Abstract

The notions of union-soft semigroups, union-soft -ideals, and union-soft -ideals are introduced, and related properties are investigated. Characterizations of a union-soft semigroup, a union-soft -ideal, and a union-soft -ideal are provided. The concepts of union-soft products and union-soft semiprime soft sets are introduced, and their properties related to union-soft -ideals and union-soft -ideals are investigated. Using the notions of union-soft -ideals and union-soft -ideals, conditions for an ordered semigroup to be regular are considered. The concepts of concave soft sets and critical soft points are introduced, and their properties are discussed.

1. Introduction

The uncertainty which has appeared in economics, engineering, environmental science, medical science and social science, and so forth is too complicated to be captured within a traditional mathematical framework. Molodtsov’s soft set theory [1] is a kind of new mathematical model for coping with uncertainty from a parameterization point of view. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields. At present, works on the soft set theory with algebraic applications are progressing rapidly (see [2–5]). Mainly, Kehayopulu et al. studied ordered semigroups (see [6–10]). Feng et al. discussed soft relations in semigroups (see [11]) and explored decomposition of fuzzy soft sets with finite value spaces (see [12]). Also, Feng and Li [13] considered soft product operations. Jun et al. [14] applied the concept of soft set theory to ordered semigroups. They applied the notion of soft sets by Molodtsov to ordered semigroups and introduced the notions of (trivial, whole) soft ordered semigroups, soft ordered subsemigroups, soft -ideals, soft -ideals, and -idealistic and -idealistic soft ordered semigroups. They investigated various related properties.

The aim of this paper is to lay a foundation for providing a soft algebraic tool (in ordered semigroups) in considering many problems that contain uncertainties. We introduce the notions of union-soft semigroups, union-soft -ideals, and union-soft -ideals and investigate their properties. We consider characterizations of a union-soft semigroup, a union-soft -ideal, and a union-soft -ideal. We introduce the concepts of union-soft products and union-soft semiprime soft sets and investigate their properties related to union-soft -ideals and union-soft -ideals. Using the notions of union-soft -ideals and union-soft -ideals, we provide conditions for an ordered semigroup to be regular. We also introduce the concepts of concave soft sets and critical soft points and discuss their properties.

2. Preliminaries

2.1. Basic Results on Ordered Semigroups

An ordered semigroup (or, po-semigroup) is an ordered set which is a semigroup such that

For , we denote For any , denote by (resp., ) the right (resp., left) ideal of generated by . Note that and .

An ordered semigroup is said to be(i)left (resp., right) regular if it satisfies (ii)regular if it satisfies (iii)intraregular if it satisfies

Lemma 1 (see [9]). An ordered semigroup is regular if and only if

A nonempty subset of an ordered semigroup is called a left (resp., right) ideal of if

2.2. Basic Results on Soft Sets

A soft set theory is introduced by Molodtsov [1], and Çağman and Enginoğlu [15] 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 denotes the power set of and .

Definition 2 (see [1, 15]). A soft set over is defined to be the set of ordered pairs where such that if .

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

For any soft sets and over , we define The soft union, denoted by , of and is defined to be the soft set over in which is defined by The soft intersection, denoted by , of and is defined to be the soft set over in which is defined by

3. Union-Soft Ideals

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

Definition 3. A soft set over is called a union-soft semigroup over if it satisfies

Theorem 4. A soft set over is a union-soft semigroup over if and only if the nonempty -exclusive set of is a subsemigroup of for all .

Proof. Assume that over is a union-soft semigroup over . Let be such that . Let . Then and . It follows from (13) that so that . Thus is a subsemigroup of .
Conversely, suppose that the nonempty -exclusive set of is a subsemigroup of for all . Let be such that and . Taking implies that . Hence , and so . Therefore is a union-soft semigroup over .

Definition 5. For a left ideal of , a soft set over is called a union-soft -ideal over related to if it satisfies

Definition 6. For a right ideal of , a soft set over is called a union-soft -ideal over related to if it satisfies (16) and

A union-soft -ideal (resp., union-soft -ideal) over related to is called a union-soft -ideal (resp., union-soft -ideal) over .

If a soft set over is both a union-soft -ideal and a union-soft -ideal over , we say that is a union-soft ideal over .

Example 7. Let be an ordered semigroup with the following Cayley table and order (see [10]): (1)Let be a soft set over in which is given as follows: Routine calculations show that is a union-soft ideal over .(2)Let be a soft set over in which is given as follows: Then is a union-soft -ideal over . But it is not a union-soft -ideal over since .

Example 8. Let be an ordered semigroup with Cayley table and Hasse diagramas shown in Figure 1. In Figure 1, is a soft set over in which is given as follows: Routine calculations show that is a union-soft -ideal over . But it is not a union-soft -ideal over since .


Figure 1 

For a nonempty subset of , the characteristic soft set is defined to be the soft set over in which is given as follows: The soft set , where for all , is called the identity soft set over . For the characteristic soft set over , the soft set over is given as follows:

Theorem 9. For any nonempty subset of , the following are equivalent. (1) is a left (resp., right) ideal of .(2)The soft set over is a union-soft -ideal (resp., union-soft -ideal) over .

Proof. Assume that is a left ideal of . Let be such that . If , then and so . If , then . Since and is a left ideal of , we have and thus . For any , if then . If , then since is a left ideal of . Hence . Therefore is a union-soft -ideal over . Similarly, is a union-soft -ideal over when is a right ideal of .
Conversely suppose that is a union-soft -ideal over . Let and . Then , and so ; that is, . Thus . Let and be such that . Then , and thus . Therefore is a left ideal of . Similarly, we can show that if is a union-soft -ideal over , then is a right ideal of .

Corollary 10. For any nonempty subset of , the following are equivalent. (1) is an ideal of .(2)The soft set over is a union-soft ideal over .

Theorem 11. If a soft set over is a union-soft -ideal (resp., union-soft -ideal) over , then the nonempty -exclusive set of is a left (resp., right) ideal of for all .

Proof. Assume that is a union-soft -ideal over . Let be such that . Let and . Then , and thus by (15). Hence . Let and be such that . Using (16), we have . Thus . Therefore is a left ideal of . The right case can be seen in a similar way.

Corollary 12. If a soft set over is a union-soft -ideal (resp., union-soft -ideal) over , then the set is a left (resp., right) ideal of for every .

Theorem 13. Let be a soft set over in which is a chain. If the nonempty -exclusive set of is a left (resp., right) ideal of for all , then is a union-soft -ideal (resp., union-soft -ideal) over .

Proof. Let be such that . If , then and by taking . This is a contradiction, and so . Assume that for some . Then and by taking . This is a contradiction, and thus for all . Therefore is a union-soft -ideal over . The right case can be seen in a similar way.

Question. In Theorem 13, can we delete the condition “ is a chain”?

For any soft sets and over , the union-soft product, denoted by , of and is defined to be the soft set over in which is a mapping from to given by where .

Proposition 14. Let and be soft sets over where and are nonempty subsets of . Then the following properties hold: (1).(2).

Proof. (1) Let . If , then or . Thus we have If , then and . Hence we have Therefore .
(2) For any , suppose . Then for some and , and so . Thus we have and so . Since , we get . Suppose . Then . If , then and . Assume that . Then We now prove that for all . Let . Then . If and , then and so . This is impossible. Thus we have or . If , then and so . Similarly, if then . In any case, we have .

Proposition 15. Let , , , and be soft sets over . If and , then

Proof. For any , if then clearly . Assume that . Then and so .

Proposition 16. Let and be a union-soft -ideal and a union-soft -ideal over , respectively. Then

Proof. Let . If , then . Suppose that . Then . Let be such that . Then . Since is a union-soft -ideal over , it follows that . Since is a union-soft -ideal over , we have . Hence for all , and so Therefore .

Proposition 17. Let and be soft sets over . If is regular and is a union-soft -ideal over , then

Proof. Let be a union-soft -ideal over . Let . Then there exists such that since is regular. Thus ; that is, , and so On the other hand, since is a union-soft -ideal over , we have Hence . Since , it follows that Therefore .

In a similar way we prove the following.

Proposition 18. Let and be soft sets over . If is regular and is a union-soft -ideal over , then the soft inclusion (34) is valid.

Corollary 19. Let and be a union-soft -ideal and a union-soft -ideal over , respectively. If is regular, then

We now provide a characterization of a regular ordered semigroup.

Theorem 20. An ordered semigroup is regular if and only if the soft inclusion (34) is valid for every union-soft -ideal and every union-soft -ideal over .

Proof. Assume that is regular. Proposition 17 (or Proposition 18) implies that the soft inclusion (34) is valid.
Conversely, assume that the soft inclusion (34) is valid for every union-soft -ideal and every union-soft -ideal over . Let and . Since (resp., ) is a right (resp., left) ideal of , it follows from Theorem 9 that (resp., ) is a union-soft -ideal (resp., union-soft -ideal) over . Using (34), we have Since , we have . Hence and so . Therefore . Let be such that . Suppose that or . Then , and so This is a contradiction, and thus and . Therefore , which implies that . Hence Using Lemma 1, we know that is regular.

Theorem 21. For a union-soft -ideal over , the following assertion is valid where is an empty soft set over ; that is, for all .

Proof. Suppose that is a union-soft -ideal over . Let . If , then Assume that . Then . This completes the proof.

Theorem 22. Let be the empty soft set over and be a soft set over . If satisfies the conditions (43) and (16), then is a union-soft -ideal over .

Proof. For any , we have Hence is a union-soft -ideal over .

Similarly, we have the following theorem.

Theorem 23. For the empty soft set over and a soft set over , the following assertions are equivalent: (1) is a union-soft -ideal over .(2) satisfies the conditions (16) and

Corollary 24. For the empty soft set over and a soft set over , the following assertions are equivalent: (1) is a union-soft ideal over .(2) satisfies the conditions (16) and

Lemma 25. If (resp., ) is a union-soft -ideal (resp., union-soft -ideal) over , then (1) (resp., ).(2) (resp., ).

Proof. (1) Assume that is a union-soft -ideal over and let . If , then . Assume that and let be such that . Then , and so by (16) and (17). Hence for all . Therefore Consequently, . Similarly, for each union-soft -ideal over .
(2) Note that and . Using Proposition 15 and (1), we have . In a similar way, we obtain for each union-soft -ideal over .

Proposition 26. Let be a regular ordered semigroup. If (resp., ) is a union-soft -ideal (resp., union-soft -ideal) over , then

Proof. Let be a union-soft -ideal over and let . Then there exists such that . Hence since . Thus since . Therefore . Similarly we have for each union-soft -ideal over .

We say that a soft set over is soft idempotent if .

By Lemma 25 and Proposition 26 we have the following result.

Proposition 27. If is a regular ordered semigroup, then every union-soft -ideal (resp., union-soft -ideal) over is soft idempotent.

Definition 28. A soft set over is said to be union-soft semiprime if it satisfies

Theorem 29. If is left regular, then every union-soft -ideal is a union-soft semiprime.

Proof. Let be a union-soft -ideal over and let . Then for some since is left regular. It follows from (16) and (15) that Hence is union-soft semiprime.

In a similar way, we have the following theorem.

Theorem 30. If is right regular, then every union-soft -ideal is union-soft semiprime.

Theorem 31. If is intraregular, then every union-soft ideal is union-soft semiprime.

Proof. Let be a union-soft ideal over and let . Then for some since is intraregular. It follows from (16), (15), and (17) that Hence is union-soft semiprime.

Corollary 32. If is intraregular, then every union-soft ideal over satisfies the following equality:

Proof. Using Theorem 31, we have for all . This completes the proof.

4. Concave Soft Sets and Critical Soft Points

For any soft set over , consider a soft set over where Since for all , we have for all . Hence .

A soft set over is said to be concave if , and hence .

Theorem 33. For a soft set over , the following are equivalent: (1) is concave.(2)()  ().

Proof. Assume that is concave. Let be such that . Then
Conversely, if (2) is valid, then for all . Hence ; that is, is concave.

Proposition 34. For any soft sets , , and over , we have (1)if , then .(2).(3) is concave.

Proof. (1) If , then for all . Thus for all . Therefore .
(2) Let . If , then . If , then for some . Thus Therefore .
(3) Let be such that . Then It follows from Theorem 33 that is concave.

Let be a soft set over . For any and any proper subset of , a critical soft point, denoted by , over is defined to be a soft set over where

Proposition 35. For any proper subsets and of , if and are critical soft points over , then .

Proof. Let . If , then and so Note that for all . Hence . It follows that For , assume that . Then and so for some with . Thus and ; that is, and . It follows that and that . This is a contradiction. Therefore . Consequently, we know that for all ; that is, .

Corollary 36. For any proper subsets and of , if and are critical soft points over , then

Proof. It is straightforward.

Proposition 37. If is a concave soft set over , then for any and a proper subset of .

Proof. Let be a critical soft point over such that . Then for all , and so for all . Hence On the other hand, let for . Then . In fact, if then . If , then and . Since is concave, it follows from Theorem 33 that . Therefore , and so Hence

Theorem 38. For any soft sets , , and over , the following items are valid: (1)if is a union-soft ideal over , then it is concave; that is, ;(2)if and are union-soft -ideals (resp., union-soft -ideals) over , then so are and .

Proof. (1) Since every union-soft ideal over satisfies the condition (16), it follows from Theorem 33.
(2) Assume that and are union-soft -ideals over . For any with , we have Theorem 21 implies that It follows from Theorem 22 that is a union-soft -ideal over .
It is easy to verify that . Let be such that . Then and . Hence Therefore is a union-soft -ideal over by Theorem 22. Similarly, one can prove that and are union-soft -ideals over when and are union-soft -ideals over .

Theorem 39. For any critical soft point over , let be a soft set over in which is given as follows: Then is the greatest union-soft -ideal over which is contained in the critical soft point .

Proof. Let . If , then it is clear that . If , then and . Thus for some , and so . Hence , and thus . Assume that . If then . If then since . Thus . Consequently, is a union-soft -ideal over . For each , if then and so . If then . Therefore . Let be a union-soft -ideal over such that . If , then there exists such that . Hence If , then . Therefore . This completes the proof.

Similarly, we have the following theorem.

Theorem 40. For any critical soft point over , let be a soft set over in which is given as follows: Then is the greatest union-soft -ideal over which is contained in the critical soft point .

Theorem 41. Let be a critical soft points over . Then is a union-soft ideal over , and for all .

Proof. Let . If , then there exist such that . Hence On the other hand, for all , and so for all . It follows that for all . Assume that . If there exist such that , then If , then there exist such that and . Thus which leads a contradiction. Therefore . If there does not exist such that , then . Now, it is easy to verify that Let be such that . Obviously, for all . If , then . If , then and so . Therefore is a union-soft ideal over by Theorems 22 and 23.

Similarly, we have the following theorems.

Theorem 42. Let be a critical soft points over . Then is a union-soft -ideal over , and for all .

Theorem 43. Let be critical soft points over . Then is a union-soft -ideal over , and for all .

Proposition 44. For any critical soft points and over , we have

Proof. If , then . Hence by Theorem 41.
Conversely, assume that and . For any , if then . It follows from Theorem 41 that . If , then . Therefore This completes the proof.

For any subset of and a proper subset of , let and be soft sets over given as follows: respectively. Obviously, if then .

Proposition 45. For any nonempty subsets and of and any proper subset of , we have the following assertions. (1).(2).(3).(4)

Proof. (1) If then and for some and . Hence ; that is, , and thus Since and for all , we get . Therefore .
If then . For the case , we have The case implies that for all . If and , then and so . This is impossible, and thus or . If , then and thus . Similarly, if then . Therefore Consequently, (1) is true.
(2) Let . If , then and , and so Assume that . Then . If , then and so Similarly, if then . Therefore
(3) Let . If , then or , which implies that or . Hence Suppose that . Then and . It follows that Therefore .
(4) Let . If , then for some . Hence Note that for any critical soft point over . Thus Conditions (100) and (101) induce If , then(i) and(ii) for all , and so for all .It follows that Therefore

Theorem 46. If is a left ideal of , then is a union-soft -ideal over .

Proof. Suppose that is a left ideal of . Let . If , then and so . If , then . Assume that . If , then and thus . If , then . Therefore is a union-soft -ideal over .

In the same way, we can verify the following result.

Theorem 47. If is a right ideal of , then is a union-soft -ideal over .

Conflict of Interests

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

Acknowledgment

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