#### Abstract

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.

It is easy to see that is a KU-algebra.

*Definition 5 (see [6]). *A subset of a KU-algebra is called subalgebra of if , whenever .

*Definition 6 (see [6]). *A nonempty subset of a KU-algebra is called a KU-ideal of if it satisfies the following conditions:(1),
(2) implies , for all .

*Example 7. *Let in which is defined by Table 2.

Clearly is a KU-algebra. It is easy to show that and are KU-ideals of .

*Definition 8 (see [6]). *Let and be KU-algebras. A homomorphism is a map satisfying , for all .

Theorem 9 (see [6]). *Let be a homomorphism of a KU-algebra into a KU-algebra , then*(1)*if is the identity in , then is the identity in ;*(2)*if is a KU-subalgebra of , then is a KU-subalgebra of ;*(3)*if is a KU-ideal of , then is a KU-ideal in ;*(4)*if is a KU-subalgebra of , then is a KU-subalgebra algebra of ;*(5)*if is a KU-ideal in , then is a KU-ideal in ;*(6)*if is a homomorphism from KU-algebra to a KU-algebra then is one to one if and only if .*

Proposition 10 (see [8]). *Suppose is a homomorphism of KU-algebras, then:*(1),
(2)if implies .

Proposition 11 (see [8]). *Let and be KU-algebras and let be a homomorphism, then is a KU-ideal of .*

Now we will recall the concept of interval-valued fuzzy sets.

An interval number is , where . Let denote the family of all closed subintervals of , that is, We define the operations “,” “,” “,” “,” and “” in case of two elements in . We consider two elements and in . Then(1) if and only if and ,(2) if and only if and ,(3) if and only if and ,(4), (5). It is obvious that () is a complete lattice with as its least element and as its greatest element. Let where . We define An interval-valued fuzzy set (briefly, IVF-set) on is defined as where , for all . Then the ordinary fuzzy sets and are called a lower fuzzy set and an upper fuzzy set of , respectively. Let , then where .

Jun et al. [12] introduced the concept of cubic sets defined on a nonempty set as objects having the form: which is briefly denoted by , where the functions and .

Denote by family of all cubic sets in .

#### 3. Cubic KU-Ideals of KU-Algebras

In this section, we will introduce a new notion called cubic KU-ideal of KU-algebras and study several properties of it.

*Definition 12. *Let be a KU-algebra. A cubic set in is called cubic KU-subalgebra of if it satisfies the following conditions:(S1) and ,(S2) and , for all .

*Definition 13. *Let be a KU-algebra. A cubic set in is called cubic KU-ideal of if it satisfies the following conditions:(A1) and ,(A2) and , for all .

*Example 14. *Let in which is defined by Table 3.

Clearly is a KU-algebra. Define a cubic set in as follows: By routine calculations it can be seen that the cubic set is a cubic KU-ideal of .

Lemma 15. *Let be a cubic KU-ideal of KU-algebra . If the inequality holds in , then and .*

*Proof. *Assume that the inequality holds in , then and by , if we put then
but
From and , we get . Similarly we can show that . This completes the proof.

Lemma 16. *If is a cubic KU-ideal of KU-algebra and if then and .*

*Proof. *If then . This together with and also , we get
Also
This completes the proof.

Let and be two cubic sets in a KU-algebra , then

Proposition 17. *Let be a family of cubic KU-ideals of a KU-algebra , then is a cubic KU-ideal of .*

*Proof. *Let be a family of cubic KU-ideals of a KU-algebra , then for any ,
Also
This completes the proof.

For any and , let be a cubic set in a KU-algebra , then the set is called the cubic level set of .

Theorem 18. *Let be a cubic subset in then is a cubic KU-ideal of if and only if for all and , the set is either empty or a KU-ideal of .*

*Proof. *Assume that is a cubic KU-ideal of , let and , be such that , and let be such that , then and . By we get
also
Thus, . Now letting , it implies that
also
So that . Hence, is a KU-ideal of .

Conversely, suppose that is a KU-ideal of and let be such that
taking
and taking
we have and , and
It follows that and . This is a contradiction and therefore is cubic KU-ideal of .

Proposition 19. *If is a cubic KU-ideal of KU-algebra , then
*

*Proof. *Taking in and using (ku2) and , we get
Thus, we get and . This completes the proof.

#### 4. Image and Preimage of Cubic KU-Ideals

In this section we will present some results on images and preimages of cubic KU-ideals in KU-algebras.

*Definition 20. *Let be a mapping from a set to a set . If is a cubic subset of , then the cubic subset of is defined by
is said to be the image of under .

Similarly if is a cubic subset of , then the cubic subset in (i.e., the cubic subset defined by and for all ) is called the preimage of under .

Theorem 21. *An onto homomorphic preimage of cubic KU-ideal is also cubic KU-ideal.*

*Proof. *Let be an onto homomorphism of KU-algebras, a cubic KU-ideal of , and the preimage of under , then and for all . Let , then
Now let , then
This completes the proof.

*Definition 22. *A cubic subset of has sup and inf properties if for any subset of , there exist such that and .

Theorem 23. *Let be a homomorphism between KU-algebras and . For every cubic KU-ideal in is cubic KU-ideal of .*

*Proof. *By definition and for all and and . We have to prove that and for all . Let be an onto homomorphism of KU-algebras, a cubic KU-ideal of with sup and inf properties, and the image of under . Since is cubic KU-ideal of , we have
Note that where are the zero of and , respectively. Thus,
which implies that and for any . For any , let , and be such that
Also
Then
Hence, is cubic KU-ideal of .

#### 5. Cartesian Product of Cubic KU-Ideals

In this section we will provide some new definitions on cartesian product of cubic KU-ideals in KU-algebras.

*Definition 24. *Let and be two cubic subsets of KU-algebras and , respectively. Then cartesian product of cubic subsets and is denoted by and is defined as
for all .

*Remark 25. *Let and be KU-algebras. We define on by for every belong to , then clearly is a KU-algebra.

*Definition 26. *A cubic subset of is called a cubic KU-subalgebra of if(CP1) and ,(CP2) ,
(CP3) ,
for all .

*Definition 27. *A cubic subset of is called a cubic KU-ideal of if(CP4) and ,(CP5) ,
(CP6) ,
for all .

Theorem 28. *Let and be two cubic KU-subalgebras of KU-algebras and , respectively. Then is a cubic KU-subalgebra of KU-algebra .*

*Proof. *For any ,
For any . Then
Hence, for all is a cubic KU-subalgebra of KU-algebra .

Theorem 29. *Let and be two cubic KU-ideals of KU-algebras and , respectively. Then is a cubic KU-ideal of KU-algebra .*

*Proof. *For any ,
Now for any ,
Hence, for all is a cubic KU-ideal of KU-algebra .

Lemma 30. *If is a cubic KU-ideal of KU-algebra and if , we have and , for all .*

*Proof. *Let , such that . This together with and also . Consider
This shows that and , for all .

Lemma 31. *If is a cubic KU-ideal of KU-algebra and if holds in , then we have and , for all .*

*Proof. *Let and let holds in , then
Now for and from
we have
and from
we have
This completes the proof.

*Definition 32. *Let be a cubic subset of KU-algebra , and for any and , the set
is called the cubic level set of .

Theorem 33. *Let be a cubic subset of KU-algebra . Then is a cubic KU-subalgebra of KU-algebra if and only if for any and , the set is either empty or a KU-subalgebra of .*

*Proof. *The proof is straightforward.

Theorem 34. *Let *