The notions of a C-energetic subset and (anti) permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions for an element t in [0, 1] to be an (anti) permeable C-value are provided. Also conditions for a subset to be a C-energetic subset are discussed. We decompose BCK-algebra by a partition which consists of a C-energetic subset and a commutative ideal

Jun, Young Bae

Roh, Eun Hwan

Song, Seok Zun

Bulletin of the Section of Logic

English

The notions of a C-energetic subset and (anti) permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions for an element t in [0, 1] to be an (anti) permeable C-value are provided. Also conditions for a subset to be a C-energetic subset are discussed. We decompose BCK-algebra by a partition which consists of a C-energetic subset and a commutative ideal.</jats:p

Young Bae Jun

Eun Hwan Roh

Seok Zun Song

Crossref

Commutative Energetic Subsets of BCK-Algebras

Biblioteka Nauki - repozytorium artykuÅÃ³w

Repozytorium Uniwersytetu Łódzkiego (University of Lodz Repository)

Bulletin of the Section of LogicVolume 45/1 (2016), pp. 53–63http://dx.doi.org/10.18778/0138-0680.45.1.04Young Bae Jun, Eun Hwan Roh and Seok Zun Song∗COMMUTATIVE ENERGETIC SUBSETS OFBCK-ALGEBRASAbstractThe notions of a C-energetic subset and (anti) permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions foran element t in [0, 1] to be an (anti) permeable C-value are provided. Also con-ditions for a subset to be a C-energetic subset are discussed. We decomposeBCK-algebra by a partition which consists of a C-energetic subset and a com-mutative ideal.Keywords: S-energetic subset, I-energetic subset, C-energetic subset, (anti)fuzzy commutative ideal, (anti) permeable I-value, (anti) permeableC-value.2010 Mathematics Subject Classification. 06F35, 03G25, 08A72.1. IntroductionJun et al. [3] introduced the notions of energetic (resp. right vanished, rightstable) subsets and (anti) permeable values in BCK/BCI-algebras. Usingthe notion of (anti) fuzzy subalgebras/ideals of BCK/BCI-algebras, theyinvestigated relations among subalgebras/ideals, energetic subsets, (anti)permeable values, right vanished subsets and right stable subsets.∗Corresponding author.54 Young Bae Jun, Eun Hwan Roh and Seok Zun SongIn this article, we introduce the notions of a C-energetic subset and(anti) permeable C-value in BCK-algebras, and investigate related prop-erties. We provide conditions for an element t in [0, 1] to be an (anti) per-meable C-value. We also discuss conditions for a subset to be a C-energeticsubset. We show that a BCK-algebra is decomposed by a partition whichconsists of a C-energetic subset and a commutative ideal.2. PreliminariesA BCK/BCI-algebra is an important class of logical algebras introducedby K. Ise´ki and was extensively investigated by several researchers.An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra it satisfiesthe following conditions(I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0),(II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0),(III) (∀x ∈ X) (x ∗ x = 0),(IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y).If a BCI-algebra X satisfies the following identity(V) (∀x ∈ X) (0 ∗ x = 0),then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies thefollowing axioms(∀x ∈ X) (x ∗ 0 = x) , (2.1)(∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) , (2.2)(∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) , (2.3)(∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.4)where x ≤ y if and only if x∗y = 0. A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. A subsetI of a BCK/BCI-algebra X is called an ideal of X if it satisfies0 ∈ I, (2.5)(∀x ∈ X) (∀y ∈ I) (x ∗ y ∈ I ⇒ x ∈ I) . (2.6)A subset I of a BCK-algebra X is called a commutative ideal (see [5])of X if it satisfies (2.5) and(∀x, y ∈ X)(∀z ∈ I) ((x ∗ y) ∗ z ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I) . (2.7)Commutative Energetic Subsets of BCK-Algebras 55Observe that every commutative ideal is an ideal, but the converse is nottrue (see [6]).We refer the reader to the books [2, 6] for further information regardingBCK/BCI-algebras.The concept of fuzzy sets was introduced by Zadeh [7]. Let X be a set.The mapping f : X → [0, 1] is called a fuzzy set in X.A fuzzy set f in a BCK/BCI-algebra X is called a fuzzy subalgebra ofX if it satisfies(∀x, y ∈ X) (f(x ∗ y) ≥ min{f(x), f(y)}) . (2.8)A fuzzy set f in a BCK/BCI-algebra X is called a fuzzy ideal of X ifit satisfies(∀x ∈ X) (f(0) ≥ f(x)) . (2.9)(∀x, y ∈ X) (f(x) ≥ min{f(x ∗ y), f(y)}) . (2.10)Note that every fuzzy ideal f of a BCK/BCI-algebra X satisfies(∀x, y ∈ X) (x ≤ y ⇒ f(x) ≥ f(y)) . (2.11)A fuzzy set f in a BCK-algebra X is called a fuzzy commutative ideal(see [4]) of X if it satisfies (2.9) and(∀x, y, z ∈ X) (f(x ∗ (y ∗ (y ∗ x))) ≥ min{f((x ∗ y) ∗ z), f(z)}) . (2.12)For a fuzzy set f in X and t ∈ [0, 1], the (strong) upper (resp. lower)t-level sets are defined as follows:U(f ; t) := {x ∈ X | f(x) ≥ t}, U∗(f ; t) := {x ∈ X | f(x) > t},L(f ; t) := {x ∈ X | f(x) ≤ t}, L∗(f ; t) := {x ∈ X | f(x) < t}.3. Commutative energetic subsetsIn what follows, let X denote a BCK-algebra unless otherwise specified.Definition 3.1 ([3]). A non-empty subset A of X is said to be S-energeticif it satisfies(∀a, b ∈ X) (a ∗ b ∈ A ⇒ {a, b} ∩A 6= ∅) . (3.1)Definition 3.2 ([3]). A non-empty subset A of X is said to be I-energeticif it satisfies(∀x, y ∈ X) (y ∈ A ⇒ {x, y ∗ x} ∩A 6= ∅) . (3.2)56 Young Bae Jun, Eun Hwan Roh and Seok Zun SongLemma 3.3 ([3]). For any subset A of X, if X \ A is an ideal of X, thenA is I-energetic.Definition 3.4. A non-empty subset A of X is said to be commutativeenergetic (briefly, C-energetic) if it satisfies(∀x, y, z ∈ X) (x ∗ (y ∗ (y ∗ x)) ∈ A ⇒ {z, (x ∗ y) ∗ z} ∩A 6= ∅) . (3.3)Example 3.5. Let X = {0, 1, 2, 3, 4} be a BCK-algebra with the followingCayley table∗ 0 1 2 3 40 0 0 0 0 01 1 0 0 0 02 2 1 0 0 03 3 3 3 0 04 4 4 4 3 0It is routine to verify that A := {3, 4} is a C-energetic subset of X.We consider relations between an I-energetic subset and a C-energeticsubset.Theorem 3.6. Every C-energetic subset is I-energetic.Proof: Let A be a C-energetic subset of X. Let x, y ∈ X be such thaty ∈ A. Then y ∗ (0 ∗ (0 ∗ y)) = y ∈ A, and so {x, (y ∗ 0) ∗ x} ∩ A 6= ∅ by(3.3). It follows from (2.1) that {x, y∗x}∩A 6= ∅. Hence A is an I-energeticsubset of X.The converse of Theorem 3.6 is not true as seen in the following exam-ples.Example 3.7. Let X = {0, 1, 2, 3, 4} be a BCK-algebra with the followingCayley table∗ 0 1 2 3 40 0 0 0 0 01 1 0 0 1 02 2 1 0 2 03 3 3 3 0 34 4 4 4 4 0Take A := {1, 2, 4}. Then X \ A = {0, 3} is an ideal of X. Hence, byLemma 3.3, A is an I-energetic subset of X. But it is not C-energeticsinceCommutative Energetic Subsets of BCK-Algebras 571 ∗ (4 ∗ (4 ∗ 1)) = 1 ∈ A and {3, (1 ∗ 4) ∗ 3} ∩A = ∅.Theorem 3.8. For any nonempty subset A of X, if X \A is a commutativeideal of X, then A is C-energetic.Proof: Assume that A is not C-energetic. Then for any x, y ∈ X withx ∗ (y ∗ (y ∗ x)) ∈ A,there exists z ∈ X such that {z, (x ∗ y) ∗ z} ∩A = ∅. It follows that(x ∗ y) ∗ z ∈ X \A and z ∈ X \A.Since X \A is a commutative ideal of X, we have x ∗ (y ∗ (y ∗ x)) ∈ X \A,that is, x∗(y∗(y∗x)) /∈ A. This is a contradiction, and so A is a C-energeticsubset of X.Corollary 3.9. For any nonempty subset A of X, if X \A is a commu-tative ideal of X, then A is I-energetic.Theorem 3.10. Let A be a nonempty subset of X with 0 /∈ A. If A isC-energetic, then X \A is a commutative ideal of X.Proof: Obviously 0 ∈ X \A. Let x, y, z ∈ X be such that z ∈ X \A and(x∗y)∗z ∈ X\A. Assume that x∗(y∗(y∗x)) ∈ A. Then {z, (x∗y)∗z}∩A 6= ∅by (3.3), which implies that z ∈ A or (x∗y)∗z ∈ A. This is a contradiction,and so x ∗ (y ∗ (y ∗ x)) ∈ X \ A. This shows that X \ A is a commutativeideal of X.Corollary 3.11. Let A be a nonempty subset of X with 0 /∈ A. If A isC-energetic, then X \A is an ideal and hence a subalgebra of X.Theorem 3.12. If f is a fuzzy commutative ideal of X, then the nonemptylower t-level set L(f ; t) is a C-energetic subset of X.Proof: Assume that L(f ; t) 6= ∅ for t ∈ [0, 1]. Let x, y ∈ X be such thatx ∗ (y ∗ (y ∗ x)) ∈ L(f ; t). Thent ≥ f(x ∗ (y ∗ (y ∗ x))) ≥ min{f((x ∗ y) ∗ z), f(z)}for all z ∈ X, which implies that f((x ∗ y) ∗ z) ≤ t or f(z) ≤ t, that is,(x ∗ y) ∗ z ∈ L(f ; t) or z ∈ L(f ; t). Thus {z, (x ∗ y) ∗ z} ∩ L(f ; t) 6= ∅, andtherefore L(f ; t) is a C-energetic subset of X.Corollary 3.13. If f is a fuzzy commutative ideal of X, then the nonemptystrong lower t-level set L∗(f ; t) is a C-energetic subset of X.Since L(f ; t)∪U∗(f ; t) = X and L(f ; t)∩U∗(f ; t) = ∅ for all t ∈ [0, 1],we have the following corollary.58 Young Bae Jun, Eun Hwan Roh and Seok Zun SongCorollary 3.14. If f is a fuzzy commutative ideal of X, then U∗(f ; t) isempty or a commutative ideal of X for all t ∈ [0, 1].Definition 3.15 ([1]). A fuzzy set f in X is called an anti fuzzy ideal ofX if it satisfies(∀x ∈ X) (f(0) ≤ f(x)) . (3.4)(∀x, y ∈ X) (f(x) ≤ max{f(x ∗ y), f(y)}) . (3.5)Definition 3.16. A fuzzy set f in X is called an anti fuzzy commutativeideal of X if it satisfies (3.4) and(∀x, y, z ∈ X) (f(x ∗ (y ∗ (y ∗ x))) ≤ max{f((x ∗ y) ∗ z), f(z)}) . (3.6)Example 3.17. Consider a BCK-algebra X = {0, a, b, c} with the follow-ing Cayley table∗ 0 a b c0 0 0 0 0a a 0 0 ab b a 0 bc c c c 0Define a fuzzy set f in X as followsf : X → [0, 1], x 7→t0 if x = 0,t1 if x = c,t2 if x ∈ {a, b}where t0 < t1 < t2 in [0, 1]. It is routine to verify that f is an anti fuzzycommutative ideal of X.Theorem 3.18. Every anti fuzzy commutative ideal is an anti fuzzy ideal.Proof: Let f be an anti fuzzy commutative ideal of X. If we put y = 0in (3.6), thenmax{f(x ∗ z), f(z)} = max{f((x ∗ 0) ∗ z), f(z)}≥ f(x ∗ (0 ∗ (0 ∗ x))) = f(x).Hence f is an anti fuzzy ideal of X.The converse of Theorem 3.18 is not true as seen in the following ex-ample.Commutative Energetic Subsets of BCK-Algebras 59Example 3.19. Let X = {0, 1, 2, 3, 4} be a BCK-algebra with the followingCayley table∗ 0 1 2 3 40 0 0 0 0 01 1 0 1 0 02 2 2 0 0 03 3 3 3 0 04 4 4 4 3 0Define a fuzzy set f in X as followsf : X → [0, 1], x 7→s0 if x = 0,s1 if x = 1,s2 if x ∈ {2, 3, 4}where s0 < s1 < s2 in [0, 1]. Then f is an anti fuzzy ideal of X. But it isnot an anti fuzzy commutative ideal of X sincef(2 ∗ (3 ∗ (3 ∗ 2)))  max{f(0), f((2 ∗ 3) ∗ 0)}.We provide a characterization of an anti fuzzy commutative ideal.Theorem 3.20. For a fuzzy set f in X, the following are equivalent.(1) f is an anti fuzzy commutative ideal of X.(2) f is an anti fuzzy ideal of X satisfying the following condition(∀x, y ∈ X) (f(x ∗ (y ∗ (y ∗ x))) ≤ f(x ∗ y)) . (3.7)Proof: Assume that f is an anti fuzzy commutative ideal of X. Then fis an anti fuzzy ideal of X (see Theorem 3.18). Taking z = 0 in (3.6) andusing (3.4) and (2.1), we have (3.7).Conversely, suppose that (2) is valid. Thenf(x ∗ y) ≤ max{f((x ∗ y) ∗ z), f(z)} (3.8)for all x, y, z ∈ X. Combining (3.7) and (3.8), we get (3.6). The proof iscomplete.Definition 3.21 ([3]). Let f be a fuzzy set in X. A number t ∈ [0, 1] iscalled a permeable I-value for f if U(f ; t) 6= ∅ and the following assertionis valid.(∀x, y ∈ X) (f(y) ≥ t ⇒ max{f(y ∗ x), f(x)} ≥ t) . (3.9)60 Young Bae Jun, Eun Hwan Roh and Seok Zun SongDefinition 3.22. Let f be a fuzzy set in X. A number t ∈ [0, 1] is calleda permeable C-value for f if U(f ; t) 6= ∅ and the following assertion isvalid.f(x ∗ (y ∗ (y ∗ x))) ≥ t ⇒ max{f((x ∗ y) ∗ z), f(z)} ≥ t (3.10)for all x, y, z ∈ X.Example 3.23. Consider a BCK-algebra X = {0, a, b, c} which is givenin Example 3.17. Let f be a fuzzy set in X defined by f(0) = 0.3, f(a) =f(b) = 0.7 and f(c) = 0.5. If t ∈ (0.5, 0.7], then U(f ; t) = {a, b} and it iseasy to check that t is a permeable C-value for f .Theorem 3.24. Let f be a fuzzy commutative ideal of X. If t ∈ [0, 1] isa permeable C-value for f , then the nonempty upper t-level set U(f ; t) isa C-energetic subset of X.Proof: Assume that U(f ; t) 6= ∅ for t ∈ [0, 1]. Let x, y ∈ X be such thatx ∗ (y ∗ (y ∗ x)) ∈ U(f ; t).Then f(x ∗ (y ∗ (y ∗x))) ≥ t, and so max{f((x ∗ y) ∗ z), f(z)} ≥ t by (3.10).It follows that f((x ∗ y) ∗ z) ≥ t or f(z) ≥ t, that is, (x ∗ y) ∗ z ∈ U(f ; t)or z ∈ U(f ; t). Hence {z, (x ∗ y) ∗ z} ∩ U(f ; t) 6= ∅, and therefore U(f ; t) isa C-energetic subset of X.Since U(f ; t)∪L∗(f ; t) = X and U(f ; t)∩L∗(f ; t) = ∅ for all t ∈ [0, 1],we have the following corollary.Corollary 3.25. Let f be a fuzzy commutative ideal of X. If t ∈ [0, 1] isa permeable C-value for f , then L∗(f ; t) is empty or a commutative idealof X.Theorem 3.26. For a fuzzy set f in X, if there exists a subset K of [0, 1]such that {U(f ; t), L∗(f ; t)} is a partition of X and L∗(f ; t) is a commuta-tive ideal of X for all t ∈ K, then t is a permeable C-value for f .Proof: Assume that f(x ∗ (y ∗ (y ∗ x))) ≥ t for any x, y ∈ X. Thenx ∗ (y ∗ (y ∗ x)) ∈ U(f ; t),and so {z, (x ∗ y) ∗ z} ∩ U(f ; t) 6= ∅ since U(f ; t) is a C-energetic subset ofX. It follows that z ∈ U(f ; t) or (x ∗ y) ∗ z ∈ U(f ; t) and so thatmax{f((x ∗ y) ∗ z), f(z)} ≥ t.Therefore t is a permeable C-value for f .Theorem 3.27. Let f be a fuzzy set in X with U(f ; t) 6= ∅ for t ∈ [0, 1]. Iff is an anti fuzzy commutative ideal of X, then t is a permeable C-valuefor f .Commutative Energetic Subsets of BCK-Algebras 61Proof: Let x, y, z ∈ X be such that f(x ∗ (y ∗ (y ∗ x))) ≥ t. Thent ≤ f(x ∗ (y ∗ (y ∗ x))) ≤ max{f((x ∗ y) ∗ z), f(z)}by (3.6). Hence t is a permeable C-value for f .Theorem 3.28. If f is an anti fuzzy commutative ideal of X, then(∀t ∈ [0, 1]) (U(f ; t) 6= ∅ ⇒ U(f ; t) is a C-energetic subset of X) .Proof: Let x, y, z ∈ X be such that x ∗ (y ∗ (y ∗ x)) ∈ U(f ; t). Thenf(x ∗ (y ∗ (y ∗ x))) ≥ t,which implies from (3.6) thatt ≤ f(x ∗ (y ∗ (y ∗ x))) ≤ max{f((x ∗ y) ∗ z), f(z)}.Hence f((x∗y)∗z) ≥ t or f(z) ≥ t, that is, (x∗y)∗z ∈ U(f ; t) or z ∈ U(f ; t).Thus {z, (x ∗ y) ∗ z} ∩ U(f ; t) 6= ∅, and therefore U(f ; t) is a C-energeticsubset of X.Theorem 3.29. For any fuzzy set f in X, every permeable C-value for fis a permeable I-value for f .Proof: Let t ∈ [0, 1] be a permeable C-value for f . Assume that f(y) ≥ tfor all y ∈ X. Thent ≤ f(y) = f((y ∗ (0 ∗ (0 ∗ y)))by (V) and (2.1), and sot ≤ max{f((y ∗ 0) ∗ z), f(z)} = max{f(y ∗ z), f(z)}for all y, z ∈ X by (3.10) and (2.1). Therefore t is a permeable I-value forf .Definition 3.30 ([3]). Let f be a fuzzy set in X. A number t ∈ [0, 1]is called an anti permeable I-value for f if L(f ; t) 6= ∅ and the followingassertion is valid.(∀x, y ∈ X) (f(y) ≤ t ⇒ min{f(y ∗ x), f(x)} ≤ t) . (3.11)Theorem 3.31. Let f be a fuzzy set in X with L(f ; t) 6= ∅ for t ∈ [0, 1]. Iff is a fuzzy ideal of X, then t is an anti permeable I-value for f .Proof: Let f(y) ≤ t for y ∈ X. Thenmin{f(y ∗ x), f(x)} ≤ f(y) ≤ t62 Young Bae Jun, Eun Hwan Roh and Seok Zun Songfor all x ∈ X by (2.10). Hence t is an anti permeable I-value for f .Definition 3.32. Let f be a fuzzy set in X. A number t ∈ [0, 1] is calledan anti permeable C-value for f if L(f ; t) 6= ∅ and the following assertionis valid.f(x ∗ (y ∗ (y ∗ x))) ≤ t ⇒ min{f((x ∗ y) ∗ z), f(z)} ≤ t (3.12)for all x, y, z ∈ X.Theorem 3.33. Let f be a fuzzy set in X with L(f ; t) 6= ∅ for t ∈ [0, 1]. Iff is a fuzzy commutative ideal of X, then t is an anti permeable C-valuefor f .Proof: Let x, y ∈ X be such that f(x ∗ (y ∗ (y ∗ x))) ≤ t. Thenmin{f((x ∗ y) ∗ z), f(z)} ≤ f(x ∗ (y ∗ (y ∗ x))) ≤ tfor all z ∈ X by (2.12). Hence t is an anti permeable C-value for f .Theorem 3.34. Let f be an anti fuzzy commutative ideal of X. If t ∈ [0, 1]is an anti permeable C-value for f , then the lower t-level set L(f ; t) is aC-energetic subset of X.Proof: Let x, y ∈ X be such that x ∗ (y ∗ (y ∗ x)) ∈ L(f ; t). Then f(x ∗(y ∗ (y ∗ x))) ≤ t and so min{f((x ∗ y) ∗ z), f(z)} ≤ t by (3.12). It followsthat (x ∗ y) ∗ z ∈ L(f ; t) or z ∈ L(f ; t). Hence {z, (x ∗ y) ∗ z} ∩L(f ; t) 6= ∅,and therefore L(f ; t) is a C-energetic subset of X.Corollary 3.35. Let f be an anti fuzzy commutative ideal of X. Ift ∈ [0, 1] is an anti permeable C-value for f , then U∗(f ; t) is empty ora commutative ideal of X.Theorem 3.36. For a fuzzy set f in X, if there exists a subset K of[0, 1] such that {U∗(f ; t), L(f ; t)} is a partition of X and U∗(f ; t) is acommutative ideal of X for all t ∈ K, then t is an anti permeable C-valuefor f .Proof: Assume that f(x ∗ (y ∗ (y ∗ x))) ≤ t for any x, y ∈ X. Thenx ∗ (y ∗ (y ∗ x)) ∈ L(f ; t),and so {z, (x∗y)∗z}∩L(f ; t) 6= ∅ for all z ∈ X since L(f ; t) is a C-energeticsubset of X. It follows that f(z) ≤ t or f((x ∗ y) ∗ z) ≤ t, and so thatmin{f((x ∗ y) ∗ z), f(z)} ≤ t.Therefore t is an anti permeable C-value for f .Commutative Energetic Subsets of BCK-Algebras 63AcknowledgementThe authors wish to thank the anonymous reviewer(s) for the valuablesuggestions.References[1] S. M. Hong and Y. B. Jun, Anti fuzzy ideals in BCK-algebras, KyungpookMath. J. 38 (1998), pp. 145–150.[2] Y. Huang, BCI-algebra, Science Press, Beijing, 2006.[3] Y. B. Jun, S. S. Ahn and E. H. Roh, Energetic subsets and permeable valueswith applications in BCK/BCI-algebras, Appl. Math. Sci. 7 (2013), no.89, pp. 4425–4438.[4] Y. B. Jun and E. H. Roh, Fuzzy commutative ideals of BCK-algebras, FuzzySets and Systems 64 (1994), pp. 401–405.[5] J. Meng, Commutative ideals in BCK-algebras, Pure Appl. Math. (inChina) 9 (1991), pp. 49–53.[6] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoonsa Co. Seoul, Korea(1994).[7] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), pp. 338–353.Department of Mathematics Education (and RINS)Gyeongsang National University, Jinju 52828, Koreae-mail: skywine@gmail.comDepartment of Mathematics Education,Chinju National University of Education, Jinju 660-756, Koreae-mail: idealmath@gmail.comDepartment of Mathematics,Jeju National University, Jeju 690-756, Koreae-mail: szsong@jejunu.ac.kr

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Commutative Energetic Subsets of BCK-Algebras

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Commutative Energetic Subsets of BCK-Algebras

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