On Cubic KU-Ideals of KU-Algebras

We introduce the notion of cubic KU-ideals of KU-algebras and several results are presented in this regard. The image, preimage, and cartesian product of cubic KU-ideals of KU-algebras are defined. 1. Introduction BCK-algebras form an important class of logical algebras introduced by Iséki and were extensively investigated by several researchers. The class of all BCK-algebras is quasivariety. Iséki and Tanaka introduced two classes of abstract algebras, BCK-algebras and BCI-algebras [1–3]. In connection with this problem, Komori [4] introduced a notion of BCC-algebras. Prabpayak and Leerawat [5] introduced a new algebraic structure which is called KU-algebra. They gave the concept of homomorphisms of KU-algebras and investigated some related properties in [5, 6]. Zadeh [7] introduced the notion of fuzzy sets. At present this concept has been applied to many mathematical branches, such as groups, functional analysis, probability theory, and topology. Mostafa et al. [8] introduced the notion of fuzzy KU-ideals of KU-algebras and then they investigated several basic properties which are related to fuzzy KU-ideals, also see [9]. Abdullah et al. [10, 11] introduced the concept of direct product of intuitionistic fuzzy sets in BCK-algebras. Jun et al. [12] introduced the notion of cubic subalgebras/ideals in BCK/BCI-algebras, and then they investigated several properties. They discussed the relationship between a cubic subalgebra and a cubic ideal. Also, they provided characterizations of a cubic subalgebra/ideal and considered a method to make a new cubic subalgebra from an old one, also see [13–17]. In this paper, we introduce the notion of cubic KU-ideals of KU-algebras and then we study the homomorphic image and inverse image of cubic KU-ideals. 2. Preliminaries In this section we will recall some concepts related to KU-algebra and cubic sets. Definition 1 (see [5]). By a KU-algebra we mean an algebra of type with a single binary operation that satisfies the following identities: for any ,(ku1) , (ku2) , (ku3) , (ku4) implies . In what follows, let denote a -algebra unless otherwise specified. For brevity we also call a -algebra. In we can define a binary relation by: if and only if . Definition 2 (see [5]). is a KU-algebra if and only if it satisfies(ku5) , (ku6) , (ku7) implies ,(ku8) if and only if . Definition 3 (see [8]). In a KU-algebra, the following identities are true:(1) , (2) , (3) imply ,(4) , for all ,(5) . Example 4 (see [8]). Let in which is defined by Table 1. Table 1 It is easy to see that is a KU-algebra. Definition 5 (see [6]). A subset