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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 910450, 13 pages
http://dx.doi.org/10.1155/2012/910450
Research Article

Ideal Theory in BCK/BCI-Algebras Based on Soft Sets and -Structures

1Department of Mathematics Education and RINS, Gyeongsang 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 14 December 2011; Accepted 1 February 2012

Academic Editor: Lee-Chae Jang

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

Based on soft sets and -structures, the notion of (closed) -ideal over a BCI-algebra is introduced, and related properties are investigated. Relations between -BCI-algebras and -ideals are established. Characterizations of a (closed) -ideal over a BCI-algebra are provided. Conditions for an -ideal to be an -BCI-algebra are considered.

1. Introduction

To solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as the probability theory, the theory of (intuitionistic) fuzzy sets, the theory of vague sets, the theory of interval mathematics, and the theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [1]. Maji et al. [2] and Molodtsov [1] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [2] described the application of soft set theory to a decision-making problem. Maji et al. [3] also studied several operations on the theory of soft sets. Chen et al. [4] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [5]. Roy and Maji [6] presented some results on an application of fuzzy soft sets in decision making problem. Feng et al. [7] provided a framework to combine fuzzy sets, rough sets, and soft sets all together, which gives rise to several interesting new concepts such as rough soft sets, soft rough sets, and soft rough fuzzy sets. Feng et al. [8] gave deeper insights into decision making based on fuzzy soft sets. Feng et al. [9] initiated the notion of soft rough sets, which can be seen as a generalized rough set model based on soft sets. Aygünoğlu et al. [10] introduced the notion of fuzzy soft group and studied its properties. Ali et al. [11] discussed new operations in soft set theory. Jun [12] applied the notion of soft set to BCK/BCI-algebras, and Jun et al. [13] considered applications of soft set theory in the ideals of -algebras. Han et al. [14] discussed the fuzzy set theory of fated filters in -algebras based on fuzzy points.

A (crisp) set in a universe can be defined in the form of its characteristic function yielding the value 1 for elements belonging to the set and the value 0 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. [15] 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. [16] considered closed ideals in BCH-algebras based on -structures. Jun et al. [17] introduced the notion of -soft sets which are a soft set based on -structures, and then they applied it to both a decision-making problem and a BCK/BCI-algebra.

In this paper, we introduce the notion of (closed) -ideal over a BCI-algebra based on soft sets and -structures and investigate related properties. We establish relations between -BCI-algebras and -ideals. We also provide characterizations of a (closed) -ideal over a BCI-algebra and consider conditions for an -ideal to be an -BCI-algebra.

2. Preliminaries

Let be the class of all algebras of type . An element is called a BCI-algebra if it satisfies the following axioms:(a1), (a2), (a3), (a4)

for all . If a BCI-algebra satisfies(a5) for all ,

then we say that is a BCK-algebra.

We can define a partial ordering  ≤  on by In a BCK/BCI-algebra , the following hold:(b1), (b2), (b3), (b4)

for all .

A nonempty subset of a BCK/BCI-algebra is called a subalgebra of if for all . A subset of a BCK/BCI-algebra is called an ideal of if it satisfies(c1), (c2).

We refer the reader to the books [18, 19] for further information regarding BCK/BCI-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 .

Definition 2.1 (see [15]). By a subalgebra of a BCK/BCI-algebra based on -function (briefly, -subalgebra of ) are means an -structure in which satisfies the following assertion:

Definition 2.2 (see [15]). By an ideal of a BCK/BCI-algebra based on -function (briefly, -ideal of ) are means an -structure in which satisfies the following assertion:

Definition 2.3 (see [15]). Let be a BCI-algebra. An -ideal is said to be closed if it is also an -subalgebra of .

3. -SoftBCK/BCI-Algebras and -Soft Ideals

In what follows let denote a set of attributes unless otherwise specified. We will use the terminology “soft machine” which means that it produces a BCK/BCI-algebra.

Definition 3.1 (see [17]). Let be an initial universe set and a set of attributes. By an -soft set over we mean a pair where and is a mapping from to ; that is, for each is an -function on .

Denote by the collection of all -soft sets over with attributes from , and we call it an -soft class.

Definition 3.2 (see [17]). Let and be -soft sets in . Then is called an -soft subset of , denoted by , if it satisfies(i), (ii).

Definition 3.3 (see [17]). Let be an -soft set over a BCK/BCI-algebra where is a subset of . If there exists an attribute for which the -structure is an -subalgebra of , then we say that is an -soft -algebra related to the attribute (briefly, -soft -algebra). If is an -soft BCK/BCI-algebra for all , we say that is an -soft BCK/BCI-algebra.

Definition 3.4. Let be an -soft set over a BCK/BCI-algebra where is a subset of . If there exists an attribute for which the -structure is an -ideal of , then we say that is an -soft ideal of related to the attribute (briefly, -soft ideal). If is an -soft ideal of for all , we say that is an -soft ideal of .

Example 3.5. Let be a universe, and consider a soft machine which produces the following products: Then is a BCK-algebra under the soft machine . Consider a set of attributes and let be an -soft set over the BCK-algebra with the tabular representation which is given by Table 1. Then , and are -soft ideals over . But is not an -soft ideal over since In general, we know that horses like carrots best of all. In the above example, we know that is not an -soft ideal over based on attribute “horse.” This means that if a horse like a carrot better than the others, then cannot be an -soft ideal over .

tab1
Table 1: Tabular representation of .

Obviously, every -soft ideal is an -soft BCK-algebra in a BCK-algebra, but the converse is not true as seen in the following example.

Example 3.6. Let be a universe, and consider a soft machine which produces the following products: Then is a BCK-algebra under the soft machine . Consider a set of parameters Let be an -soft set over the -algebra with the tabular representation which is given by Table 2. It is easy to verify that is an -soft -algebra. But it is not an -soft ideal over because is not an -soft ideal since
We discuss characterizations of an -soft ideal.

tab2
Table 2: Tabular representation of .

Theorem 3.7. For a BCK/BCI-algebra , let be an -soft set in such that for all and . Then is an -soft ideal of if and only if the following assertion is valid:

Proof. Suppose that is an -soft ideal of . Let and be such that . Then , and so It follows that .
Conversely, assume that (3.8) holds. Note that for all . It follows from (3.8) that for all and . Hence is an -soft ideal of for all , and so is an -soft ideal of .

Lemma 3.8 (see [17]). Every -soft BCK/BCI-algebra over a BCK/BCI-algebra satisfies the following inequality:

Theorem 3.9. Let be an -BCK/BCI-algebra of a BCK/BCI-algebra . Then is an -soft ideal of if and only if it satisfies (3.8).

Proof. Necessity is by Theorem 3.7. Conversely, assume that assertion (3.8) is valid. Since for all , it follows that for all and . Combining this and Lemma 3.8, we know that is an -soft ideal of .

Proposition 3.10. Every -soft ideal of a BCI-algebra satisfies the following inequality:

Proof. Let be an -soft ideal of a -algebra . Then for all and .

The following example shows that there exists an attribute such that an -soft ideal of a BCI-algebra may not be an -soft BCI-algebra.

Example 3.11. Let be a universe where is the set of all rational numbers. Let be a soft machine which is established by Then is a BCI-algebra under the soft machine . Let be an -soft set in which is defined by Then is an -soft ideal over , but it is not an -soft BCI-algebra over since

Definition 3.12. Let be an -soft set over a BCI-algebra , where is a subset of . If there exists an attribute for which the -structure is a closed -ideal of , then we say that is a closed -soft ideal over related to the attribute (briefly, closed -soft ideal). If is a closed -soft ideal over for all , we say that is a closed -soft ideal over .

Example 3.13. Suppose there are five colors in the universe , that is, Let be a soft machine to mix two colors according to order in such a way that we have the following results: Then is a BCI-algebra under the soft machine . Consider a set of attributes and let be an -soft set over with the tabular representation which is given by Table 3. Then is a closed -soft ideal over .

tab3
Table 3: Tabular representation of .

Theorem 3.14. A closed -soft ideal of a BCI-algebra related to an attribute is an -soft BCI-algebra over related to the same attribute.

Proof. Let be a closed -soft ideal over a BCI-algebra related to an attribute . Then for all . It follows that for all . Hence is an -soft BCI-algebra over .

We provide a characterization of a closed -soft ideal.

Theorem 3.15. For an -soft ideal over a BCI-algebra , the following are equivalent:(1) is closed,(2) is an -soft BCI-algebra over .

Proof. (1) (2). It follows from Theorem 3.14.
(2) (1). Suppose that is an -soft BCI-algebra over . Then for all and , and so Thus is a closed -soft ideal over for all , and therefore is a closed -soft ideal over .

We consider a condition for an -soft ideal to be closed.

Theorem 3.16. Given an attribute , if an -soft ideal over a BCI-algebra satisfies the following assertion: then is a closed -soft ideal over .

Proof. For all , we have . Hence Thus is an -soft BCI-algebra over . It follows from Theorem 3.15 that is a closed -soft ideal over .

Corollary 3.17. If an -soft ideal over a BCI-algebra satisfies the following assertion: then is a closed -soft ideal over .

Definition 3.18 (see [17]). Let and be two -soft sets in . The union of and is defined to be the -soft set in satisfying the following conditions:(i), (ii) for all , In this case, we write .

Lemma 3.19 (see [15]). If and are -ideals of a BCK/BCI -algebra , then the union of and is an -ideal of .

Theorem 3.20. If and are -soft ideals over a BCK/BCI-algebra , then the union of and is an -soft ideal over .

Proof. Let be the union of and . Then . For any , if (resp., ), then (resp., ) is an -ideal of . If , then is an -ideal of for all by Lemma 3.19. Therefore is an -soft ideal over a BCK/BCI-algebra .

Definition 3.21 (see [17]). Let and be two -soft sets in . The intersection of and is the -soft set in , where and for every ,

In this case, we write .

Theorem 3.22. Let and be -soft ideals over a BCK/BCI-algebra . If and are disjoint, then the intersection of and is an -soft ideal over .

Proof. Let be the intersection of and . Then . Since , if , then either or . If , then is an -ideal of . If , then is an -ideal of . Hence is an -soft ideal over a -algebra .

The following example shows that Theorem 3.22 is not valid if and are not disjoint.

Example 3.23. Let be an initial universe set that consists of “white,” “blackish,” “reddish,” “green” and “yellow.” Consider a soft machine “’’ which produces the following products: Then is a BCI-algebra under the soft machine . Consider sets of attributes: Let and be -soft sets over with the tabular representations which are given by Tables 4 and 5, respectively. Then and are -soft ideals over . But we have and so is not an -soft ideal over .

tab4
Table 4: Tabular representation of .
tab5
Table 5: Tabular representation of .

4. Conclusions

We have introduced the notion of (closed) -ideal over a BCI-algebra based on soft sets and -structures. We have investigated several properties and established relations between --algebras and -ideals. We have provided characterizations of a (closed) -ideal over a BCI-algebra and considered conditions for an -ideal to be an -BCI-algebra. Based on these results, we will apply the -structure to the other type of ideals/filters in BCK/BCI-algebras, MV-algebras, MTL-algebras, -algebras BL-algebras, and so forth, in the future study.

Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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