#### Abstract

We introduce the notion of an ordered quasi-ideal of an ordered semiring and show that ordered quasi-ideals and ordered bi-ideals coincide in regular ordered semirings. Then we give characterizations of regular ordered semirings, regular ordered duo-semirings, and left (right) regular ordered semirings by their ordered quasi-ideals.

#### 1. Introduction

The concept of a quasi-ideal was defined first by Steinfeld for semigroups and for rings [1–3] as a generalization of a right ideal and a left ideal. Then Iséki [4] introduced the notion of a quasi-ideal in a semiring without zero and investigated some of its properties. In 1994, Dönges [5] studied quasi-ideals of a semiring with zero, investigated connections between left (right) ideals, bi-ideals, and quasi-ideals and characterized regular semirings using their quasi-ideals. Later, Shabir et al. [6] have studied some properties of quasi-ideals, using quasi-ideals to characterize regular and intraregular semirings and regular duo-semirings. As a generalization of quasi-ideals of semirings the quasi-ideals of -semirings were investigated by many authors; see, for example, [7–9].

In 2011, the notion of an ordered semiring was introduced by Gan and Jiang [10] as a semiring with a partially ordered relation on the semiring such that the relation is compatible to the operations of the semiring. In the paper, the concept of a left (right) ordered ideal, a minimal ordered ideal, and a maximal ordered ideal was defined. Then Mandal [11] studied fuzzy ideals in an ordered semiring with the least element zero and gave a characterization of regular ordered semirings by their fuzzy ideals.

In this paper, we introduce the notion of an ordered quasi-ideal of an ordered semiring and show that ordered quasi-ideals and ordered bi-ideals coincide in regular ordered semirings. Then characterizations of regular ordered semirings, regular ordered duo-semirings, and left (right) regular ordered semirings by their ordered quasi-ideals have been investigated.

#### 2. Preliminaries

An ordered semiring is a system consisting of a nonempty set such that is a semiring, is a partially ordered set, and for any the following conditions are satisfied:(i)if then and ;(ii)if then and .

An ordered semiring is said to be* additively commutative* if for all . An element is said to be an* absorbing zero* if and for all . In this paper we assume that is an additively commutative ordered semiring with an absorbing zero .

For any subsets of and , we denote

Now, we mention some properties of finite sums on an ordered semiring.

*Remark 1. *For any subsets of , the following statements hold:(i);(ii);(iii) and ;(iv) and ;(v).

We note that, for any , if and only if ( is a subsemigroup of ).

Now, we give the basic properties of the operator which are not difficult to verify.

Lemma 2. *Let be subsets of an ordered semiring . Then the following statements hold: *(i)* and ;*(ii)* If then ;*(iii)* and ;*(iv)* and ;*(v)* and ;*(vi)*;*(vii)*.*

In (vii) of the above lemma, we have when and .

Lemma 3. *Let be an ordered semiring and . If then .*

*Proof. *Assume that . Then

*Definition 4 (see [10]). *Let be an ordered semiring and . Then is said to be a* left ordered ideal (right ordered ideal)* if the following conditions are satisfied. (1) is a left ideal (right ideal) of .(2) If for some then (i.e., ).We call an* ordered ideal* if it is both left ordered ideal and right ordered ideal of .

*Example 5 (see [10]). *Let be the unit interval of real numbers. Define binary operations and on by letting ,and an ordered relation is the natural order on real numbers. It is easy to show that is an ordered semiring. Let . Then we can prove that is an ordered ideal of .

Lemma 6. *Let be a nonempty subset of an ordered semiring . Then *(i)* is a left ordered ideal of ;*(ii)* is a right ordered ideal of ;*(iii)* is an ordered ideal of .*

*Proof. *(i) Let . Then and for some . It is clear that , and so . By Remark 1 and Lemma 2, we obtain . We have . Hence, is a left ordered ideal of .

(ii) and (iii) can be proved similar to (i).

Corollary 7. *Let be an ordered semiring. Then, for any , *(i)* is a left ordered ideal of ;*(ii)* is a right ordered ideal of ;*(iii)* is an ordered ideal of .*

Let be a nonempty subset of an ordered semiring . We denote and as the smallest left ordered ideal, right ordered ideal, and ordered ideal of containing , respectively. In particular, we can show that if is a left ideal (right ideal, ideal) of then is the smallest left ordered ideal (resp., right ordered ideal and ordered ideal) of containing .

Lemma 8. *Let be a nonempty subset of an ordered semiring . Then *(i)*;*(ii)*;*(iii)*.*

*Proof. *(i) Since has an absorbing zero, we have, for every , . Hence, . Let . Then and for some . Thus and for some and . It is easy to show that and . It follows that , and so . By Remark 1 and Lemma 2, we obtain Since , is a left ordered ideal of . Let be any left ordered ideal of containing . It turns out and , so . It follows that . Therefore, is the smallest left ordered ideal of containing .

(ii) and (iii) can be proved similar to (i).

As a special case of Lemma 8, if then we have the following corollary.

Corollary 9. *Let be an ordered semiring. Then, for any ,*(i)*;*(ii)*;*(iii)*.*

An element of an ordered semiring is said to be an* identity* if for all . If has an identity, then we denote as the identity of .

It is not difficult to show that if has an identity, then and for any . In particular case, we have and for any .

#### 3. Ordered Quasi-Ideals in Ordered Semirings

Here, we present a notion of an ordered quasi-ideal of an ordered semiring. Then, in ordered semiring with an identity, we show that every ordered quasi-ideal can be expressed as an intersection of an ordered left ideal and an ordered right ideal.

*Definition 10. *Let be an ordered semiring and let be a subsemigroup of . Then is said to be an* ordered quasi-ideal* of if the following conditions are satisfied: (1);(2)if for some then (i.e., ).

It is clear that every left ordered ideal (right ordered ideal and ordered ideal) of an ordered semiring is an ordered quasi-ideal of . Moreover, each ordered quasi-ideal of is a subsemiring of ; indeed, .

*Example 11. *Let . Define binary operations + and on by the following equations: Then is an additively commutative semiring with an absorbing zero . Define a binary relation on by We give the covering relation “” and the figure of :

Now, is an ordered semiring. Let . We have and . Hence, is an ordered quasi-ideal of but is not a left ordered ideal of , since .

Lemma 12. *Let be an ordered semiring and let be a family of ordered quasi-ideals of . Then is an ordered quasi-ideal of .*

Let be a nonempty subset of an ordered semiring . We denote the smallest ordered quasi-ideal of containing .

Theorem 13. *Let be an ordered semiring and let be a nonempty subset of . Then .*

*Proof. *Let . Since has an absorbing zero, we have for every . Hence, . Let . Then and for some . Thus and for some and . Clearly, and . It follows that , and so . By Remark 1 and Lemma 2, we obtain Similarly, we can show that . Thus . Since , we obtain that is an ordered quasi-ideal of containing . Let be any ordered quasi-ideal of containing . It follows that . So . Hence, . Therefore, is the smallest ordered quasi-ideal of containing .

As a special case of Theorem 13, if then we have the following corollary.

Corollary 14. *Let be an ordered semiring. Then for any .*

If has an identity, then it is easy to check that for any . In particular case, we have for any .

Let be the set of all ordered quasi-ideals of an ordered semiring . Using Lemma 12, we define the operations and on by letting , Then we obtain the following theorem.

Theorem 15. *Let be an ordered semiring. Then is a complete lattice.*

Theorem 16. *The intersection of a left ordered ideal and a right ordered ideal of an ordered semiring is an ordered quasi-ideal of .*

*Proof. *It is easy to show that is a subsemigroup of . By Remark 1 and Lemma 2, we obtain Hence, . Let such that for some . Then .

The converse of Theorem 16 is not true as Example page 8 in [2] given by A. H. Clifford.

Corollary 17. *Let be an ordered semiring. Then the following statements hold. *(i)* is an ordered quasi-ideal of , for any .*(ii)* is an ordered quasi-ideal of , for any .*

*Proof. *(i) By Lemma 6, we have and a left and a right ordered ideal of , respectively. Then by Theorem 16, we have that is an ordered quasi-ideal of .

(ii) It is a particular case of (i).

Now, we will show that the converse of Theorem 16 is true if contains an identity as the following theorem.

Theorem 18. *Let be an ordered semiring with identity. Then every ordered quasi-ideal of can be written in the form for some right ordered ideal and left ordered ideal of .*

*Proof. *Assume that has an identity. Let be an ordered quasi-ideal of . Then and . We obtain and . Hence, .

#### 4. Regular Ordered Semirings

In this section, we show that in regular ordered semirings the converse of Theorem 16 is true and ordered quasi-ideals coincide with ordered bi-ideals. Then we give characterizations of regular ordered semirings, regular ordered duo-semirings, and left regular and right regular ordered semirings by their ordered quasi-ideals.

*Definition 19 (see [11]). *An element of an ordered semiring is said to be* regular* if for some . An ordered semiring is said to be* regular* if every element is regular.

The following lemma is characterizations of regular ordered semiring which directly follows Definition 19.

Lemma 20. *Let be an ordered semiring. Then the following statements are equivalent: *(i)* is regular;*(ii)* for each ;*(iii)* for any .*

Now, we will show that the converse of Theorem 16 is true in regular ordered semirings.

Theorem 21. *Every ordered quasi-ideal of a regular ordered semiring can be written in the form for some right ordered ideal and left ordered ideal of .*

*Proof. *Let be an ordered quasi-ideal of . By Lemma 8, we have and . Now, . Let . Since is regular, there exists such that . So . Since , . It follows that This implies that . Similarly, we can show that . Hence, . Therefore, .

*Definition 22. *Let be an ordered semiring. A subsemigroup of is said to be an* ordered bi-ideal* of if the following conditions hold: (1);(2) if for some , then (i.e., ).

We note that condition (1) of Definition 22 is equivalent to .

Theorem 23. *Every ordered quasi-ideal of an ordered semiring is an ordered bi-ideal of .*

*Proof. *Let be an ordered quasi-ideal of . Then and . So, . Hence, is an ordered bi-ideal of .

The converse of Theorem 23 is not generally true as the following example.

*Example 24. *Let . Define binary operations + and by the following equations: Then is an additively commutative semiring with an absorbing zero . Define a binary relation on by We give the covering relation “” and the figure of :

Then is an ordered semiring but not regular, since for any . Let . It is easy to show that is an ordered bi-ideal but not an ordered quasi-ideal of , since

Now, we show that in regular ordered semirings, ordered bi-ideals and ordered quasi-ideals coincide as the following theorem.

Theorem 25. *Let be a regular ordered semiring. Then ordered bi-ideals and ordered quasi-ideals coincide in .*

*Proof. *By Theorem 23, we have that every ordered quasi-ideal of is an ordered bi-ideal of . Now, we show that every ordered bi-ideal of is an ordered quasi-ideal of . Let be an ordered bi-ideal of . Let . By Lemma 20, Remark 1, and Lemma 2, we obtain . Hence, is an ordered quasi-ideal of .

Theorem 26. *Let be an ordered semiring. Then the following statements are equivalent: *(i)* is regular;*(ii)* for every right ordered ideal and left ordered ideal of ;*(iii)* for each ordered bi-ideal of ;*(iv)* for each ordered quasi-ideal of .*

*Proof. *: assume that is regular and let and be a right ordered ideal and a left ordered ideal of , respectively. So, and . Hence, . Let . Since is regular, for some . Since , . It follows that . This means . Therefore, .

: assume that (ii) holds. Let be an ordered bi-ideal of . It is clear that . By assumption, . By Lemma 8, Remark 1, and Lemmas 2 and 3, we have : it follows from Theorem 23.

: let . Then . By Corollary 14, Remark 1, and Lemma 2, we have By Lemma 20, is regular.

Theorem 27. *Let be a regular ordered semiring. Then the following statements hold: *(i)*every ordered quasi-ideal of can be written in the form for some right ordered ideal and left ordered ideal of ;*(ii)* for each ordered quasi-ideal of .*

*Proof. *(i) It is obvious by Theorems 21 and 26.

(ii) Let be an ordered quasi-ideal of . Clearly, . Let . Then for some . Since is regular, there exists such that . Hence, Therefore,

Theorem 28. *Let be an ordered semiring. Then is regular if and only if for every ordered bi-ideal , every ordered ideal , and every left ordered ideal of .*

*Proof. *Let , and be an ordered bi-ideal, an ordered ideal, and a left ordered ideal of , respectively. Let . Since is regular, . Hence, .

Conversely, assume that for every ordered bi-ideal , every ordered ideal , and every left ordered ideal of . Then we obtain for every right ordered ideal and left ordered ideal of . On the other hand, we have . Hence, . By Theorem 26, is regular.

*Definition 29. *An ordered semiring is said to be an* ordered duo-semiring* if every one-sided (right or left) ordered ideal of is an ordered ideal of .

We note that every multiplicatively commutative ordered semiring is an ordered duo-semiring, but the converse is not generally true. Now, we give an example of a multiplicatively noncommutative ordered semiring which is an ordered duo-semiring.

*Example 30. *Let . Define binary operations + and by the following equations: Then is an additively commutative semiring with an absorbing zero . Define a binary relation on by We give the covering relation “” and the figure of :

Then is an ordered semiring which is not multiplicatively commutative, since . We have all one-sided ordered ideals of which are as follows: It is not difficult to check that all of them are ordered ideals of . This shows that is an ordered duo-semiring.

Lemma 31. *Let be an ordered semiring. Then the following conditions are equivalent: *(i)* is an ordered duo-semiring;*(ii)* for each ;*(iii)* for each .*

*Proof. * and are obvious.

: let be a left ordered ideal of and let . By assumption, we have . It follows that is a right ordered ideal of . Similarly, we have that every right ordered ideal of is a left ordered ideal of . Hence, is an ordered duo-semiring.

Theorem 32. *Let be an ordered duo-semiring. Then is regular if and only if for each two ordered quasi-ideals and of .*

*Proof. *Assume that is a regular ordered semiring. Let and be ordered quasi-ideals of . By Theorem 21, and can be written in the forms for some and which are right ordered ideals and left ordered ideals of , respectively. Since is an ordered duo-semiring, , and are ordered ideals of . It follows that and are ordered ideals of . By Theorem 26, we have .

Conversely, assume that for each two ordered quasi-ideals and of . Let . By assumption, . By Theorem 13, Remark 1, and Lemmas 2 and 3, we have By Lemma 20, is a regular ordered semiring.

Theorem 33. *Let be an ordered duo-semiring. Then the following conditions are equivalent: *(i)* is regular;*(ii)* and for each two left ordered ideals and right ordered ideals of ;*(iii)*, for each right ordered ideal and left ordered ideal of .*

*Proof. *It is obvious by Theorem 26.

*Definition 34. *Let be an ordered semiring. Then an element is said to be* left regular (right regular)* if for some . An ordered semiring is said to be* left regular (right regular)* if every element is left regular (right regular).

*Example 35. *Let . Define binary operations + and on by the following equations: Then is a semiring with an absorbing zero . Define a binary relation on by We give the covering relation “” and the figure of :

Now, is an ordered semiring. Clearly, , and are left regular. We consider . This implies that is left regular. Since there does not exist such that , is not regular.

*Example 36. *Consider the ordered semiring where is the natural order relation on numbers. Since for any , we get a regular, left regular, and right regular ordered semiring.

The following lemmas can be easily proved using Definition 34.

Lemma 37. *Let be an ordered semiring. Then the following statements are equivalent: *(i)* is left regular;*(ii)* for each ;*(iii)* for each .*

Lemma 38. *Let be an ordered semiring. Then the following statements are equivalent: *(i)* is right regular;*(ii)* for each ;*(iii)* for each .*

*Definition 39. *Let be a nonempty subset of an ordered semiring . Then is said to be* semiprime* if for any , if , then .

We note that a nonempty subset of is semiprime if and only if, for any , implies . Because, if is semiprime, , and then and so ; that is, .

Theorem 40. *Let be an ordered semiring. Then is left regular and right regular if and only if every ordered quasi-ideal of is semiprime.*

*Proof. *Let be an ordered quasi-ideal of . Let be a nonempty subset of such that . Since is left regular and by Lemma 37, we have . Since is right regular and by Lemma 38, we have . Hence, . Therefore, is semiprime.

Conversely, assume that every ordered quasi-ideal of is semiprime. Let . By Theorem 13, we have . Since is semiprime, . Then we obtain In case , we get and so . By Lemma 37, is left regular. Similarly, in case , we get . By Lemma 38, is right regular.

#### Conflict of Interests

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

#### Acknowledgment

This work has been supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.