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The Scientific World Journal
Volume 2015, Article ID 198672, 6 pages
http://dx.doi.org/10.1155/2015/198672
Research Article

Int-Soft (Generalized) Bi-Ideals of Semigroups

1Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea
2Department of Mathematics, Jeju National University, Jeju 690-756, Republic of Korea

Received 14 August 2014; Accepted 23 December 2014

Academic Editor: Guilong Liu

Copyright © 2015 Young Bae Jun and Seok-Zun Song. 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 notions of int-soft semigroups and int-soft left (resp., right) ideals in semigroups are studied in the paper by Song et al. (2014). In this paper, further properties and characterizations of int-soft left (right) ideals are studied, and the notion of int-soft (generalized) bi-ideals is introduced. Relations between int-soft generalized bi-ideals and int-soft semigroups are discussed, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are considered. Given a soft set over U, int-soft (generalized) bi-ideals generated by are established.

1. Introduction

As a new mathematical tool for dealing with uncertainties, the notion of soft sets is introduced by Molodtsov [1]. The author 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. Çağman and Enginoglu [4] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision making method based on FP-soft set theory and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Feng [5] considered the application of soft rough approximations in multicriteria group decision making problems. Aktaş and Çağman [6] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. In general, it is well known that the soft set theory is a good mathematical model to deal with uncertainty. Nevertheless, it is also a new notion worth applying to abstract algebraic structures. So, we can provide the possibility of a new direction of soft sets based on abstract algebraic structures in dealing with uncertainty. In fact, in the aspect of algebraic structures, the soft set theory has been applied to rings, fields, and modules (see [7, 8]), groups (see [6]), semirings (see [9]), (ordered) semigroups (see [10, 11]), and hypervector spaces (see [12]). Song et al. [11] introduced the notion of int-soft semigroups and int-soft left (resp., right) ideals and investigated several properties.

Our aim in this paper is to apply the soft sets to one of abstract algebraic structures, the so-called semigroup. So, we take a semigroup as the parameter set for combining soft sets with semigroups. This paper is a continuation of [11]. We first study further properties and characterizations of int-soft left (right) ideals. We introduce the notion of int-soft (generalized) bi-ideals and provide relations between int-soft generalized bi-ideals and int-soft semigroups. We discuss characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals. Given a soft set over , we establish int-soft (generalized) bi-ideals generated by .

2. Preliminaries

Let be a semigroup. Let and be subsets of . Then the multiplication of and is defined as follows:

A semigroup is said to be regular if for every there exists such that .

A semigroup is said to be left (resp., right) zero if (resp., ) for all .

A semigroup is said to be left (right) simple if it contains no proper left (right) ideal.

A semigroup is said to be simple if it contains no proper two-sided ideal.

A nonempty subset of is called(i)a subsemigroup of if , that is, for all ,(ii)a left (resp., right) ideal of if (resp., ), that is, (resp., ) for all and ,(iii)a two-sided ideal of if it is both a left and a right ideal of ,(iv)a generalized bi-ideal of if ,(v)a bi-ideal of if it is both a semigroup and a generalized bi-ideal of .

A soft set theory is introduced by Molodtsov [1], and Çağman and Enginoğlu [13] provided new definitions and various results on soft set theory.

In what follows, let be an initial universe set and be a set of parameters. Let denote the power set of and .

Definition 1 (see [1, 13]). A soft set over is defined to be the set of ordered pairs where such that if .

The function is called approximate function of the soft set . The subscript in the notation indicates that is the approximate function of .

For a soft set over and a subset of , the -inclusive set of , denoted by , is defined to be the set

For any soft sets and over , we define The soft union of and is defined to be the soft set over in which is defined by The soft intersection of and is defined to be the soft set over in which is defined by The int-soft product of and is defined to be the soft set over in which is a mapping from to given by

3. Int-Soft Ideals

In what follows, we take , as a set of parameters, which is a semigroup unless otherwise specified.

Definition 2 (see [11]). A soft set over is called an int-soft semigroup over if it satisfies

Definition 3 (see [11]). A soft set over is called an int-soft left (resp., right) ideal over if it satisfies

If a soft set over is both an int-soft left ideal and an int-soft right ideal over , we say that is an int-soft two-sided ideal over .

Obviously, every int-soft (resp., right) ideal over is an int-soft semigroup over . But the converse is not true in general (see [11]).

Proposition 4. Let be an int-soft left ideal over . If is a left zero subsemigroup of , then the restriction of to is constant; that is, for all .

Proof. Let . Then and . Thus and so for all .

Similarly, we have the following proposition.

Proposition 5. Let be an int-soft right ideal over . If is a right zero subsemigroup of , then the restriction of to is constant; that is, for all .

Proposition 6. Let be an int-soft left ideal over . If the set of all idempotent elements of forms a left zero subsemigroup of , then for all idempotent elements and of .

Proof. Assume that the set is a left zero subsemigroup of . For any , we have and . Hence and thus for all idempotent elements and of .

Similarly, we have the following proposition.

Proposition 7. Let be an int-soft right ideal over . If the set of all idempotent elements of forms a right zero subsemigroup of , then for all idempotent elements and of .

For a nonempty subset of and , with , define a map as follows: Then is a soft set over , which is called the -characteristic soft set. The soft set is called the -identity soft set over . The -characteristic soft set with and is called the characteristic soft set and is denoted by . The -identity soft set with and is called the identity soft set and is denoted by .

Lemma 8. Let and be -characteristic soft sets over where and are nonempty subsets of . Then the following properties hold: (1),(2).

Proof. (1) Let . If , then and . Thus we have If , then or . Hence we have Therefore .
(2) For any , suppose . Then there exist and such that . Thus we have and so . Since , we get . Suppose . Then for all and . If for some , then or . Hence If for all , then In any case, we have .

Theorem 9. For the -identity soft set , let be a soft set over such that for all . Then the following assertions are equivalent:(1) is an int-soft left ideal over ,(2).

Proof. Suppose that is an int-soft left ideal over . Let . If for some , then Otherwise, we have . Therefore .
Conversely, assume that . For any , we have Hence is an int-soft left ideal over .

Similarly, we have the following theorem.

Theorem 10. For the -identity soft set , let be a soft set over such that for all . Then the following assertions are equivalent: (1) is an int-soft right ideal over ,(2).

Corollary 11. For the -identity soft set , let be a soft set over such that for all . Then the following assertions are equivalent:(1) is an int-soft two-sided ideal over ,(2) and .

Note that the soft intersection of int-soft left (right, two-sided) ideals over is an int-soft left (right, two-sided) ideal over . In fact, the soft intersection of int-soft left (right, two-sided) ideals containing a soft set over is the smallest int-soft left (right, two-sided) ideal over .

For any soft set over , the smallest int-soft left (right, two-sided) ideal over containing is called the int-soft left (right, two-sided) ideal over generated by and is denoted by   .

Theorem 12. Let be a monoid with identity . Then , where for all .

Proof. Let . Since , we have and so . For all , we have Thus is an int-soft left ideal over . Now let be an int-soft left ideal over such that . Then for all and This implies that . Therefore .

Similarly, we have the following theorem.

Theorem 13. Let be a monoid with identity . Then , where for all .

4. Int-Soft (Generalized) Bi-Ideals

Definition 14. A soft set over is called an int-soft generalized bi-ideal over if it satisfies

We know that any int-soft generalized bi-ideal may not be an int-soft semigroup by the following example.

Example 15. Let be a semigroup with the following Cayley table: Let be a soft set over defined as follows: Then is an int-soft generalized bi-ideal over . But it is not an int-soft semigroup over since .

If a soft set over is both an int-soft semigroup and an int-soft generalized bi-ideal over , then we say that is an int-soft bi-ideal over .

We provide a condition for an int-soft generalized bi-ideal to be an int-soft semigroup.

Theorem 16. In a regular semigroup , every int-soft generalized bi-ideal is an int-soft semigroup, that is, an int-soft bi-ideal.

Proof. Let be an int-soft generalized bi-ideal over and let and be any elements of . Then there exists such that , and so Therefore is an int-soft semigroup over .

We consider characterizations of an int-soft (generalized) bi-ideal.

Lemma 17. For any nonempty subset of , the following are equivalent: (1) is a generalized bi-ideal of ,(2)the -characteristic soft set over is an int-soft generalized bi-ideal over for any with .

Proof. Assume that is a generalized bi-ideal of . Let with and . If , then and . Hence If or , then or . Hence Therefore is an int-soft generalized bi-ideal over for any with .
Conversely, suppose that the -characteristic soft set over is an int-soft generalized bi-ideal over for any with . Let be any element of . Then for some and . Then and so . Thus , which shows that . Therefore is a generalized bi-ideal of .

Theorem 18. A soft set over is an int-soft generalized bi-ideal over if and only if the nonempty -inclusive set of is a generalized bi-ideal of for all .

Proof. Assume that is an int-soft semigroup over . Let be such that . Let and . Then and . It follows from (26) that and that . Thus is a generalized bi-ideal of .
Conversely, suppose that the nonempty -inclusive set of is a generalized bi-ideal of for all . Let be such that and . Taking implies that . Hence , and so Therefore is an int-soft generalized bi-ideal over .

Lemma 19 (see [11]). A soft set over is an int-soft semigroup over if and only if the nonempty -inclusive set of is a subsemigroup of for all .

Combining Theorem 18 and Lemma 19, we have the following characterization of an int-soft bi-ideal.

Theorem 20. A soft set over is an int-soft bi-ideal over if and only if the nonempty -inclusive set of is a bi-ideal of for all .

Theorem 21. For the identity soft set and a soft set over , the following are equivalent: (1) is an int-soft generalized bi-ideal over ,(2).

Proof. Assume that is an int-soft generalized bi-ideal over . Let be any element of . If , then it is clear that . Otherwise, there exist such that and . Since is an int-soft generalized bi-ideal over , it follows from (26) that and that Therefore .
Conversely, suppose that . For any , let . Then
Therefore is an int-soft generalized bi-ideal over .

Theorem 22. Let be an int-soft semigroup over . Then is an int-soft bi-ideal over if and only if , where is the identity soft set over .

Proof. It is the same as the proof of Theorem 21.

Theorem 23. If and are int-soft generalized bi-ideals over , then so is the soft intersection .

Proof. For any , we have
Thus is an int-soft generalized bi-ideals over .

Theorem 24. If and are int-soft bi-ideals over , then so is the soft intersection .

Proof. It is the same as the proof of Theorem 23.

Theorem 25. If is a group, then every int-soft generalized bi-ideal is a constant function.

Proof. Let be a group with identity and let be an int-soft generalized bi-ideal over . For any , we have and so . Therefore is a constant function.

For any soft set over , the smallest int-soft (generalized) bi-ideal over containing is called an int-soft (generalized) bi-ideal over generated by and is denoted by .

Theorem 26. Let be a monoid with identity and let be a soft set over such that for all . Then , where for all .

Proof. Let . Since it follows from hypothesis that and that . For any , we get Since we have . Let . Then for all . Hence is an int-soft (generalized) bi-ideal over . Let be an int-soft (generalized) bi-ideal over such that . For any , we have and so . Therefore .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to thank the anonymous reviewer(s) for the valuable suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012R1A1A2042193).

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