# Paper

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Mathware & Soft Computing 13 (2006) 37-58 Dual Commutative Hyper K-Ideals of Type 1 in Hyper K-algebras of Order 3 L. Torkzadeh1 and M.M. Zahedi2 1 Dept. Math., Islamic Azad Univ. of Kerman, Kerman, Iran Dept. Math., Shahid Bahonar Univ. of Kerman, Kerman, Iran ltorkzadeh@yahoo.com zahedi mm@mail.uk.ac.ir, http://math.uk.ac.ir/∼zahedi Abstract In this note we classify the bounded hyper K-algebras of order 3, which have D1 = {1}, D2 = {1, 2} and D3 = {0, 1} as a dual commutative hyper K-ideal of type 1. In this regard we show that there are such non-isomorphic bounded hyper K-algebras. 2000 Mathematics Subject Classiﬁcation. 03B47, 06F35, 03G25 Key words and phrases: hyper K-algebra, dual commutative hyper K-ideal. 2 1 Introduction The hyperalgebraic structure theory was introduced by F. Marty [5] in 1934. Imai and Iseki [3] in 1966 introduced the notion of a BCK-algebra. Borzooei, Jun and Zahedi et.al. [2,8] applied the hyperstructure to BCK-algebras and introduced the concept of hyper K-algebra which is a generalization of BCK-algebra. In [7]we deﬁned the notions of dual commutative hyper K-ideals of type 1 and type 2 (Brieﬂy DCHKI − T 1, T 2). Now we follow it and determine all bounded hyper K-algebras of order 3 which have DCIHKI − T 1. 2 Preliminaries Deﬁnition 2.1. [2] Let H be a nonempty set and ” ◦ ” be a hyperoperation on H, that is ” ◦ ” is a function from H × H to P ∗ (H) = P(H)\{∅}. Then H is called a hyper K-algebra if it contains a constant ”0” and satisﬁes the following axioms: (HK1) (x ◦ z) ◦ (y ◦ z) < x ◦ y (HK2) (x ◦ y) ◦ z = (x ◦ z) ◦ y (HK3) x < x 37 38 L. Torkzadeh & M.M. Zahedi (HK4) x < y, y < x ⇒ x = y (HK5) 0 < x, for all x, y, z ∈ H, where x < y is deﬁned by 0 ∈ x ◦ y and for every A, B ⊆ H, A < B is deﬁned by ∃a ∈ A, ∃b ∈ B such that a < b. Note that if A, B ⊆ H, then by A ◦ B we mean the subset a ◦ b of