) quasi-coincident with ψ, denoted by ⟨a/t⟩ q ψ, ([6]) if ψ(a) + t > 1. Definition 4 ([3]). Let ψ be a fuzzy set in a set X and let ε ∈ (0, 1). A function Lεψ : X → [0, 1], x 7→ max{0, ψ(x) + ε− 1} is called the Lukasiewicz fuzzy set (of ψ) in X. For the Lukasiewicz fuzzy set Lεψ (of ψ) in X and t ∈ (0, 1], consider the sets ( Lεψ, t)∈ := {x ∈ X | ⟨x/t⟩ ∈ Lεψ}, ( Lεψ, t)q := {x ∈ X | ⟨x/t⟩ q Lεψ}, which are called the ∈-set and q-set, respectively, of Lεψ (with value t). Also, consider a set: O( Lεψ) := {x ∈ X | Lεψ(x) > 0} (8) which is called an O-set of Lεψ. It is observed that O( Lεψ) = {x ∈ X | ψ(x) + ε− 1 > 0}. 3. Lukasiewicz fuzzy ideals In this section, let ψ and ε be a fuzzy set in X and an element of (0, 1), respectively, unless otherwise specified. Definition 5. A Lukasiewicz fuzzy set Lεψ in X is called a Lukasiewicz fuzzy ideal of (X, 1)∗ if it satisfies: (∀x, y ∈ X)(∀t ∈ (0, 1]) ( ⟨y/t⟩ ∈ Lεψ ⇒ ⟨(x ∗ y)/t⟩ ∈ Lεψ ) , (9) (∀x, y, z ∈ X)(∀ta, tb ∈ (0, 1]) ( ⟨x/ta⟩ ∈ Lεψ, ⟨y/tb⟩ ∈ Lεψ ⇒ ⟨((x ∗ (y ∗ z)) ∗ z)/min{ta, tb}⟩ ∈ Lεψ ) . (10) Example 1. Let X = {1, a, b, c, d, 0} be a set with the binary operation “ ∗ ” given by the following Cayley table: ∗ 1 a b c d 0 1 1 a b c d 0 a 1 1 a c c d b 1 1 1 c c c c 1 a b 1 a b d 1 1 a 1 1 a 0 1 1 1 1 1 1 S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1310 Then (X, 1)∗ is a BE-algebra (see [5]). Let ψ be a fuzzy set in X defined as follows. ψ : X → [0, 1], x 7→  0.57 if x ∈ {1, a, b}, 0.14 if x = c, 0.33 if x = d, 0.21 if x = 0. For ε := 0.65, the Lukasiewicz fuzzy set Lεψ of ψ in X is given as follows. Lεψ : X → [0, 1], y 7→ { 0.22 if y ∈ {1, a, b}, 0.00 if y ∈ {c, d, 0}. It is routine to verify that Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗. Theorem 1. A Lukasiewicz fuzzy set Lεψ in X is a Lukasiewicz fuzzy ideal of (X, 1)∗ if and only if it satisfies: (∀x, y ∈ X) ( Lεψ(x ∗ y) ≥ Lεψ(y) ) . (11) (∀x, y, z ∈ X) ( Lεψ((x ∗ (y ∗ z)) ∗ z) ≥ min{ Lεψ(x), Lεψ(y)} ) . (12) Proof. Assume that Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗. Let x, y ∈ X. Since ⟨y/ Lεψ(y)⟩ ∈ Lεψ, we have ⟨(x ∗ y)/ Lεψ(y)⟩ ∈ Lεψ by (9), and so Lεψ(x ∗ y) ≥ Lεψ(y). Note that ⟨x/ Lεψ(x)⟩ ∈ Lεψ and ⟨y/ Lεψ(y)⟩ ∈ Lεψ for all x, y ∈ X. It follows from (10) that ⟨((x ∗ (y ∗ z)) ∗ z)/min{ Lεψ(x), Lεψ(y)}⟩ ∈ Lεψ, that is, Lεψ((x ∗ (y ∗ z)) ∗ z) ≥ min{ Lεψ(x), Lεψ(y)} for all x, y, z ∈ X. Conversely, let Lεψ be a Lukasiewicz fuzzy set satisfying (11) and (12). If ⟨y/t⟩ ∈ Lεψ for all y ∈ X and t ∈ (0, 1], then Lεψ(x ∗ y) ≥ Lεψ(y) ≥ t for all x ∈ X by (11). Hence ⟨(x∗y)/t⟩ ∈ Lεψ. Let x, y, z ∈ X and ta, tb ∈ (0, 1] be such that ⟨x/ta⟩ ∈ Lεψ and ⟨y/tb⟩ ∈ Lεψ. Then Lεψ(x) ≥ ta and Lεψ(y) ≥ tb. It follows from (12) that Lεψ((x ∗ (y ∗ z)) ∗ z) ≥ min{ Lεψ(x), Lεψ(y)} ≥ min{ta, tb}. Hence ⟨((x ∗ (y ∗ z)) ∗ z)/min{ta, tb}⟩ ∈ Lεψ, and therefore Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗. Proposition 1. Every Lukasiewicz fuzzy ideal Lεψ of (X, 1)∗ satisfies: (∀x ∈ X)(∀t ∈ (0, 1]) ( ⟨x/t⟩ ∈ Lεψ ⇒ ⟨1/t⟩ ∈ Lεψ ) . (13) (∀x, y ∈ X)(∀t ∈ (0, 1]) ( ⟨x/t⟩ ∈ Lεψ ⇒ ⟨((x ∗ y) ∗ y)/t⟩ ∈ Lεψ ) . (14) (∀x, y ∈ X)(∀t ∈ (0, 1]) ( x ≤ y, ⟨x/t⟩ ∈ Lεψ ⇒ ⟨y/t⟩ ∈ Lεψ ) . (15) (∀x, y ∈ X)(∀ta, tb ∈ (0, 1]) ( ⟨(x ∗ y)/tb⟩ ∈ Lεψ, ⟨x/ta⟩ ∈ Lεψ ⇒ ⟨y/min{ta, tb}⟩ ∈ Lεψ. ) . (16) (∀x, y, z ∈ X)(∀ta, tb ∈ (0, 1]) ( ⟨(x ∗ (y ∗ z))/ta⟩ ∈ Lεψ, ⟨y/tb⟩ ∈ Lεψ ⇒ ⟨(x ∗ z)/min{ta, tb}⟩ ∈ Lεψ. ) . (17) S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1311 Proof. The condition (13) is derived from the combination of (BE1) and (9). Let x ∈ X and t ∈ (0, 1] be such that ⟨x/t⟩ ∈ Lεψ. Then ⟨((x ∗ y) ∗ y)/t⟩ = ⟨((x ∗ (1 ∗ y)) ∗ y)/t⟩ = ⟨((x ∗ (1 ∗ y)) ∗ y)/min{t, t}⟩ ∈ Lεψ by (BE3), (10) and (13). The combination of (BE3), (1) and (14) induces (15). Let x, y ∈ X and ta, tb ∈ (0, 1] be such that ⟨(x ∗ y)/tb⟩ ∈ Lεψ and ⟨x/ta⟩ ∈ Lεψ. Then ⟨y/min{ta, tb}⟩ = ⟨(1 ∗ y)/min{ta, tb}⟩ = ⟨(((x ∗ y) ∗ (x ∗ y)) ∗ y)/min{ta, tb}⟩ ∈ Lεψ by (BE1), (BE3) and (10), which proves (16). The condition (17) is derived from the combination of (BE4) and (16). We provide conditions for the Lukasiewicz fuzzy set to be a Lukasiewicz fuzzy ideal. Theorem 2. If a Lukasiewicz fuzzy set Lεψ in X satisfies conditions (13) and (17), then it is a Lukasiewicz fuzzy ideal of (X, 1)∗. Proof. Assume that Lεψ satisfies conditions (13) and (17). Let y ∈ X and t ∈ (0, 1] be such that ⟨y/t⟩ ∈ Lεψ. Then ⟨(x ∗ (y ∗ y))/t⟩ = ⟨(x ∗ 1)/t⟩ = ⟨1/t⟩ ∈ Lεψ for all x ∈ X by (BE1), (BE2) and (13). It follows from (17) that ⟨(x ∗ y)/t⟩ ∈ Lεψ. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that ⟨x/ta⟩ ∈ Lεψ and ⟨y/tb⟩ ∈ Lεψ. Then ⟨((x ∗ z) ∗ (x ∗ z))/tb⟩ = ⟨1/tb⟩ ∈ Lεψ and so ⟨((x ∗ z) ∗ z)/min{ta, tb}⟩ ∈ Lεψ for all z ∈ X by (17). In particular, ⟨((x ∗ (y ∗ z)) ∗ (y ∗ z))/min{ta, tb}⟩ ∈ Lεψ, which implies from (17) that ⟨((x ∗ (y ∗ z)) ∗ z)/min{ta, tb}⟩ ∈ Lεψ for all z ∈ X. Hence Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗. Corollary 1. If a Lukasiewicz fuzzy set Lεψ in X satisfies (13) and (17), then it satisfies the conditions (14), (15) and (16). We discuss the relationship between fuzzy ideal and Lukasiewicz fuzzy ideal. Theorem 3. If ψ is a fuzzy ideal of (X, 1)∗, then Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗. Proof. Let y ∈ X and t ∈ (0, 1] be such that ⟨y/t⟩ ∈ Lεψ. Then Lεψ(y) ≥ t, and so Lεψ(x ∗ y) = max{0, ψ(x ∗ y) + ε− 1} ≥ max{0, ψ(y) + ε− 1} = Lεψ(y) ≥ t for all x ∈ X. Hence ⟨(x ∗ y)/t⟩ ∈ Lεψ for all x ∈ X. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that ⟨x/ta⟩ ∈ Lεψ and ⟨y/tb⟩ ∈ Lεψ. Then Lεψ(x) ≥ ta and Lεψ(y) ≥ tb. It follows that Lεψ((x ∗ (y ∗ z)) ∗ z) = max{0, ψ((x ∗ (y ∗ z)) ∗ z) + ε− 1} ≥ max{0,min{ψ(x), ψ(y)} + ε− 1} = max{0,min{ψ(x) + ε− 1, ψ(y) + ε− 1}} = min{max{0, ψ(x) + ε− 1},max{0, ψ(y) + ε− 1}} S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1312 = min{ Lεψ(x), Lεψ(y)} ≥ min{ta, tb} for all z ∈ X. Thus ⟨((x ∗ (y ∗ z)) ∗ z)/min{ta, tb}⟩ ∈ Lεψ for all z ∈ X. Therefore Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗. In Example 1, Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗. But ψ is not a fuzzy ideal of (X, 1)∗ since ψ(b ∗ 0) = ψ(c) = 0.14 ≱ 0.21 = ψ(0). Therefore, the converse of Theorem 3 may not be true. In the sense of Theorem 3, we can say that Lukasiewicz fuzzy ideal is a generalization of fuzzy ideal. We explore the conditions under which ∈-set and q-set of the Lukasiewicz fuzzy set can be ideal. Theorem 4. Let Lεψ be a Lukasiewicz fuzzy set in X. Then the ∈-set ( Lεψ, t)∈ of Lεψ with value t ∈ (0.5, 1] is an ideal of (X, 1)∗ if and only if Lεψ satisfies: (∀x, y ∈ X) ( Lεψ(y) ≤ max{ Lεψ(x ∗ y), 0.5} ) , (18) (∀x, y, z ∈ X) ( min{ Lεψ(x), Lεψ(y)} ≤ max{ Lεψ((x ∗ (y ∗ z)) ∗ z), 0.5} ) . (19) Proof. Assume that ( Lεψ, t)∈ is an ideal of (X, 1)∗ for t ∈ (0.5, 1]. If there exist a, b ∈ X such that Lεψ(b) > max{ Lεψ(a ∗ b), 0.5}, then Lεψ(b) ∈ (0.5, 1] and Lεψ(a ∗ b) < Lεψ(b). Hence ⟨b/ Lεψ(b)⟩ ∈ Lεψ, and so b ∈ ( Lεψ, L ε ψ(b))∈, but a ∗ b /∈ ( Lεψ, L ε ψ(b))∈. This is a contradiction, and thus Lεψ(y) ≤ max{ Lεψ(x ∗ y), 0.5} for all x, y ∈ X. If the condition (19) is not valid, then there exist a, b, c ∈ X such that min{ Lεψ(a), Lεψ(b)} > max{ Lεψ((a ∗ (b ∗ c)) ∗ c), 0.5}. If we take t := min{ Lεψ(a), Lεψ(b)}, then t ∈ (0.5, 1], ⟨a/t⟩ ∈ Lεψ and ⟨b/t⟩ ∈ Lεψ, but ⟨((a∗(b∗c))∗c)/t⟩ ∈ Lεψ, that is, a ∈ ( Lεψ, t)∈ and b ∈ ( Lεψ, t)∈, but (a∗(b∗c))∗c /∈ ( Lεψ, t)∈. This is a contradiction, and thus (19) is valid. Conversely, suppose that Lεψ satisfies (18) and (19), and let y ∈ ( Lεψ, t)∈ for t ∈ (0.5, 1]. Then t ≤ Lεψ(y) ≤ max{ Lεψ(x∗y), 0.5} by (18). Hence Lεψ(x∗y) ≥ t, and so x∗y ∈ ( Lεψ, t)∈. Let x, y ∈ X and t ∈ (0.5, 1] be such that x ∈ ( Lεψ, t)∈ and y ∈ ( Lεψ, t)∈. Then Lεψ(x) ≥ t and Lεψ(y) ≥ t, which imply from (19) that 0.5 < t ≤ min{ Lεψ(x), Lεψ(y)} ≤ max{ Lεψ((x ∗ (y ∗ z)) ∗ z), 0.5} for all z ∈ X. Hence ⟨((x∗(y∗z))∗z)/t⟩ ∈ Lεψ, that is, (x∗(y∗z))∗z ∈ ( Lεψ, t)∈. Therefore ( Lεψ, t)∈ is an ideal of (X, 1)∗ for t ∈ (0.5, 1]. Theorem 5. Let Lεψ be a Lukasiewicz fuzzy set in X. Then the ∈-set ( Lεψ, t)∈ of Lεψ with value t ∈ (0.5, 1] is an ideal of (X, 1)∗ if and only if Lεψ satisfies: (∀x ∈ X) ( Lεψ(x) ≤ max{ Lεψ(1), 0.5} ) , (20) (∀x, y, z ∈ X) ( min{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)} ≤ max{ Lεψ(x ∗ z), 0.5} ) . (21) S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1313 Proof. Assume that ( Lεψ, t)∈ is an ideal of (X, 1)∗ for t ∈ (0.5, 1]. If there exist a ∈ X such that Lεψ(a) > max{ Lεψ(1), 0.5}, then Lεψ(a) ∈ (0.5, 1] and Lεψ(1) < Lεψ(a). Hence ⟨a/ Lεψ(a)⟩ ∈ Lεψ, and so a ∈ ( Lεψ, L ε ψ(a))∈, but 1 /∈ ( Lεψ, L ε ψ(a))∈. This is a contradiction, and thus Lεψ(x) ≤ max{ Lεψ(1), 0.5} for all x ∈ X. If the condition (21) is not valid, then there exist a, b, c ∈ X such that min{ Lεψ(a ∗ (b ∗ c)), Lεψ(b)} > max{ Lεψ(a ∗ c), 0.5}. If we take t := min{ Lεψ(a ∗ (b ∗ c)), Lεψ(b)}, then t ∈ (0.5, 1], ⟨(a ∗ (b ∗ c))/t⟩ ∈ Lεψ and ⟨b/t⟩ ∈ Lεψ, but ⟨(a ∗ c)/t⟩ ∈ Lεψ, that is, a ∗ (b ∗ c) ∈ ( Lεψ, t)∈ and b ∈ ( Lεψ, t)∈, but a ∗ c /∈ ( Lεψ, t)∈. This is a contradiction, and thus (21) is valid. Conversely, suppose that Lεψ satisfies (20) and (21), and let t ∈ (0.5, 1]. For every x ∈ ( Lεψ, t)∈, we have t ≤ Lεψ(x) ≤ max{ Lεψ(1), 0.5} by (20). Hence Lεψ(1) ≥ t, and so 1 ∈ ( Lεψ, t)∈. Let x, y, z ∈ X and t ∈ (0.5, 1] be such that x ∗ (y ∗ z) ∈ ( Lεψ, t)∈ and y ∈ ( Lεψ, t)∈. Then Lεψ(x ∗ (y ∗ z)) ≥ t and Lεψ(y) ≥ t, which imply from (21) that 0.5 < t ≤ min{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)} ≤ max{ Lεψ(x ∗ z), 0.5}. Hence ⟨(x ∗ z)/t⟩ ∈ Lεψ, that is, x ∗ z ∈ ( Lεψ, t)∈. Therefore ( Lεψ, t)∈ is an ideal of (X, 1)∗ for t ∈ (0.5, 1] by Lemma 1. Remark 1. In Theorems 4 and 5, if t /∈ (0.5, 1], that is, there exists at least one t ≤ 0.5, then Theorems 4 and 5 are incorrect as shown in the following example. Example 2. Consider the BE-algebra (X, 1)∗ in Example 1 and let ψ be a fuzzy set in X defined as follows. ψ : X → [0, 1], x 7→  0.92 if x = 1, 0.66 if x = a, 0.66 if x = b, 0.77 if x = c, 0.81 if x = d, 0.95 if x = 0. For ε := 0.61, the Lukasiewicz fuzzy set Lεψ of ψ in X is given as follows. Lεψ : X → [0, 1], y 7→  0.53 if y = 1, 0.27 if y ∈ {a, b}, 0.38 if y = c, 0.42 if y = d, 0.56 if y = 0. Then ( Lεψ, 0.41)∈ = {1, d, 0} is not an ideal of (X, 1)∗ because of b∗0 = c /∈ ( Lεψ, 0.41)∈. In this case, we know that Lεψ(0) = 0.56 ≰ 0.5 = max{ Lεψ(b ∗ 0), 0.5} and Lεψ(0) = 0.56 ≰ 0.53 = max{ Lεψ(1), 0.5}. S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1314 Theorem 6. If a Lukasiewicz fuzzy set Lεψ in X satisfies: (∀x ∈ X)(∀t ∈ (0.5, 1]) ( ⟨x/t⟩ q Lεψ ⇒ ⟨1/t⟩ ∈ Lεψ ) , (22) (∀x, y, z ∈ X)(∀ta, tb ∈ (0.5, 1]) ( ⟨(x ∗ (y ∗ z))/ta⟩ q Lεψ, ⟨y/tb⟩ q Lεψ ⇒ ⟨(x ∗ z)/max{ta, tb}⟩ ∈ Lεψ ) , (23) then the non-empty ∈-set ( Lεψ,max{ta, tb})∈ of Lεψ is an ideal of (X, 1)∗ for all ta, tb ∈ (0.5, 1]. Proof. Let ta, tb ∈ (0.5, 1] and assume that the ∈-set ( Lεψ,max{ta, tb})∈ of Lεψ is non-empty. Then there exists x ∈ ( Lεψ,max{ta, tb})∈, and so Lεψ(x) ≥ max{ta, tb} > 1 − max{ta, tb}, i.e., ⟨x/max{ta, tb}⟩ q Lεψ. Hence ⟨1/max{ta, tb}⟩ ∈ Lεψ by (22), and thus 1 ∈ ( Lεψ,max{ta, tb})∈. Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ ( Lεψ,max{ta, tb})∈ and y ∈ ( Lεψ,max{ta, tb})∈. Then Lεψ(x ∗ (y ∗ z)) ≥ max{ta, tb} > 1 − max{ta, tb} and Lεψ(y) ≥ max{ta, tb} > 1 − max{ta, tb}, that is, ⟨(x ∗ (y ∗ z))/max{ta, tb}⟩ q Lεψ and ⟨y/max{ta, tb}⟩ q Lεψ. It follows from (23) that ⟨(x ∗ z)/max{ta, tb}⟩ ∈ Lεψ. Hence x ∗ z ∈ ( Lεψ,max{ta, tb})∈, and therefore ( Lεψ,max{ta, tb})∈ is an ideal of (X, 1)∗ for all ta, tb ∈ (0.5, 1] by Lemma 1. Theorem 7. If a Lukasiewicz fuzzy set Lεψ in X satisfies (22) and (∀x, y, z ∈ X)(∀ta, tb ∈ (0.5, 1]) ( ⟨(x ∗ (y ∗ z))/ta⟩ q Lεψ, ⟨y/tb⟩ q Lεψ ⇒ ⟨(x ∗ z)/min{ta, tb}⟩ ∈ Lεψ ) , (24) then the non-empty ∈-set ( Lεψ,min{ta, tb})∈ of Lεψ is an ideal of (X, 1)∗ for all ta, tb ∈ (0.5, 1]. Proof. It can be verified through a process similar to the proof in Theorem 6. Theorem 8. If a Lukasiewicz fuzzy set Lεψ in X satisfies: (∀x, y ∈ X)(∀t ∈ (0.5, 1]) ( ⟨y/t⟩ q Lεψ ⇒ ⟨(x ∗ y)/t⟩ ∈ Lεψ ) , (25) and ⟨x/ta⟩ q Lεψ, ⟨y/tb⟩ q Lεψ ⇒ ⟨((x ∗ (y ∗ z)) ∗ z)/max{ta, tb}⟩ ∈ Lεψ, (26) for all x, y, z ∈ X and ta, tb ∈ (0.5, 1], then the non-empty ∈-set ( Lεψ,max{ta, tb})∈ of Lεψ is an ideal of (X, 1)∗ for all ta, tb ∈ (0.5, 1]. Proof. Let y ∈ ( Lεψ,max{ta, tb})∈ for ta, tb ∈ (0.5, 1]. Then Lεψ(y) ≥ max{ta, tb} > 1 − max{ta, tb}, and so ⟨y/max{ta, tb}⟩ q Lεψ. Hence ⟨(x ∗ y)/max{ta, tb}⟩ ∈ Lεψ for all x ∈ X by (25), which implies that x ∗ y ∈ ( Lεψ,max{ta, tb})∈ for all x ∈ X. Let x, y ∈ ( Lεψ,max{ta, tb})∈ for ta, tb ∈ (0.5, 1]. Then Lεψ(x) ≥ max{ta, tb} > 1 − max{ta, tb} and Lεψ(y) ≥ max{ta, tb} > 1−max{ta, tb}, that is, ⟨x/max{ta, tb}⟩ q Lεψ and ⟨y/max{ta, tb}⟩ q Lεψ. It follows from (26) that ⟨((x ∗ (y ∗ z)) ∗ z)/max{ta, tb}⟩ ∈ Lεψ for all z ∈ X. Hence (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ,max{ta, tb})∈ for all z ∈ X. Therefore ( Lεψ,max{ta, tb})∈ of Lεψ is an ideal of (X, 1)∗ for all ta, tb ∈ (0.5, 1]. S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1315 Lemma 2. Every Lukasiewicz fuzzy ideal Lεψ of (X, 1)∗ satisfies: (∀x, y, z ∈ X) ( Lεψ(x ∗ z) ≥ max{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)} ) . Proof. Note that ⟨(x ∗ (y ∗ z))/ Lεψ(x ∗ (y ∗ z))⟩ ∈ Lεψ and ⟨y/ Lεψ(y)⟩ ∈ Lεψ for all x, y, z ∈ X. It follows from (17) that ⟨(x ∗ z)/min{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)}⟩ ∈ Lεψ, that is, Lεψ(x ∗ z) ≥ min{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)} for all x, y, z ∈ X. Theorem 9. If Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗, then its q-set ( Lεψ, t)q is an ideal of (X, 1)∗ for all t ∈ (0, 1]. Proof. Let Lεψ be a Lukasiewicz fuzzy ideal of (X, 1)∗ and let t ∈ (0, 1]. If 1 /∈ ( Lεψ, t)q, then ⟨1/t⟩ q Lεψ, i.e., Lεψ(1) + t ≤ 1. Since ⟨x/ Lεψ(x)⟩ ∈ Lεψ for all x ∈ X, we get ⟨1/ Lεψ(x)⟩ ∈ Lεψ for all x ∈ X by (13). Hence Lεψ(1) ≥ Lεψ(x) for x ∈ ( Lεψ, t)q, and so 1 − t ≥ Lεψ(1) ≥ Lεψ(x). This shows that ⟨x/t⟩ q Lεψ, that is, x /∈ ( Lεψ, t)q, a contradiction. Thus 1 ∈ ( Lεψ, t)q. Let x, y, z ∈ X be such that x∗ (y ∗ z) ∈ ( Lεψ, t)q and y ∈ ( Lεψ, t)q. Then ⟨(x ∗ (y ∗ z))/t⟩ q Lεψ and ⟨y/t⟩ q Lεψ, that is, Lεψ(x ∗ (y ∗ z)) > 1 − t and Lεψ(y) > 1 − t. It follows from Lemma 2 that Lεψ(x ∗ z) ≥ max{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)} > 1 − t. Hence ⟨(x ∗ z)/t⟩ q Lεψ, and so x ∗ z ∈ ( Lεψ, t)q. Therefore ( Lεψ, t)q is an ideal of (X, 1)∗ by Lemma 1. Corollary 2. If ψ is a fuzzy ideal of (X, 1)∗, then the q-set of Lεψ is an ideal of (X, 1)∗. Proposition 2. For the Lukasiewicz fuzzy set Lεψ in X, if the q-set of Lεψ is an ideal of (X, 1)∗, then the following arguments are satisfied. 1 ∈ ( Lεψ, t)∈, (27) ⟨x/ta⟩ q Lεψ, ⟨y/tb⟩ q Lεψ ⇒ (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ,max{ta, tb})∈, (28) ⟨(x ∗ (y ∗ z))/ta⟩ q Lεψ, ⟨y/tb⟩ q Lεψ ⇒ x ∗ z ∈ ( Lεψ,max{ta, tb})∈ (29) for all x, y, z ∈ X and t, ta, tb ∈ (0, 0.5]. Proof. Assume that the q-set ( Lεψ, t)q of Lεψ is an ideal of (X, 1)∗. Then 1 ∈ ( Lεψ, t)q by Lemma 1. If 1 /∈ ( Lεψ, t)∈ for some t ∈ (0, 0.5], then ⟨1/t⟩ ∈ Lεψ. Hence Lεψ(1) < t ≤ 1 − t since t ∈ (0, 0.5], and so ⟨1/t⟩ q Lεψ, i.e., 1 /∈ ( Lεψ, t)q. This is a conradiction, and thus 1 ∈ ( Lεψ, t)∈. Let x, y ∈ X and ta, tb ∈ (0, 0.5] be such that ⟨x/ta⟩ q Lεψ and ⟨y/tb⟩ q Lεψ. Then x ∈ ( Lεψ, ta)q ⊆ ( Lεψ,max{ta, tb})q and y ∈ ( Lεψ, tb)q ⊆ ( Lεψ,max{ta, tb})q, from which (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ,max{ta, tb})q is derived. Hence Lεψ((x ∗ (y ∗ z)) ∗ z) > 1 − max{ta, tb} ≥ max{ta, tb}, S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1316 i.e., ⟨((x ∗ (y ∗ z)) ∗ z)/max{ta, tb}⟩ ∈ Lεψ. Hence (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ,max{ta, tb})∈. Let x, y, z ∈ X and ta, tb ∈ (0, 0.5] be such that ⟨(x ∗ (y ∗ z))/ta⟩ q Lεψ and ⟨y/tb⟩ q Lεψ. Then x ∗ (y ∗ z) ∈ ( Lεψ, ta)q ⊆ ( Lεψ,max{ta, tb})q and y ∈ ( Lεψ, tb)q ⊆ ( Lεψ,max{ta, tb})q, from which x ∗ z ∈ ( Lεψ,max{ta, tb})q is derived by Lemma 1. Hence Lεψ(x ∗ z) > 1 − max{ta, tb} ≥ max{ta, tb}, i.e., ⟨(x ∗ z)/max{ta, tb}⟩ ∈ Lεψ. Therefore x ∗ z ∈ ( Lεψ,max{ta, tb})∈. Theorem 10. If a Lukasiewicz fuzzy set Lεψ in X satisfies (∀x, y ∈ X)(∀t ∈ (0, 1]) ( ⟨y/t⟩ ∈ Lεψ ⇒ ⟨(x ∗ y)/t⟩ q Lεψ ) , (30) and ⟨x/ta⟩ ∈ Lεψ, ⟨y/tb⟩ ∈ Lεψ ⇒ ⟨((x ∗ (y ∗ z)) ∗ z)/min{ta, tb}⟩ q Lεψ (31) for all x, y, z ∈ X and ta, tb ∈ (0, 1], then the q-set ( Lεψ,min{ta, tb})q of Lεψ is an ideal of (X, 1)∗ for all ta, tb ∈ (0, 0.5]. Proof. Let t := min{ta, tb} for all ta, tb ∈ (0, 0.5]. If y ∈ ( Lεψ, t)q, then Lεψ(y) > 1− t ≥ t since t ≤ 0.5, and so ⟨y/t⟩ ∈ Lεψ. Thus ⟨(x ∗ y)/t⟩ q Lεψ by (30), that is, x ∗ y ∈ ( Lεψ, t)q = ( Lεψ,min{ta, tb})q for all x ∈ X. Let x, y ∈ X be such that x, y ∈ ( Lεψ,min{ta, tb})q. Then Lεψ(x) + ta ≥ Lεψ(x) + min{ta, tb} > 1 and Lεψ(y) + tb ≥ Lεψ(y) + min{ta, tb} > 1, which implies that Lεψ(x) > 1 − ta ≥ ta and Lεψ(y) > 1 − tb ≥ tb, that is, ⟨x/ta⟩ ∈ Lεψ and ⟨y/tb⟩ ∈ Lεψ. It follows from (31) that ⟨((x ∗ (y ∗ z)) ∗ z)/min{ta, tb}⟩ q Lεψ for all z ∈ X. Hence (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ,min{ta, tb})q for all z ∈ X. Therefore ( Lεψ,min{ta, tb})q is an ideal of (X, 1)∗ for all ta, tb ∈ (0, 0.5]. Theorem 11. If a Lukasiewicz fuzzy set Lεψ in X satisfies: (∀x ∈ X)(∀t ∈ (0, 1]) ( ⟨x/t⟩ ∈ Lεψ ⇒ ⟨1/t⟩ q Lεψ ) , (32) and ⟨(x ∗ (y ∗ z))/ta⟩ ∈ Lεψ, ⟨y/tb⟩ ∈ Lεψ ⇒ ⟨(x ∗ z)/min{ta, tb}⟩ q Lεψ (33) for all x, y, z ∈ X and ta, tb ∈ (0, 1], then the non-empty q-set ( Lεψ,min{ta, tb})q of Lεψ is an ideal of (X, 1)∗ for all ta, tb ∈ (0, 0.5]. S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1317 Proof. Let ta, tb ∈ (0, 0.5]. If ( Lεψ,min{ta, tb})q is non-empty, then there exists x ∈ ( Lεψ,min{ta, tb})q. Hence Lεψ(x) > 1 − min{ta, tb} ≥ min{ta, tb}, which shows that ⟨x/min{ta, tb}⟩ ∈ Lεψ. It follows from (32) that ⟨1/min{ta, tb}⟩ q Lεψ. Thus 1 ∈ ( Lεψ,min{ta, tb})q. Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ ( Lεψ,min{ta, tb})q and y ∈ ( Lεψ,min{ta, tb})q. Then Lεψ(x ∗ (y ∗ z)) > 1 − min{ta, tb} ≥ min{ta, tb} and Lεψ(y) > 1−min{ta, tb} ≥ min{ta, tb}. Thus ⟨(x∗(y∗z))/min{ta, tb}⟩ ∈ Lεψ and ⟨y/min{ta, tb}⟩ ∈ Lεψ. It follows from (33) that ⟨(x∗z)/min{ta, tb}⟩ q Lεψ, i.e., x∗z ∈ ( Lεψ,min{ta, tb})q. Therefore ( Lεψ,min{ta, tb})q is an ideal of (X, 1)∗ by Lemma 1. Theorem 12. If a Lukasiewicz fuzzy set Lεψ in X satisfies (27) and (29) for all x, y, z ∈ X and t, ta, tb ∈ (0.5, 1], then the q-set ( Lεψ, t)q of Lεψ is an ideal of (X, 1)∗ for all t ∈ (0.5, 1]. Proof. Assume that Lεψ satisfies (27) and (29) for all x, y, z ∈ X and t, ta, tb ∈ (0.5, 1]. The condition (27) induces Lψ(1) + t ≥ 2t > 1, i.e., ⟨1/t⟩ q Lεψ. Hence 1 ∈ ( Lεψ, t)q. Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ ( Lεψ, t)q and y ∈ ( Lεψ, t)q. Then ⟨(x ∗ (y ∗ z))/t⟩ q Lεψ and ⟨y/t⟩ q Lεψ. It follows from (29) that x ∗ z ∈ ( Lεψ,min{t, t})∈ = ( Lεψ, t)∈. Hence Lεψ(x ∗ z) ≥ t > 1 − t, that is, x ∗ z ∈ ( Lεψ, t)q. Therefore ( Lεψ, t)q is an ideal of (X, 1)∗ for all t ∈ (0.5, 1] by Lemma 1. Theorem 13. If a Lukasiewicz fuzzy set Lεψ in X satisfies (28) for all x, y, z ∈ X and ta, tb ∈ (0.5, 1], and (∀x, y ∈ X)(∀t ∈ (0.5, 1]) ( ⟨y/t⟩ q Lεψ ⇒ ⟨(x ∗ y)/t⟩ ∈ Lεψ ) , (34) then the q-set ( Lεψ, t)q of Lεψ is an ideal of (X, 1)∗ for all t ∈ (0.5, 1]. Proof. Let x, y ∈ X and t ∈ (0.5, 1] be such that y ∈ ( Lεψ, t)q. Then ⟨y/t⟩ q Lεψ, and so ⟨(x ∗ y)/t⟩ ∈ Lεψ by (34). Thus Lεψ(x ∗ y) ≥ t > 1 − t, that is, ⟨(x ∗ y)/t⟩ q Lεψ. Hence x ∗ y ∈ ( Lεψ, t)q. Let x, y ∈ X and t ∈ (0.5, 1] be such that x ∈ ( Lεψ, t)q and y ∈ ( Lεψ, t)q. Then Lεψ(x) ≥ t > 1 − t and Lεψ(y) ≥ t > 1 − t, i.e., ⟨x/t⟩ q Lεψ and ⟨y/t⟩ q Lεψ. It follows from (28) that ⟨((x ∗ (y ∗ z)) ∗ z)/t⟩ = ⟨((x ∗ (y ∗ z)) ∗ z)/min{t, t}⟩ q Lεψ. This shows that (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ, t)q. Therefore the q-set ( Lεψ, t)q of Lεψ is an ideal of (X, 1)∗ for all t ∈ (0.5, 1]. Theorem 14. If ψ is a fuzzy ideal of (X, 1)∗, then the non-empty O-set of Lεψ is an ideal of (X, 1)∗. Proof. If ψ is a fuzzy ideal of (X, 1)∗, then Lεψ is a Lukasiewicz fuzzy ideal of (X, 1)∗ (see Theorem 3). It is clear that 1 ∈ O( Lεψ). Let x, y, z ∈ X be such that y ∈ O( Lεψ) and x ∗ (y ∗ z) ∈ O( Lεψ). Then Lεψ(x ∗ (y ∗ z)) > 0 and Lεψ(y) > 0. Since ⟨(x ∗ (y ∗ z))/ Lεψ(x ∗ (y ∗ z))⟩ ∈ Lεψ and ⟨y/ Lεψ(y)⟩ ∈ Lεψ, we have ⟨(x ∗ z)/min { Lεψ(x ∗ (y ∗ z)), Lεψ(y) } ⟩ ∈ Lεψ by (17). It follows that Lεψ(x ∗ z) ≥ min { Lεψ(x ∗ (y ∗ z)), Lεψ(y) } > 0. S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1318 Hence x ∗ z ∈ O( Lεψ), and therefore O( Lεψ) is an ideal of (X, 1)∗ by Lemma 1. Theorem 15. If a Lukasiewicz fuzzy set Lεψ in X satisfies (13) and (∀x, y, z ∈ X)(∀ta, tb ∈ (0, 1]) ( ⟨(x ∗ (y ∗ z))/ta⟩ ∈ Lεψ, ⟨y/tb⟩ ∈ Lεψ ⇒ ⟨(x ∗ z)/max{ta, tb}⟩ q Lεψ ) . (35) then the non-empty O-set of Lεψ is an ideal of (X, 1)∗. Proof. Let O( Lεψ) be a non-empty O-set of Lεψ. Then there exists x ∈ O( Lεψ), and so t := Lεψ(x) > 0, i.e., ⟨x/t⟩ ∈ Lεψ for t > 0. Hence ⟨1/t⟩ ∈ Lεψ by (13), and thus Lεψ(1) ≥ t > 0. Thus 1 ∈ O( Lεψ). Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ O( Lεψ) and y ∈ O( Lεψ). Then ψ(x ∗ (y ∗ z)) + ε > 1 and ψ(y) + ε > 1. Since ⟨(x ∗ (y ∗ z))/ Lεψ(x ∗ (y ∗ z))⟩ ∈ Lεψ and ⟨y/ Lεψ(y)⟩ ∈ Lεψ, it follows from (35) that ⟨(x ∗ z)/max{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)}⟩ q Lεψ. If x ∗ z /∈ O( Lεψ), then Lεψ(x ∗ z) = 0, and so Lεψ(x ∗ z) + max{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)} = max{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)} = max{max{0, ψ(x ∗ (y ∗ z)) + ε− 1}, max{0, ψ(y) + ε− 1}} = max{ψ(x ∗ (y ∗ z)) + ε− 1, ψ(y) + ε− 1} = max{ψ(x ∗ (y ∗ z)), ψ(y)} + ε− 1 ≤ 1 + ε− 1 ≤ 1. Hence ⟨(x ∗ z)/max{ Lεψ(x ∗ (y ∗ z)), Lεψ(y)}⟩ q Lεψ, a contradiction. Thus x ∗ z ∈ O( Lεψ), and therefore O( Lεψ) is an ideal of (X, 1)∗ by Lemma 1. Theorem 16. If a Lukasiewicz fuzzy set Lεψ in X satisfies (∀x, y ∈ X)(∀t ∈ (0, 1]) ( ⟨y/t⟩ ∈ ψ ⇒ ⟨(x ∗ y)/t⟩ q Lεψ ) , (36) and ⟨x/ta⟩ ∈ ψ, ⟨y/tb⟩ ∈ ψ ⇒ ⟨((x ∗ (y ∗ z)) ∗ z)/max{ta, tb}⟩ q Lεψ (37) for all x, y, z ∈ X and ta, tb ∈ (0, 1], then the O-set of Lεψ is an ideal of (X, 1)∗. Proof. If y ∈ O( Lεψ), then ψ(y) > 1−ε, i.e., ⟨y/(1−ε)⟩ ∈ ψ. Hence ⟨(x∗y)/(1 − ε)⟩ q Lεψ for all x ∈ X by (36), and thus Lεψ(x ∗ y) + 1 − ε > 1. Thus Lεψ(x ∗ y) > ε > 0, which shows that x ∗ y ∈ O( Lεψ) for all x ∈ X. Let x, y, z ∈ X be such that x, y ∈ O( Lεψ). Then ψ(x) > 1 − ε and ψ(y) > 1 − ε, that is, ⟨x/(1 − ε)⟩ ∈ ψ and ⟨y/(1 − ε)⟩ ∈ ψ. It follows from (37) that ⟨((x ∗ (y ∗ z)) ∗ z)/(1 − ε)⟩ = ⟨((x ∗ (y ∗ z)) ∗ z)/max{1 − ε, 1 − ε}⟩ q Lεψ. Thus Lεψ((x ∗ (y ∗ z)) ∗ z) + 1 − ε > 1, and so Lεψ((x ∗ (y ∗ z)) ∗ z) > ε > 0. Hence (x ∗ (y ∗ z)) ∗ z ∈ O( Lεψ), and therefore O( Lεψ) is an ideal of (X, 1)∗. S. S. Ahn, E. H. Roh and Y. B. Jun / Eur. J. Pure Appl. Math, 15 (3) (2022), 1307-1320 1319 Theorem 17. Let Lεψ be a Lukasiewicz fuzzy set in X that satisfies ⟨1/ε⟩ q ψ and (∀x, y, z ∈ X) ( ⟨(x ∗ (y ∗ z))/ε⟩ q ψ, ⟨y/ε⟩ q ψ ⇒ ⟨(x ∗ z)/ε⟩ ∈ Lεψ ) . (38) Then the O-set of Lεψ is an ideal of (X, 1)∗. Proof. Let O( Lεψ) be the O-set of Lεψ. If ⟨1/ε⟩ q ψ, then ψ(1) + ε > 1 and so Lεψ(1) = max{0, ψ(1) + ε− 1} = ψ(1) + ε− 1 > 0. Hence 1 ∈ O( Lεψ). Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ O( Lεψ) and y ∈ O( Lεψ). Then ψ(x ∗ (y ∗ z)) + ε > 1 and ψ(y) + ε > 1, i.e., ⟨(x ∗ (y ∗ z))/ε⟩ q ψ and ⟨y/ε⟩ q ψ. It follows from (38) that ⟨(x ∗ z)/ε⟩ ∈ Lεψ, which shows Lεψ(x ∗ z) ≥ ε > 0. Hence x ∗ z ∈ O( Lεψ), and therefore O( Lεψ) is an ideal of (X, 1)∗ by Lemma 1. Theorem 18. Let Lεψ be a Lukasiewicz fuzzy set in X that satisfies: (∀x, y ∈ X)(∀t ∈ [ε, 1]) ( ⟨y/t⟩ q ψ ⇒ ⟨(x ∗ y)/ε⟩ ∈ Lεψ ) , (39) (∀x, y, z ∈ X)(∀ta, tb ∈ [ε, 1]) ( ⟨x/ta⟩ q ψ, ⟨y/tb⟩ q ψ ⇒ (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ, ε)∈ ) . (40) Then the O-set of Lεψ is an ideal of (X, 1)∗. Proof. Let t ∈ [ε, 1], x ∈ X and y ∈ O( Lεψ). Then ψ(y) + t ≥ ψ(y) + ε > 1, and so ⟨y/t⟩ q ψ, which implies that ⟨(x ∗ y)/ε⟩ ∈ Lεψ by (39). Hence Lεψ(x ∗ y) ≥ ε > 0, i.e., x ∗ y ∈ O( Lεψ). Let ta, tb ∈ [ε, 1] and x, y, z ∈ X be such that x ∈ O( Lεψ) and y ∈ O( Lεψ). Then ψ(x)+ta ≥ ψ(x)+ε > 1 and ψ(y)+tb ≥ ψ(y)+ε > 1. Thus ⟨x/ta⟩ q ψ and ⟨y/tb⟩ q ψ Using (40) leads to (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ, ε)∈. Hence Lεψ((x ∗ (y ∗ z)) ∗ z) ≥ ε > 0, and so (x ∗ (y ∗ z)) ∗ z ∈ O( Lεψ). Consequently, O( Lεψ) is an ideal of (X, 1)∗. Corollary 3. Let Lεψ be a Lukasiewicz fuzzy set in X that satisfies: (∀x, y ∈ X) ( ⟨y/ε⟩ q ψ ⇒ ⟨(x ∗ y)/ε⟩ ∈ Lεψ ) , (41) (∀x, y, z ∈ X) ( ⟨x/ε⟩ q ψ, ⟨y/ε⟩ q ψ ⇒ (x ∗ (y ∗ z)) ∗ z ∈ ( Lεψ, ε)∈ ) . (42) Then the O-set of Lεψ is an ideal of (X, 1)∗. 4. Conclusions and future work The concept of Lukasiewicz fuzzy sets using Lukasiewicz t-norm was introduced by Y. B. Jun. In this paper, Lukasiewicz fuzzy set has been applied to the ideal in BE- algebra, and introducing the concept of Lukasiewicz fuzzy ideal and examining several properties. We discussed the characterization of Lukasiewicz fuzzy ideal and considered the relationship between fuzzy ideal and Lukasiewicz fuzzy ideal. We provided conditions REFERENCES 1320 under which Lukasiewicz fuzzy set can be Lukasiewicz fuzzy ideal, and further explored conditions under which three subsets, ∈-set, q-set, and O-set, will be ideal The ideas and results obtained in this paper will be applied to the relevant algebraic systems in the future, further examining their usability as a mathematical tool applicable to decision theory, medical diagnosis systems, and automation systems etc. Acknowledgements The authors are very grateful to anonymous reviewers for their valuable comments. References [1] S. S. Ahn and K. S. So. On ideals and upper sets in be-algebras. Sci. Math. Jpn., 68(2), 2008. [2] G.Dymek and A. 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