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Journal of Applied Mathematics
Volume 2012, Article ID 938910, 12 pages
http://dx.doi.org/10.1155/2012/938910
Research Article

Generalizations of -Subalgebras in BCK/BCI-Algebras Based on Point -Structures

1Department of Mathematics Education (and RINS), Gyengsang National University, Jinju 660-701, Republic of Korea
2Department of Mathematics Education, Hannam University, Daejeon 306-791, Republic of Korea
3Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea

Received 13 June 2012; Accepted 31 July 2012

Academic Editor: Nicola Mastronardi

Copyright © 2012 Young Bae Jun 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

The aim of this article is to obtain more general forms than the papers of (Jun et al. (2010); Jun et al. (in press)). The notions of -subalgebras of types , and are introduced, and the concepts of -support and -support are also introduced. Several related properties are investigated. Characterizations of -subalgebra of type are discussed, and conditions for an -subalgebra of type to be an -subalgebra of type are considered.

1. Introduction

A (crisp) set in a universe can be defined in the form of its characteristic function yielding the value for elements belonging to the set and the value for elements excluded from the set . So far most of the generalizations of the crisp set have been conducted on the unit interval , and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fits the crisp point into the interval . Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. [1] introduced a new function which is called negative-valued function and constructed -structures. They applied -structures to BCK/BCI-algebras and discussed -subalgebras and -ideals in BCK/BCI-algebras. Jun et al. [2] considered closed ideals in BCH-algebras based on -structures. To obtain more general form of an -subalgebra in BCK/BCI-algebras, Jun et al. [3] defined the notions of -subalgebras of types , and and investigated related properties. They also gave conditions for an -structure to be an -subalgebra of type . Jun et al. provided a characterization of an -subalgebra of type (see [3, 4]).

In this paper, we try to have more general form of the papers [3, 4]. We introduce the notions of -subalgebras of types , and . We also introduce the concepts of -support and -support and investigate several properties. We discuss characterizations of -subalgebra of type . We consider conditions for an -subalgebra of type to be an -subalgebra of type . The important achievement of the study of -subalgebras of types , and is that the notions of -subalgebras of types , and are a special case of -subalgebras of types , and , and thus so many results in the papers [3, 4] are corollaries of our results obtained in this paper.

2. Preliminaries

Let be the class of all algebras with type . By a BCI-algebra, we mean a system in which the following axioms hold:(i), (ii), (iii), (iv), for all . If a BCI-algebra satisfies for all , then we say that is a BCK-algebra. We can define a partial ordering by

In a BCK/BCI-algebra , the following hold:(a1) ,(a2) ,for all .

A nonempty subset of a BCK/BCI-algebras is called a subalgebra of if for all . For our convenience, the empty set is regarded as a subalgebra of .

We refer the reader to the books [5, 6] for further information regarding BCK/BCI-algebras.

For any family of real numbers, we define

Denote by the collection of functions from a set to . We say that an element of is a negative-valued function from to (briefly, on ). By an , we mean an ordered pair of and an -function on . In what follows, let denote a BCK/BCI-algebras and an -function on unless otherwise specified.

Definition 2.1 (see [1]). By a subalgebra of based on -function (briefly, -subalgebra of ), we mean an -structure in which satisfies the following assertion:
For any -structure and , the set is called a closed of , and the set is called an open of .
Using the similar method to the transfer principle in fuzzy theory (see [7, 8]), Jun et al. [2] considered transfer principle in -structures as follows.

Theorem 2.2 (see [2]; -transfer principle). An -structure satisfies the property if and only if for all ,

Lemma 2.3 (see [1]). An -structure is an -subalgebra of if and only if every open -support of is a subalgebra of for all .

3. General Form of -Subalgebras with Type

In what follows, let and denote arbitrary elements of and , respectively, unless otherwise specified.

Let be an -structure in which is given by In this case, is denoted by , and we call a point -structure. For any -structure , we say that a point -structure is an -subset (resp., -subset) of if (resp., ). If a point -structure is an -subset of or an -subset of , we say is an -subset of . We say that a point -structure is an -subset of if . Clearly, every -subset with is an -subset. Note that if with , then every -subset is an -subset.

Definition 3.1. An -structure is called an -subalgebra of type (i) (resp., and ) if whenever two point -structures and are -subsets of then the point -structure is an -subset (resp., -subset and -subset) of .(ii) (resp., and ) if whenever two point -structures and are -subsets of then the point -structure is an -subset (resp., -subset and -subset) of .

Definition 3.2. An -structure is called an -subalgebra of type (resp., ()) if whenever two point -structures and are -subsets (resp., -subsets) of then the point -structure is an -subset of .

Example 3.3. Consider a -algebra with the following Cayley table: Let be an -structure in which is defined by It is routine to verify that is an -subalgebra of type .
Note that if with , then every -subalgebra of type is an -subalgebra of type , but the converse is not true as seen in the following example.

Example 3.4. The -subalgebra of type in Example 3.3 is not of type since and are -subsets of , but is not an -subset of .

Theorem 3.5. Every -subalgebra of type is of type .

Proof. Straightforward.

Taking in Theorem 3.5 induces the following corollary.

Corollary 3.6. Every -subalgebra of type is of type .

The converse of Theorem 3.5 is not true as seen in the following example.

Example 3.7. Consider the -subalgebra of type which is given in Example 3.3. Then is not an -subalgebra of type since and are -subsets of , but is not an -subset of .

Definition 3.8. An -structure is called an -subalgebra of type if whenever two point -structure and are -subsets of then the point -structure is an -subset of .

Theorem 3.9. Every -subalgebra of type is of type .

Proof. Straightforward.

Taking in Theorem 3.9 induces the following corollary.

Corollary 3.10. Every -subalgebra of type is of type .

The converse of Theorem 3.9 is not true as seen in the following example.

Example 3.11. Consider the -subalgebra of type which is given in Example 3.3. Then and are -subsets of , but is not an -subset of for since .

We consider a characterization of an -subalgebra of type .

Theorem 3.12. An -structure is an -subalgebra of type if and only if it satisfies

Proof. Let be an -structure of type . Assume that (3.6) is not valid. Then there exists such that If , then . Hence for some . It follows that point -structures and are -subsets of , but the point -structure is not an -subset of . Moreover, and so is not an -subset of . Consequently, is not an -subset of . This is a contradiction. If , then and . Thus and are -subsets of , but is not an -subset of . Also, that is, is not an -subset of . Hence is not an -subset of , a contradiction. Therefore (3.6) is valid.
Conversely, suppose that (3.6) is valid. Let and be such that two point -structures and are -subsets of . Then Assume that or . Then , and so is an -subset of . Now suppose that and . Then , and thus that is, is an -subset of . Therefore is an -subset of and consequently is an -subalgebra of type .

Corollary 3.13 (see [3]). An -structure is an -subalgebra of type if and only if it satisfies

Proof. It follows from taking in Theorem 3.12.

We provide conditions for an -structure to be an -subalgebra of type .

Theorem 3.14. Let be a subalgebra of and let be an -structure such that(a), (b). Then is an -subalgebra of type .

Proof. Let and be such that two point -structures and are -subsets of . Then and . Thus because if it is impossible, then or . Thus or , and so or . This is a contradiction. Hence . If , then and so the point -structure is an -subset of . If , then and so the point -structure is an -subset of . Therefore the point -structure is an -subset of . This shows that is an -subalgebra of type .

Taking in Theorem 3.14, we have the following corollary.

Corollary 3.15 (see [3]). Let be a subalgebra of and let be an -structure such that(a), (b). Then is an -subalgebra of type .

Theorem 3.16. Let be an -subalgebra of type . If is not constant on the open -support of , then for some . In particular, .

Proof. Assume that for all . Since is not constant on the open -support of , there exists such that . Then either or . For the case , choose such that . Then the point -structure is an -subset of . Since is an -subset of . It follows from (a1) that the point -structure is an -subset of . But, implies that the point -structure is not an -subset of . Also, implies that the point -structure is not an -subset of . This is a contradiction. Assume that and take such that . Then is an -subset of . Since is not an -subset of . Since is not an -subset of . Hence is not an -subset of , which is a contradiction. Therefore for some . We now prove that . Assume that . Note that there exists such that and so . Choose such that . Then , and thus the point -structure is an -subset of . Now we have and . Hence is not an -subset of . This is a contradiction, and therefore .

Corollary 3.17 (see [3]). Let be an -subalgebra of type . If is not constant on the open -support of , then for some . In particular, .

Theorem 3.18. An -structure is an -subalgebra of type if and only if for every the nonempty closed -support of is a subalgebra of .

Proof. Assume that is an -subalgebra of type and let be such that . Let . Then and . It follows from Theorem 3.12 that so that . Therefore is a subalgebra of .
Conversely, let be an -structure such that the nonempty closed -support of is a subalgebra of for all . If there exist such that , then we can take such that Thus and . Since is a subalgebra of , it follows that so that . This is a contradiction, and therefore for all . Using Theorem 3.12, we conclude that is an -subalgebra of type .

Taking in Theorem 3.18, we have the following corollary.

Corollary 3.19 (see [4]). An -structure is an -subalgebra of type if and only if for every the nonempty closed -support of is a subalgebra of .

Theorem 3.20. Let be a subalgebra of . For any , there exists an -subalgebra of type for which is represented by the closed -support of .

Proof. Let be an -structure in which is given by for all where . Assume that for some . Since the cardinality of the image of is , we have and . Since , it follows that so that . Since is a subalgebra of , we obtain and so . This is a contradiction. Therefore for all . Using Theorem 3.12, we conclude that is an -subalgebra of type . Obviously, is represented by the closed -support of .

Corollary 3.21 (see [4]). Let be a subalgebra of . For any , there exists an -subalgebra of type for which is represented by the closed -support of .

Proof. It follows from taking in Theorem 3.20.

Note that every -subalgebra of type is an -subalgebra of type , but the converse is not true in general (see Example 3.7). Now, we give a condition for an -subalgebra of type to be an -subalgebra of type .

Theorem 3.22. Let be an -subalgebra of type such that   for all . Then is an -subalgebra of type .

Proof. Let and be such that and are -subsets of . Then and . It follows from Theorem 3.12 and the hypothesis that so that is an -subset of . Therefore is an -subalgebra of type .

Corollary 3.23 (see [4]). Let be an -structure of type such that for all . Then is an -subalgebra of type .

Proof. It follows from taking in Theorem 3.22.

Theorem 3.24. Let be a family of -subalgebras of type . Then is an -subalgebra of type , where is an -function on given by for all .

Proof. Let and be such that and are -subsets of . Assume that is not an -subset of . Then is neither an -subset nor an -subset of . Hence and which imply that Let is an -subset of and is an is not an . Then and . If , then is an -subset of for all , that is, for all . Thus . This is a contradiction. Hence , and so for every , we have and . It follows that . Since is an -subset of , we have for all . Similarly, for all . Next suppose that . Taking , we know that and are -subsets of , but is not an -subset of . This contradicts that is an -subalgebra of type . Hence for all , and so which contradicts (3.22). Therefore is an -subset of and consequently is an -subalgebra of type .

For any -structure and , the and the of related to are defined to be the sets (see [4]) respectively. Note that the -support is the union of the closed support and the -support, that is, The -support and the -support of related to are defined to be the sets respectively. Clearly, for all .

Theorem 3.25. An -structure is an -subalgebra of type if and only if the -support of related to is a subalgebra of for all .

Proof. Suppose that is an -subalgebra of type . Let for . Then and are -subsets of . Hence or , and or . Then we consider the following four cases: (c1) and ,(c2) and ,(c3) and ,(c4) and . Combining (3.6) and (c1), we have . If , then and so is an -subset of . Hence . If , then and so , that is, is an -subset of . Therefore . For the case (c2), assume that . Then and so . Thus is an -subset of . If , then and thus or . Consequently, . For the case (c3), it is similar to the case (c2). Finally, for the case (c4), if , then . Hence which implies that . If , then . Therefore that is, , which means that is an -subset of . Consequently, the -support of related to is a subalgebra of for all .
Conversely, let be an -structure for which the -support of related to is a subalgebra of for all . Assume that there exist such that . Then for some . It follows that but . Also, , that is, . Thus which is a contradiction. Therefore for all . Using Theorem 3.12, we conclude that is an -subalgebra of type .

If we take in Theorem 3.25, we have the following corollary.

Corollary 3.26 (see [4]). An -structure is an -subalgebra of type if and only if the -support of related to is a subalgebra of for all .

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