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ISRN Algebra
Volume 2013 (2013), Article ID 935905, 10 pages
http://dx.doi.org/10.1155/2013/935905
Research Article

On Cubic KU-Ideals of KU-Algebras

1Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
2Department of Mathematics, Ain Shams University, Roxy, Cairo, Egypt
3Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

Received 30 August 2012; Accepted 19 October 2012

Academic Editors: K. P. Shum and A. Vourdas

Copyright © 2013 Naveed Yaqoob et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [13]. 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 [1317].

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.

tab1
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.

tab2
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.

tab3
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 be a cubic subset of KU-algebra . Then is a cubic KU-ideal of KU-algebra if and only if for any and , the set is either empty or a KU-ideal of .

Proof. Let be a cubic KU-ideal of KU-algebra . For any and , define the sets Since , let . This implies and . So