The subalgebra of BCK/BCI-algebra using Łukasiewicz anti fuzzy set
introduced by Jun is studied in this article. The concept of Łukasiewicz anti
fuzzy subalgebra of a BCK/BCI-algebra is introduced, and several properties are
investigated. The relationship between anti fuzzy subalgebra and Łukasiewicz anti
fuzzy subalgebra is given, and characterization of a Łukasiewicz anti fuzzy subalgebra
is discussed. Conditions are found in which a Lukasiewicz anti fuzzy set is a
Lukasiewicz anti fuzzy subalgebra Finally, conditions under which ⋖-subset, Υsubset, and anti-subset become subalgebra are explored

Bordbar, Hashem

Kang, Jeong Gi

English

Repository of University of Nova Gorica

Kragujevac Journal of MathematicsVolume 49(5) (2025), Pages 687–696.ŁUKASIEWICZ ANTI FUZZY SUBALGEBRAS OFBCK/BCI-ALGEBRASJEONG GI KANG1 AND HASHEM BORDBAR2Abstract. The subalgebra of BCK/BCI-algebra using Łukasiewicz anti fuzzy setintroduced by Jun is studied in this article. The concept of Łukasiewicz antifuzzy subalgebra of a BCK/BCI-algebra is introduced, and several properties areinvestigated. The relationship between anti fuzzy subalgebra and Łukasiewicz antifuzzy subalgebra is given, and characterization of a Łukasiewicz anti fuzzy subalgebrais discussed. Conditions are found in which a Lukasiewicz anti fuzzy set is aLukasiewicz anti fuzzy subalgebra Finally, conditions under which ⋖-subset, Υ-subset, and anti-subset become subalgebra are explored.1. IntroductionIn [1], Biswas introduced the concept of anti fuzzy subgroups of groups. ModifyingBiswas’ idea, Hong and Jun [3] applied the idea to BCK-algebras. They introducedthe notions of anti fuzzy subalgebras and anti fuzzy ideals of BCK-algebras andinvestigated several properties. Using anti fuzzy notion and the idea of Łukasiewiczt-conorm, Jun [7] constructed the concept of Łukasiewicz anti fuzzy sets and appliedit to BE-algebras. He introduced the notion of Łukasiewicz anti fuzzy BE-ideal andinvestigated its properties. He discussed the relationship between anti fuzzy BE-idealand Łukasiewicz anti fuzzy BE-ideal and provided conditions for Łukasiewicz antifuzzy set to be Łukasiewicz anti fuzzy BE-ideal. He also gives three types of subsetsso called ⋖-subset, Υ-subset, and anti subset, and then he considered the conditionsunder which they can be BE-ideals.Key words and phrases. Anti fuzzy subalgebra, Łukasiewicz anti fuzzy set, Łukasiewicz anti fuzzysubalgebra, ⋖-subset, Υ-subset, anti subset.2020 Mathematics Subject Classification. Primary: 03G25. Secondary: 06F35, 08A72.DOI 10.46793/KgJMat2505.687KReceived: August 05, 2022.Accepted: December 01, 2022.687688 J. G. KANG AND H. BORDBARWe would like to study the subalgebra of BCK/BCI-algebra using Łukasiewiczanti fuzzy set introduced by Jun. We introduce Łukasiewicz anti fuzzy subalgebraof a BCK/BCI-algebra and investigate several properties. We give the relationshipbetween anti fuzzy subalgebra and Łukasiewicz anti fuzzy subalgebra. We discussa characterization of a Łukasiewicz anti fuzzy subalgebra. We find conditions for aLukasiewicz anti fuzzy set to be a Lukasiewicz anti fuzzy subalgebra. We finally findthe condition that ⋖-subset, Υ-subset, and anti subset become subalgebra.2. PreliminariesThis section lists the known default content that will be used later.A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki(see [5] and [6]) and was extensively investigated by several researchers.We recall the definitions and basic results required in this paper. See the books[4, 8] for further information regarding BCK-algebras and BCI-algebras.If a set X has a special element “0” and a binary operation “ ∗ ” satisfying theconditions:(I1) (∀a, b, c ∈ X) (((a ∗ b) ∗ (a ∗ c)) ∗ (c ∗ b) = 0);(I2) (∀a, b ∈ X) ((a ∗ (a ∗ b)) ∗ b = 0);(I3) (∀a ∈ X) (a ∗ a = 0);(I4) (∀a, b ∈ X) (a ∗ b = 0, b ∗ a = 0 ⇒ a = b),then we say that X is a BCI-algebra. If a BCI-algebra X satisfies the followingidentity:(K) (∀a ∈ X) (0 ∗ a = 0),then X is called a BCK-algebra.The order relation “≤” in a BCK/BCI-algebra X is defined as follows:(∀a, b ∈ X)(a ≤ b ⇔ a ∗ b = 0).(2.1)Every BCK/BCI-algebra X satisfies the following conditions (see [4, 8]):(∀a ∈ X) (a ∗ 0 = a) ,(2.2)(∀a, b, c ∈ X) (a ≤ b ⇒ a ∗ c ≤ b ∗ c, c ∗ b ≤ c ∗ a) ,(2.3)(∀a, b, c ∈ X) ((a ∗ b) ∗ c = (a ∗ c) ∗ b) .(2.4)Every BCI-algebra X satisfies (see [4]):(∀a, b ∈ X) (a ∗ (a ∗ (a ∗ b)) = a ∗ b) ,(2.5)(∀a, b ∈ X) (0 ∗ (a ∗ b) = (0 ∗ a) ∗ (0 ∗ b)) .(2.6)A subset K of a BCK/BCI-algebra X is called a subalgebra of X (see [4, 8]) if itsatisfies:(∀a, b ∈ K)(a ∗ b ∈ K).(2.7)ŁUKASIEWICZ ANTI FUZZY SUBALGEBRAS OF BCK/BCI-ALGEBRAS 689A fuzzy set g in a set X of the formg(b) :={s ∈ [0, 1), if b = a,1, if b ̸= a,(2.8)is called an anti fuzzy point with support a and value s, and is denoted by as. A fuzzyset g in a set X is said to be non-unit if there exists a ∈ X such that g(a) ̸= 1.For a fuzzy set g in a set X, we say that an anti fuzzy point asis said to(i) beside in g, denoted by as⋖ g (see [2]) if g(a) ≤ s;(ii) be non-quasi coincident with g, denoted by asΥ g (see [2]) if g(a) + s < 1.If as⋖ g or asΥ g (resp., as⋖ g and asΥ g), we say that as⋖∨Υ g (resp., as⋖∧Υ g).Given β ∈ {⋖, Υ}, to indicate asβ g means that asβ g is not established.A fuzzy set f in a BCK/BCI-algebra X is called• an anti fuzzy subalgebra of X (see [3]) if it satisfies:(∀a, b ∈ X)(f(a ∗ b) ≤ max{f(a), f(b)});(2.9)• an anti fuzzy ideal of X (see [3]) if it satisfies:(∀a ∈ X)(f(0) ≤ f(a)),(2.10)(∀a, b ∈ X)(f(a) ≤ max{f(a ∗ b), f(b)}).(2.11)Let ε be an element of the unit interval [0, 1] and let g be a fuzzy set in a set X. Afunction Łεg : X → [0, 1], x 7→ min{1, g(x) + ε}, is called a Łukasiewicz anti fuzzy setof g in X (see [7]).Let Łεg be a Łukasiewicz anti fuzzy set of a fuzzy set g in X. If ε = 0, thenŁεg(x) = min{1, g(x) + ε} = min{1, g(x)} = g(x) for all x ∈ X. This shows that ifε = 0, then the Łukasiewicz anti fuzzy set of a fuzzy set g in X is the classical fuzzyset g itself in X. If ε = 1, then Łεg(x) = min{1, g(x) + ε} = min{1, g(x) + 1} = 1for all x ∈ X, that is, if ε = 1, then the Łukasiewicz anti fuzzy set is the constantfunction with value 1. Therefore, in handling the Łukasiewicz anti fuzzy set, the valueof ε can always be considered to be in (0, 1).Let g be a fuzzy set in a set X and ε ∈ (0, 1). If g(x) + ε ≥ 1 for all x ∈ X, thenthe Łukasiewicz anti fuzzy set Łεg of g in X is the constant function with value 1,that is, Łεg(x) = 1 for all x ∈ X. Therefore, for the Łukasiewicz anti fuzzy set tohave a meaningful shape, a fuzzy set g in X and ε ∈ (0, 1) shall be set to satisfy thecondition “g(x) + ε < 1 for some x ∈ X”.Given a Łukasiewicz anti fuzzy set Łεg of a fuzzy set g in X and s ∈ [0, 1), considerthe sets:(Łεg, s)⋖ := {y ∈ X | ys ⋖ Łεg} and (Łεg, s)Υ := {y ∈ X | ys Υ Łεg},which are called the ⋖-subset and Υ-subset of Łεg in X. Also, we consider the followingsetAnti (Łεg) := {y ∈ X | Łεg(y) < 1}690 J. G. KANG AND H. BORDBARand it is called the anti subset of Łεg in X. It is observed thatAnti (Łεg) = {y ∈ X | g(y) + ε < 1}.3. Łukasiewicz Anti Fuzzy SubalgebrasIn this section, let f and γ be a fuzzy set in X and an element of (0, 1), respectively,unless otherwise specified.Definition 3.1. A Łukasiewicz anti fuzzy set Łγf in a BCK/BCI-algebra X is calleda Łukasiewicz anti fuzzy subalgebra of X if it satisfies(∀x, y ∈ X)(∀sa, sb ∈ [0, 1))(xsa⋖ Łγf , ysb ⋖ Łγf ⇒ x∗ymax{sa, sb} ⋖ Łγf).(3.1)Example 3.1. Let X = {0, b1, b2, b3, b4} be a set with a binary operation “∗” given bythe Cayley table:∗ 0 b1 b2 b3 b40 0 0 0 0 0b1 b1 0 b1 0 0b2 b2 b2 0 0 0b3 b3 b3 b3 0 0b4 b4 b3 b4 b1 0.Then X is a BCK-algebra (see [8]). Define a fuzzy set f in X as follows:f : X → [0, 1], x 7→0.24, if x = 0,0.31, if x = b1,0.37, if x = b2,0.43, if x = b3,0.58, if x = b4.Given γ := 0.58, the Łukasiewicz anti fuzzy set Łγf of f in X is given as follows:Łγf : X → [0, 1], x 7→0.82, if x = 0,0.89, if x = b1,0.95, if x = b2,1.00, if x = b3,1.00, if x = b4.It is routine to verify that Łγf is a Łukasiewicz anti fuzzy subalgebra of X.Theorem 3.1. If f is an anti fuzzy subalgebra of a BCK/BCI-algebra X, then it’sŁukasiewicz anti fuzzy set Łγf in X is a Łukasiewicz anti fuzzy subalgebra of X.Proof. Assume that f is an anti fuzzy subalgebra of a BCK/BCI-algebra X. Letx, y ∈ X and sa, sb ∈ [0, 1) be such that xsa ⋖ Łγf and ysb ⋖ Łγf . Then, Łγf(x) ≤ sa andŁUKASIEWICZ ANTI FUZZY SUBALGEBRAS OF BCK/BCI-ALGEBRAS 691Łγf (y) ≤ sb. Hence,Łγf (x ∗ y) = min{1, f(x ∗ z) + γ} ≤ min{1, max{f(x), f(y)} + γ}= min{1, max{f(x) + γ, f(y) + γ}}= max{min{1, f(x) + γ}, min{1, f(y) + γ}}= max{Łγf (x), Łγf (y)} ≤ max{sa, sb},which implies that x∗ymax{sa, sb}⋖Łγf . Therefore, Łγf is a Łukasiewicz anti fuzzy subalgebraof X. □The following example shows that the converse of Theorem 3.1 may not be true.Example 3.2. Let X = {0, b1, b2, b3, b4} be a set with a binary operation “∗”∗ 0 b1 b2 b3 b40 0 0 b2 b3 b4b1 b1 0 b2 b3 b4b2 b2 b2 0 b4 b3b3 b3 b3 b4 0 b2b4 b4 b4 b3 b2 0.Then, X is a BCI-algebra (see [4]). Define a fuzzy set f in X as follows:f : X → [0, 1], x 7→0.28, if x = 0,0.32, if x = b1,0.39, if x = b2,0.43, if x = b3,0.61, if x = b4.Given γ := 0.58, the γ-Łukasiewicz fuzzy set Łγf of f in X is given as follows:Łγf : X → [0, 1], x 7→0.86, if x = 0,0.90, if x = b1,0.97, if x = b2,1.00, if x = b3,1.00, if x = b4.It is routine to verify that Łγf is a Łukasiewicz anti fuzzy subalgebra of X. But f isnot an anti fuzzy subalgebra of X because off(b2 ∗ b3) = f(b4) = 0.61 ≰ 0.43 = max{f(b2), f(b3)}.We explore a characterization of a Łukasiewicz anti fuzzy subalgebra.Theorem 3.2. Let f be a fuzzy set in a BCK/BCI-algebra X. Then its Łukasiewiczanti fuzzy set Łγf in X is a Łukasiewicz anti fuzzy subalgebra of X if and only if itsatisfies(∀x, y ∈ X)(Łγf (x ∗ y) ≤ max{Łγf (x), Łγf (y)}).(3.2)692 J. G. KANG AND H. BORDBARProof. Suppose that Łγf is a Łukasiewicz anti fuzzy subalgebra of X. Let x, y ∈ X.Since xŁγf(x) ⋖ Łγf and yŁγf(y) ⋖ Łγf , it follows from (3.1) that x∗ymax{Łγf(x), Łγf(y)} ⋖ Łγf . Hence,Łγf (x ∗ y) ≤ max{Łγf (x), Łγf (y)}.Conversely, assume that Łγf satisfies (3.2). Let x, y ∈ X and sa, sb ∈ [0, 1) be suchthat xsa⋖ Łγf and ysb ⋖ Łγf . Then Łγf (x) ≤ sa and Łγf (y) ≤ sb, and soŁγf (x ∗ y) ≤ max{Łγf (x), Łγf (y)} ≤ max{sa, sb}.Thus, x∗ymax{sa, sb} ⋖ Łγf , and therefore, Łγf is a Łukasiewicz anti fuzzy subalgebra ofX. □Lemma 3.1 ([7]). If f is a fuzzy set in a set X, then it’s Łukasiewicz anti fuzzy setŁγf satisfies(∀x, y ∈ X)(f(x) ≥ f(y) ⇒ Łγg(x) ≥ Łγg(y)).(3.3)Lemma 3.2. If f is an anti fuzzy subalgebra of a BCK/BCI-algebra X, then it’sŁukasiewicz anti fuzzy set Łγf satisfies(∀x ∈ X)(Łγf (0) ≤ Łγf (x)).(3.4)Proof. If f is an anti fuzzy subalgebra of a BCK/BCI-algebra X, thenf(0) = f(x ∗ x) ≤ max{f(x), f(x)} = f(x),for all x ∈ X. It follows from (3.3) that Łγf (0) ≤ Łγf (x) for all x ∈ X. □Proposition 3.1. If f is an anti fuzzy subalgebra of a BCK/BCI-algebra X, then it’sŁukasiewicz fuzzy set Łγf satisfies:(∀x, y ∈ X)(Łγf (x) = Łγf (0) ⇔ Łγf (x ∗ y) ≤ Łγf (y)).(3.5)Proof. Let f be an anti fuzzy subalgebra of a BCK/BCI-algebra X. Then Łγf is aŁukasiewicz anti fuzzy subalgebra of X (see Theorem 3.1). Assume that Łγf (x) = Łγf (0)for all x ∈ X. Then,Łγf (x ∗ y) ≤ max{Łγf (x), Łγf (y)} = max{Łγf (0), Łγf (y)} = Łγf (y),for all x, y ∈ X, by Theorem 3.2 and Lemma 3.2.Conversely, suppose that Łγf(x ∗ y) ≤ Łγf(y) for all x, y ∈ X. Using (2.2) inducesŁγf (x) = Łγf (x ∗ 0) ≤ Łγf (0), and so Łγf (x) = Łγf (0) for all x ∈ X, by Lemma 3.2. □Proposition 3.2. If f is an anti fuzzy subalgebra of a BCI-algebra X, then itsŁukasiewicz fuzzy set Łγf satisfies(∀x ∈ X)(Łγf (0 ∗ x) ≤ Łγf (x)).(3.6)Proof. If f is an anti fuzzy subalgebra of a BCI-algebra X, thenf(0 ∗ x) ≤ max{f(0), f(x)} = f(x),for all x ∈ X. Hence, Łγf (0 ∗ x) ≤ Łγf (x) for all x ∈ X, by Lemma 3.1. □ŁUKASIEWICZ ANTI FUZZY SUBALGEBRAS OF BCK/BCI-ALGEBRAS 693Proposition 3.3. If f is an anti fuzzy subalgebra of a BCI-algebra X, then itsŁukasiewicz fuzzy set Łγf satisfies(∀x, y ∈ X)(∀sa, sb ∈ [0, 1))(xsa⋖ Łγf , ysb ⋖ Łγf ⇒x∗(0∗y)max{sa,sb}] ⋖ Łγf).(3.7)Proof. Let x, y ∈ X and sa, sb ∈ [0.1) be such that xsa ⋖ Łγf and ysb ⋖ Łγf . Then,Łγf (x) ≤ sa and Łγf (y) ≤ sb, and thus,Łγf (x ∗ (0 ∗ y)) = min{1, f(x ∗ (0 ∗ y)) + γ}≤ min{1, max{f(x), f(0 ∗ y)} + γ}≤ min{1, max{f(x), max{f(0), f(y)}} + γ}= min{1, max{f(x), f(y)} + γ}= min{1, max{f(x) + γ, f(y) + γ}}= max{min{1, f(x) + γ}, min{1, f(y) + γ}}= max{Łγf (x), Łγf (y)}≤ max{sa, sb}.Hence, x∗(0∗y)max{sa,sb} ] ⋖ Łγf . □We give conditions for a Lukasiewicz anti fuzzy set to be a Lukasiewicz anti fuzzysubalgebra.Theorem 3.3. Let f be a fuzzy set in a BCK/BCI-algebra X. If it’s Łukasiewiczanti fuzzy set Łγf satisfies(∀x, y ∈ X)(∀sb, sc ∈ [0, 1))(z ≤ x, ysb⋖ Łγf , zsc ⋖ Łγf ⇒ x∗ymax{sb,sc} ⋖ Łγf),(3.8)then Łγf is a Łukasiewicz anti fuzzy subalgebra of X.Proof. It is straightforward by (I3) and (3.8). □Proposition 3.4. Let f be a fuzzy set in a BCI-algebra X. Then every Łukasiewiczfuzzy subalgebra Łγf of X satisfies(∀x, y ∈ X)(∀sa, sb ∈ [0, 1))(xsa⋖ Łγf , ysb ⋖ Łγf ⇒x∗(0∗y)max{sa,sb}⋖ Łγf).(3.9)Proof. Let x, y ∈ X and sa, sb ∈ [0, 1) be such that xsa ⋖ Łγf and ysb ⋖ Łγf . ThenŁγf (x) ≤ sa and Łγf (y) ≤ sb. Using Theorem 3.2 and Proposition 3.2, we haveŁγf (x ∗ (0 ∗ y)) ≤ max{Łγf (x), Łγf (0 ∗ y)} ≤ max{Łγf (x), Łγf (y)} ≤ max{sa, sb},and so, x∗(0∗y)max{sa,sb} ⋖ Łγf . □Corollary 3.1. If f is an anti fuzzy subalgebra of a BCI-algebra X, then its Łukasie-wicz fuzzy set Łγf satisfies the condition (3.9).694 J. G. KANG AND H. BORDBARTheorem 3.4. Let Łγf be a Łukasiewicz anti fuzzy set of a fuzzy set f in a BCK/BCI-algebra X. Then the ⋖-set (Łγf , s)⋖ of Łγf is a subalgebra of X for all s ∈ [0, 0.5) ifand only if the following assertion is valid(∀x, y ∈ X)(min{Łγf (x ∗ y), 0.5} ≤ max{Łγf (x), Łγf (y)}).(3.10)Proof. Assume that the ⋖-set (Łγf , s)⋖ of Łγf is a subalgebra of X for all s ∈ [0, 0.5).If the condition (3.10) does not hold, thenmax{Łγf (a), Łγf (b)} < min{Łγf (a ∗ b), 0.5},for some a, b ∈ X. If we take s := max{Łγf(a), Łγf(b)}, then s ∈ [0, 0.5), as ⋖ Łγf andbs⋖Łγf , i.e., a, b ∈ (Łγf , s)⋖. Since (Łγf , s)⋖ is a subalgebra of X, we have a∗b ∈ (Łγf , s)⋖.But a∗bs⋖Łγf implies a ∗ b /∈ (Łγf , s)⋖, a contradiction. Hence,max{Łγf (x), Łγf (y)} ≥ min{Łγf (x ∗ y), 0.5},for all x, y ∈ X.Conversely, suppose that Łγf satisfies (3.10). Let s ∈ [0, 0.5) and x, y ∈ X be suchthat x ∈ (Łγf , s)⋖ and y ∈ (Łγf , s)⋖. Then Łγf (x) ≤ s and Łγf (y) ≤ s, which imply from(3.10) that0.5 > s ≥ max{Łγf (x), Łγf (y)} ≥ min{Łγf (x ∗ y), 0.5}.Hence, x∗ys⋖ Łγf , i.e., x ∗ y ∈ (Łγf , s)⋖. Therefore, (Łγf , s)⋖ is a subalgebra of X fors ∈ [0, 0.5). □Theorem 3.5. Let Łγf be a Łukasiewicz fuzzy set of a fuzzy set f in a BCK/BCI-algebra X. If f is an anti fuzzy subalgebra of X, then the Υ-set (Łγf , s)Υ of Łγf is asubalgebra of X for all s ∈ [0, 1).Proof. Let s ∈ [0, 1) and x, y ∈ (Łγf , s)Υ. Then xs Υ Łγf and ys Υ Łγf , that is, Łγf (x)+s < 1and Łγf (y) + s < 1. Hence,Łγf (x ∗ y) + s ≤ max{Łγf (x), Łγf (y)} + s = max{Łγf (x) + s, Łγf (y) + s} < 1,by Theorems 3.1 and 3.2. Thus, x∗ysΥ Łγf , and so, x ∗ y ∈ (Łγf , s)Υ. Therefore, (Łγf , s)Υis a subalgebra of X. □Theorem 3.6. Let f be a fuzzy set in a BCK/BCI-algebra X. For a Łukasiewicz antifuzzy set Łγf of f in X, if the Υ-set (Łγf , s)Υ is a subalgebra of X, then Łγf satisfies(∀x, y ∈ X)(∀sa, sb ∈ (0.5, 1])(xsaΥ Łγf , ysb Υ Łγf ⇒ x∗ymin{sa,sb} ⋖ Łγf).(3.11)Proof. Let x, y ∈ X and sa, sb ∈ (0.5, 1] be such that xsa Υ Łγf and ysb Υ Łγf . Thenx ∈ (Łγf , sa)Υ ⊆ (Łγf , min{sa, sb})Υ and y ∈ (Łγf , sb)Υ ⊆ (Łγf , min{sa, sb})Υ. Hence,x ∗ y ∈ (Łγf , min{sa, sb})Υ, and so,Łγf (x ∗ y) < 1 − min{sa, sb} ≤ min{sa, sb},since min{sa, sb} > 0.5. Therefore, x∗ymin{sa,sb} ⋖ Łγf . □ŁUKASIEWICZ ANTI FUZZY SUBALGEBRAS OF BCK/BCI-ALGEBRAS 695Theorem 3.7. Let Łγf be a Łukasiewicz fuzzy set of a fuzzy set f in a BCK/BCI-algebra X. If f is an anti fuzzy subalgebra of X, then the anti subset Anti (Łγf ) of Łγfis a subalgebra of X.Proof. Let x, y ∈ Anti (Łγf ). Then f(x) + γ < 1 and f(y) + γ < 1. If f is an anti fuzzysubalgebra of X, then Łγf is a Łukasiewicz anti fuzzy subalgebra of X (see Theorem3.1). It follows from Theorem 3.2 thatŁγf (x ∗ y) ≤ max{Łγf (x), Łγf (y)} = max{f(x) + γ, f(y) + γ} < 1.Hence, x ∗ y ∈ Anti (Łγf ), and therefore, Anti (Łγf ) is a subalgebra of X. □Theorem 3.8. Let f be a fuzzy set in a BCK/BCI-algebra X. If a Łukasiewicz antifuzzy set Łγf of f in X satisfies(∀x, y ∈ X)(∀sa, sb ∈ [0, 1))(xsa⋖ Łγf , ysb ⋖ Łγf ⇒ x∗ymin{sa,sb} Υ Łγf),(3.12)then the anti subset Anti (Łγf ) of Łγf is a subalgebra of X.Proof. Assume that Łγf satisfies the condition (3.12) for all x, y ∈ X and sa, sb ∈ [0, 1).Let x, y ∈ Anti (Łγf). Then f(x) + γ < 1 and f(y) + γ < 1. Since xŁγf(x) ⋖ Łγf andyŁγf(y) ⋖ Łγf , it follows from (3.12) thatx∗ymin{Łγf(x),Łγf(y)} Υ Łγf .(3.13)If x ∗ y /∈ Anti (Łγf ), then Łγf (x ∗ y) = 1, and so,Łγf (x ∗ y) + min{Łγf (x), Łγf (y)} = 1 + min{Łγf (x), Łγf (y)}= 1 + min{min{1, f(x) + γ}, min{1, f(y) + γ}}= 1 + min{f(x) + γ, f(y) + γ}= 1 + min{f(x), f(y)} + γ≥ 1 + γ > 1,which shows that (3.13) is not valid. This is a contradiction, and thus, x∗y ∈ Anti (Łγf ).Hence, Anti (Łγf ) is a subalgebra of X. □References[1] R. Biswas, Fuzzy subgroups and anti fuzzy subgroups, Fuzzy Sets and Systems 35(1) (1990),121–124. https://doi.org/10.1016/0165-0114(90)90025-2[2] A. B. Saeid and Y. B. Jun, Redefined fuzzy subalgebras of BCK/BCI-algebras, Iran. J. Fuzzy Syst.5(2) (2008), 63–70. https://doi.org/10.22111/ijfs.2008.334[3] S. M. Hong and Y. B. Jun, Anti fuzzy ideals in BCK-algebras, Kyungpook Math. J. 38(1) (1998),145–150.[4] Y. S. Huang, BCI-Algebra, Science Press, Beijing, China, 2006.[5] K. Iséki, On BCI-algebras, Mathematics Seminar Notes 8(1) (1980), 125–130.[6] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Mathematica Japonica23(1) (1978), 121–124. https://doi.org/10.1016/0165-0114(90)90025-2696 J. G. KANG AND H. BORDBAR[7] Y. B. Jun, Łukasiewicz anti fuzzy set and its application in BE-algebras, Transactions on FuzzySets and Systems 1(2) (2022), 37–45. http://doi.org/10.30495/tfss.2022.1960391.1037[8] J. Meng and Y. B. Jun, BCK-Algebras, Kyungmoonsa Co., Seoul, Korea, 1994.1Department of Mathematics Education,Gyeongsang National University,Jinju 52828, KoreaEmail address: jeonggikang@gmail.com2Center for Information Technologies and Applied Mathematics,University of Nova Gorica,Nova Gorica 5000, SloveniaEmail address: hashem.bordbar@ung.si

Łukasiewicz anti fuzzy subalgebras of BCK/BCI-algebras

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Łukasiewicz anti fuzzy subalgebras of BCK/BCI-algebras

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