The Scientific World Journal

Volume 2014 (2014), Article ID 136424, 7 pages

http://dx.doi.org/10.1155/2014/136424

Research Article

Ideal Theory in Semigroups Based on Intersectional Soft Sets

Seok Zun Song,¹ Hee Sik Kim,² and Young Bae Jun³

¹Department of Mathematics, Jeju National University, Jeju 690-756, Republic of Korea

²Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea

³Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 18 March 2014; Accepted 22 May 2014; Published 23 June 2014

Academic Editor: Feng Feng

Copyright © 2014 Seok Zun Song 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 notions of int-soft semigroups and int-soft left (resp., right) ideals are introduced, and several properties are investigated. Using these notions and the notion of inclusive set, characterizations of subsemigroups and left (resp., right) ideals are considered. Using the notion of int-soft products, characterizations of int-soft semigroups and int-soft left (resp., right) ideals are discussed. We prove that the soft intersection of int-soft left (resp., right) ideals (resp., int-soft semigroups) is also int-soft left (resp., right) ideals (resp., int-soft semigroups). The concept of int-soft quasi-ideals is also introduced, and characterization of a regular semigroup is discussed.

1. Introduction

Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties. He pointed out several directions for the applications of soft set theory. At present, works on 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 Enginoğlu [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. Decision making based on soft sets was further developed in [5–8]. It is worth noting that soft sets are closely related to many other soft computing models such as rough sets and fuzzy sets. Feng and Li [9] initiated soft approximation spaces and soft rough sets, which extended Pawlak’s rough sets. Moreover, Feng [10] considered the application of soft rough approximations in multicriteria group decision making problems. Akta and Çağman [11] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. After that, many algebraic properties of soft sets were studied by several researchers (see [12–26]). Recently, Feng et al. [27] investigated the relationships among five different types of soft subsets. They also explored free soft algebras associated with soft product operations, showing that soft sets have some nonclassical algebraic properties. In this paper, we introduce the notion of int-soft semigroups and int-soft left (resp., right) ideals. Using these notions, we provide characterizations of subsemigroups and left (resp., right) ideals. Using the notion of inclusive set, we also establish characterizations of subsemigroups and left (resp., right) ideals. Using the notion of int-soft products, we give characterizations of int-soft semigroups and int-soft left (resp., right) ideals. We show that the soft intersection of int-soft left (resp., right) ideals (resp., int-soft semigroups) is also int-soft left (resp., right) ideals (resp., int-soft semigroups). We also introduce the concept of int-soft quasi-ideals and discuss a characterization of a regular semigroup by using the notion of int-soft quasi-ideals.

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 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 quasi-ideal of if .

A soft set theory was introduced by Molodtsov [1], and Çağman and Enginoğlu [5] provided new definitions and various results on soft set theory.

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

Definition 1 (see [1, 5]). 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 .

Definition 2 (see [28]). Assume that has a binary operation . For any nonempty subset of , a soft set over is said to be intersectional over if it satisfies

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

3. Intersectional Soft Ideals

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

Definition 3. A soft set over is called an intersectional soft semigroup (briefly, int-soft semigroup) over if it satisfies

Definition 4. A soft set over is called an intersectional soft left (resp., right) ideal (briefly, 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 intersectional soft two-sided ideal (briefly, int-soft two-sided ideal) over .

Example 5. Let be a semigroup with the following Cayley table:

Let be a soft set over defined as follows:

where , and are subsets of with . Then is an int-soft two-sided ideal over .

Obviously, every int-soft left (resp., right) ideal over is an int-soft semigroup over . But the converse is not true as seen in the following example.

Example 6. Let be a semigroup with the following Cayley table:

(1)Let be a soft set over defined as follows:

where , and are subsets of with . Then is an int-soft semigroup over . But it is not an int-soft left ideal over since .(2)Let be a soft set over defined as follows:

where , and are subsets of with . Then is an int-soft semigroup over . But it is not an int-soft right ideal over since .

For a nonempty subset of , 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 .

Theorem 7. For any nonempty subset of  , the following are equivalent.

(1) is a left (resp., right) ideal of  .(2)The characteristic soft set over   is an int-soft left (resp., right) ideal over  .

Proof. Assume that is a left ideal of . For any , if then . If , then since is a left ideal of . Hence . Therefore is an int-soft left ideal over . Similarly, is an int-soft right ideal over when is a right ideal of .

Conversely suppose that is an int-soft left ideal over . Let and . Then , and so ; that is, . Thus and therefore is a left ideal of . Similarly, we can show that if is an int-soft right ideal over , then is a right ideal of .

Corollary 8. For any nonempty subset of  , the following are equivalent.

(1) is a two-sided ideal of  .(2)The characteristic soft set over    is an int-soft two-sided ideal over  .

Theorem 9. 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 .

Proof. Assume that over is an int-soft semigroup over . Let be such that . Let . Then and . It follows from (5) that

so that . Thus is a subsemigroup of .

Conversely, suppose that the nonempty -inclusive set of is a subsemigroup of for all . Let be such that and . Taking implies that . Hence , and so . Therefore is an int-soft semigroup over .

Theorem 10. A soft set over is an int-soft left (resp., right) ideal over if and only if the nonempty -inclusive set of   is a left (resp., right) ideal of   for all .

Proof. It is the same as the proof of Theorem 9.

Corollary 11. A soft set over   is an int-soft two-sided ideal over if and only if the nonempty -inclusive set of   is a two-sided ideal of   for all .

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

Proposition 12. Let  , and   be soft sets over . If

then .

Proof. Let . If is not expressed as for , then clearly

Suppose that there exist such that . Then

Therefore .

Lemma 13. Let and be characteristic soft sets over   where and are nonempty subsets of  . Then the following properties hold:

(1),(2).

Proof. 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 14. A soft set over is an int-soft semigroup over if and only if .

Proof. Assume that and let . Then

and so is an int-soft semigroup over .

Conversely, suppose that is an int-soft semigroup over . Then for all with . Thus

for all . Hence .

Theorem 15. For the identity soft set and a soft set over  , the following 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 16. For the identity soft set over and a soft set over , the following assertions are equivalent:

(1) is an int-soft right ideal over ,(2).

Corollary 17. For the identity soft set over and a soft set over , the following assertions are equivalent:

(1) is an int-soft two-sided ideal over ,(2) and .

Theorem 18. If and are int-soft semigroups over  , then so is the soft intersection .

Proof. Let . Then

Thus is an int-soft semigroup over .

By similar manner, we can prove the following theorem.

Theorem 19. If   and are int-soft left ideals (resp., int-soft right ideals) over , then so is the soft intersection .

Corollary 20. If   and are int-soft two-sided ideals over , then so is the soft intersection .

Theorem 21. Let   and be soft sets over . If is an int-soft left ideal over , then so is the int-soft product .

Proof. Let  . If for some , then and

If is not expressible as for , then . Thus for all , and so is an int-soft left ideal over .

Similarly, we have the following theorem.

Theorem 22. Let and   be soft sets over . If is an int-soft right ideal over , then so is the int-soft product .

Corollary 23. The int-soft product of two int-soft two-sided ideals over is an int-soft two-sided ideal over .

Let be a soft set over . For a subset of with , define a soft set over by

where is a subset of with .

Theorem 24. If   is an int-soft semigroup over , then so is .

Proof. Let . If  , then since is a subsemigroup of by Theorem 9. Hence we have

If   or , then or . Thus

Therefore is an int-soft semigroup over .

By similar manner, we can prove the following theorem.

Theorem 25. If   is an int-soft left ideal (resp., int-soft right ideal) over , then so is .

Corollary 26. If   is an int-soft two-sided ideal over , then so is .

Theorem 27. If   is an int-soft right ideal over and is an int-soft left ideal over , then .

Proof. Let . If is not expressible as for , then . Assume that there exist such that . Then

In any case, we have .

If we strengthen the condition of the semigroup , then we can induce the reverse inclusion in Theorem 27 as follows.

Theorem 28. Let be a regular semigroup. If   is an int-soft right ideal over  , then for every soft set over  .

Proof. Let . Then there exists such that since is regular. Thus

On the other hand, we have

since is an int-soft right ideal over . Since , we obtain

Therefore , and so .

In a similar way we prove the following.

Theorem 29. Let be a regular semigroup. If   is an int-soft left ideal over  , then for every soft set over  .

Theorem 30. If a semigroup is regular, then for every int-soft right ideal and int-soft left ideal over  .

Proof. Assume that is a regular semigroup and let and be an int-soft right ideal and an int-soft left ideal, respectively, over . By Theorem 28, we have . Since by Theorem 27, we have .

Definition 31. A soft set over is called an int-soft quasi-ideal over if

Obviously, every int-soft left (resp., right) ideal is an int-soft quasi-ideal over , but the converse does not hold in general.

In fact, we have the following example.

Example 32. Let be a semigroup with the following Cayley table:

Let be a soft set over defined as follows:

where is a subset of . Then is an int-soft quasi-ideal over and is not an int-soft left (resp., right) ideal over .

Theorem 33. Let be a nonempty subset of  . Then is a quasi-ideal of if and only if the characteristic soft set is an int-soft quasi-ideal over  .

Proof. We first assume that is a quasi-ideal of . Let be any element of . If , then

If , then . On the other hand, assume that

Then

This implies that there exist elements , and of with such that and . Hence , which contradicts that . Thus we have and so is an int-soft quasi-ideal over .

Conversely, suppose that is an int-soft quasi-ideal over . Let be any element of . Then for some and . It follows from (39) that

and so . Thus , and hence is a quasi-ideal of .

Theorem 34. For a semigroup , the following are equivalent:

(1) is regular,(2) for every int-soft quasi-ideal over  .

Proof. Assume that is regular and let . Then for some . Hence

and so . On the other hand, since is an int-soft quasi-ideal over ,

Hence .

Conversely, suppose that (2) is valid and let be a quasi-ideal of . Then and is an int-soft quasi-ideal over . For any , we have

This implies that there exist such that and . Then

and so for some . It follows that and so that . Hence , and thus . Therefore is regular.

Conflict of Interests

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

Acknowledgment

The authors are grateful to the referee for valuable suggestions and help.

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