Research Article  Open Access
Naveed Yaqoob, Samy M. Mostafa, Moin A. Ansari, "On Cubic KUIdeals of KUAlgebras", International Scholarly Research Notices, vol. 2013, Article ID 935905, 10 pages, 2013. https://doi.org/10.1155/2013/935905
On Cubic KUIdeals of KUAlgebras
Abstract
We introduce the notion of cubic KUideals of KUalgebras and several results are presented in this regard. The image, preimage, and cartesian product of cubic KUideals of KUalgebras are defined.
1. Introduction
BCKalgebras form an important class of logical algebras introduced by Iséki and were extensively investigated by several researchers. The class of all BCKalgebras is quasivariety. Iséki and Tanaka introduced two classes of abstract algebras, BCKalgebras and BCIalgebras [1–3]. In connection with this problem, Komori [4] introduced a notion of BCCalgebras.
Prabpayak and Leerawat [5] introduced a new algebraic structure which is called KUalgebra. They gave the concept of homomorphisms of KUalgebras 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 KUideals of KUalgebras and then they investigated several basic properties which are related to fuzzy KUideals, also see [9]. Abdullah et al. [10, 11] introduced the concept of direct product of intuitionistic fuzzy sets in BCKalgebras.
Jun et al. [12] introduced the notion of cubic subalgebras/ideals in BCK/BCIalgebras, 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 KUideals of KUalgebras and then we study the homomorphic image and inverse image of cubic KUideals.
2. Preliminaries
In this section we will recall some concepts related to KUalgebra and cubic sets.
Definition 1 (see [5]). By a KUalgebra 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 KUalgebra if and only if it satisfies(ku5), (ku6), (ku7) implies ,(ku8) if and only if .
Definition 3 (see [8]). In a KUalgebra, 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 KUalgebra.
Definition 5 (see [6]). A subset of a KUalgebra is called subalgebra of if , whenever .
Definition 6 (see [6]). A nonempty subset of a KUalgebra is called a KUideal 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 KUalgebra. It is easy to show that and are KUideals of .
Definition 8 (see [6]). Let and be KUalgebras. A homomorphism is a map satisfying , for all .
Theorem 9 (see [6]). Let be a homomorphism of a KUalgebra into a KUalgebra , then(1)if is the identity in , then is the identity in ;(2)if is a KUsubalgebra of , then is a KUsubalgebra of ;(3)if is a KUideal of , then is a KUideal in ;(4)if is a KUsubalgebra of , then is a KUsubalgebra algebra of ;(5)if is a KUideal in , then is a KUideal in ;(6)if is a homomorphism from KUalgebra to a KUalgebra then is one to one if and only if .
Proposition 10 (see [8]). Suppose is a homomorphism of KUalgebras, then:(1), (2)if implies .
Proposition 11 (see [8]). Let and be KUalgebras and let be a homomorphism, then is a KUideal of .
Now we will recall the concept of intervalvalued 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 intervalvalued fuzzy set (briefly, IVFset) 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 KUIdeals of KUAlgebras
In this section, we will introduce a new notion called cubic KUideal of KUalgebras and study several properties of it.
Definition 12. Let be a KUalgebra. A cubic set in is called cubic KUsubalgebra of if it satisfies the following conditions:(S1) and ,(S2) and , for all .
Definition 13. Let be a KUalgebra. A cubic set in is called cubic KUideal 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 KUalgebra. Define a cubic set in as follows: By routine calculations it can be seen that the cubic set is a cubic KUideal of .
Lemma 15. Let be a cubic KUideal of KUalgebra . 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 KUideal of KUalgebra 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 KUalgebra , then
Proposition 17. Let be a family of cubic KUideals of a KUalgebra , then is a cubic KUideal of .
Proof. Let be a family of cubic KUideals of a KUalgebra , then for any , Also This completes the proof.
For any and , let be a cubic set in a KUalgebra , then the set is called the cubic level set of .
Theorem 18. Let be a cubic subset in then is a cubic KUideal of if and only if for all and , the set is either empty or a KUideal of .
Proof. Assume that is a cubic KUideal 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 KUideal of .
Conversely, suppose that is a KUideal 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 KUideal of .
Proposition 19. If is a cubic KUideal of KUalgebra , then
Proof. Taking in and using (ku2) and , we get Thus, we get and . This completes the proof.
4. Image and Preimage of Cubic KUIdeals
In this section we will present some results on images and preimages of cubic KUideals in KUalgebras.
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 KUideal is also cubic KUideal.
Proof. Let be an onto homomorphism of KUalgebras, a cubic KUideal 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 KUalgebras and . For every cubic KUideal in is cubic KUideal of .
Proof. By definition and for all and and . We have to prove that and for all . Let be an onto homomorphism of KUalgebras, a cubic KUideal of with sup and inf properties, and the image of under . Since is cubic KUideal 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 KUideal of .
5. Cartesian Product of Cubic KUIdeals
In this section we will provide some new definitions on cartesian product of cubic KUideals in KUalgebras.
Definition 24. Let and be two cubic subsets of KUalgebras and , respectively. Then cartesian product of cubic subsets and is denoted by and is defined as for all .
Remark 25. Let and be KUalgebras. We define on by for every belong to , then clearly is a KUalgebra.
Definition 26. A cubic subset of is called a cubic KUsubalgebra of if(CP1) and ,(CP2) , (CP3) , for all .
Definition 27. A cubic subset of is called a cubic KUideal of if(CP4) and ,(CP5) , (CP6) , for all .
Theorem 28. Let and be two cubic KUsubalgebras of KUalgebras and , respectively. Then is a cubic KUsubalgebra of KUalgebra .
Proof. For any , For any . Then Hence, for all is a cubic KUsubalgebra of KUalgebra .
Theorem 29. Let and be two cubic KUideals of KUalgebras and , respectively. Then is a cubic KUideal of KUalgebra .
Proof. For any , Now for any , Hence, for all is a cubic KUideal of KUalgebra .
Lemma 30. If is a cubic KUideal of KUalgebra 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 KUideal of KUalgebra and if holds in , then we have and , for all .
Proof. Let and let holds in , then Now for and from we have