Abstract
The notions of -subalgebras and -closed ideals in BCH-algebras are introduced, and the relation between -subalgebras and -closed ideals is considered. Characterizations of -subalgebras and -closed ideals are provided. Using special subsets, -subalgebras and -closed ideals are constructed. A condition for an -subalgebra to be an -closed ideal is discussed. Given an -structure, the greatest -closed ideal which is contained in the -structure is established.
1. Introduction
In [1, 2], Hu and Li introduced the notion of BCH-algebras which are a generalization of BCK/BCI-algebras. Ahmad [3] classified BCH-algebras, and decompositions of BCH-algebras are considered by Dudek and Thomys [4]. Jun et al. [5] discussed the notion of -structures and applied it to BCK/BCI-algebras. In [6], Chaudhry et al. studied closed ideals and filters in BCH-algebras. In this paper, we apply the -structures to the closed ideal theory in BCH-algebras. We introduced the notion of -subalgebras and -closed ideals in BCH-algebras, and investigate the relation between -subalgebras and -closed ideals. We provide characterizations of -subalgebras and -closed ideals. Using special subsets, we construct -subalgebras and -closed ideals. We provide a condition for an -subalgebra to be an -closed ideal. Given an -structure we make the greatest -closed ideal which is contained in
2. Preliminaries
By a BCH-algebra we mean an algebra of type satisfying the following axioms:
(H1)(H2) and imply (H3)for all
In a BCH-algebra the following conditions are valid (see [1, 4]).
(a1)(a2) implies (a3)(a4)A nonempty subset of a BCH-algebra is called a subalgerba of if for all A nonempty subset of a BCH-algebra is called a closed ideal of (see [7]) if it satisfies:
(1)(2)Note that every closed ideal is a subalgebra, but the converse is not true (see [7]). Since every closed ideal is a subalgebra, we know that any closed ideal contains the element Denote by and the set of all subalgebras and closed ideals of respectively.
For any family of real numbers, we define
3. -Closed Ideals of BCH-Algebras
Denote by the collection of functions from a set to We say that an element of is a negative-valued function from to (briefly, -function on ). By an -structure we mean an ordered pair of and an -function on In what follows, let denote a BCH-algebra and an -function on unless otherwise specified.
For any -structure and the set
is called a closed -cut of
Using the similar method to the transfer principle in fuzzy theory (see [8, 9]), we can consider transfer principle in -structures. Let be a subset of and satisfy the following property expressed by a first-order formula:
where and are terms of constructed by variables We note that the subset satisfies the property if, for all elements whenever For the subset we define an -structure which satisfies the following property
We establish a statement without proof, and we call it -transfer principle in -structures.
Theorem 3.1. (-transfer principle) An -structure satisfies the property if and only if for all
Definition 3.2. By an -subalgebra of we mean an -structure in which satisfies:
Theorem 3.3. For an -structure the following are equivalent: (1) is an -subalgerba of ;(2)
Proof. It follows from the -transfer principle.
Definition 3.4. By an -closed ideal of we mean an -structure in which satisfies:
It is clear that if is an -closed ideal or an -subalgebra, then for all
Theorem 3.5. Every -closed ideal is an -subalgebra.
Proof. Let be an -closed ideal of For any we have Hence is an -subalgebra of
The converse of Theorem 3.5 may not be true as seen in the following example.
Example 3.6. Consider a BCH-algebra with the Cayley table which is given in Table 1 (see [7]). Let be an -structure in which is given by It is easy to check that is an -subalgebra of but it is not an -closed ideal of since
In order to discuss the converse of Theorem 3.5 we need to strengthen some conditions. We first consider the following lemma.
Lemma 3.7. Every -subalgebra of satisfies the following inequality:
Proof. For any we get which is the desired result.
Theorem 3.8. If an -subalgerba satisfies then is an -closed ideal of
Proof. It is straightforward by Lemma 3.7.
Proposition 3.9. Let be an -closed ideal of that satisfies the following inequality Then satisfies the inequality
Proof. Using (3.12), (3.6), (a3), (H1), and (H3), we have for all
Using the -transfer principle, we have a characterization of an -closed ideal.
Theorem 3.10. For an -structure the following are equivalent: (1) is an -closed ideal of (2)
Consider two subsets of as follows:
Since and are a closed ideal and a subalgebra, respectively, the following theorems are direct results of the -transfer principle.
Theorem 3.11. Let be an -structure in which is given by for all where Then is an -closed ideal of
Theorem 3.12. Let be an -structure in which is given by for all where Then is an -subalgebra of
We provide a condition for an -subalgebra to be an -closed ideal.
Theorem 3.13. Let be an -subalgebra of in which satisfies Then is an -closed ideal of
Proof. Taking in (3.18) induces for all Using (a1), (3.18), (H1), (H3), and (3.5), we have for all Therefore is an -closed ideal of
For any -structure and any element we consider the set
Then is nonempty subset of
Theorem 3.14. If an -structure is an -closed ideal of then is a closed ideal of for all
Proof. If then which implies from (3.6) that Thus Let be such that and Then and Using (3.6), we have Therefore is a closed ideal of
Proposition 3.15. Let be an -structure such that is a closed ideal of for all Then satisfies the following assertion: for all .
Proof. Let be such that Then and Since is a closed ideal of it follows that so that This completes the proof.
Theorem 3.16. If an -structure satisfies (3.22) and for all then is a closed ideal of for all
Proof. For each let be such that and Then and which imply that It follows from (3.22) that so that If then by assumption. Hence Therefore is a closed ideal of
Theorem 3.17. Given an -structure let be an -structure in which is defined by for all Then is the greatest -closed ideal of such that where is a closed ideal of generated by
Proof. For any let for any Let Then and so for all Hence there exists such that Thus and so for all Consequently On the other hand, if then for any Therefore for all Since is arbitrary, it follows that so that Thus which is a closed ideal of Using Theorem 3.10, we conclude that is an -closed ideal of For any let Then and thus It follows that so that Hence and so Finally, let be an -closed ideal of such that Let If then clearly Assume that Then and so for all It follows that for all so that since is arbitrary. This shows that This completes the proof.
Acknowledgments
The authors wish to thank the anonymous reviewers for their valuable suggestions. The first author was supported by the fund of sabbatical year program (2009), Gyeongsang National University.