International Scholarly Research Notices

International Scholarly Research Notices / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 935905 | https://doi.org/10.1155/2013/935905

Naveed Yaqoob, Samy M. Mostafa, Moin A. Ansari, "On Cubic KU-Ideals of KU-Algebras", International Scholarly Research Notices, vol. 2013, Article ID 935905, 10 pages, 2013. https://doi.org/10.1155/2013/935905

On Cubic KU-Ideals of KU-Algebras

Academic Editor: K. P. Shum
Received30 Aug 2012
Accepted19 Oct 2012
Published26 Feb 2013

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.


·01234

001234
100234
201033
300202
400000

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.


·012345

0012345
1002245
2000145
3000045
4000105
5000000

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.


·01234

001234
100001
203034
301201
400000

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