Characterizations of Regular Ordered Semirings by Ordered Quasi-Ideals Pakorn

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 quasiideals to characterize regular and intraregular semirings and regular duo-semirings. As a generalization of quasi-ideals of semirings the quasi-ideals of Γ-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.ThenMandal [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.


Introduction
The concept of a quasi-ideal was defined first by Steinfeld for semigroups and for rings [1][2][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 quasiideals to characterize regular and intraregular semirings and regular duo-semirings.As a generalization of quasi-ideals of semirings the quasi-ideals of Γ-semirings were investigated by many authors; see, for example, [7][8][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.
An ordered semiring  is said to be additively commutative if  +  =  +  for all ,  ∈ .An element 0 ∈  is said to be an absorbing zero if 0 = 0 = 0 and  + 0 =  = 0 +  for all  ∈ .In this paper we assume that  is an additively commutative ordered semiring with an absorbing zero 0.
For any subsets ,  of  and  ∈ , we denote Now, we mention some properties of finite sums on an ordered semiring.
We note that, for any  ⊆ , Σ =  if and only if + ⊆  ((, +) is a subsemigroup of (, +)).Now, we give the basic properties of the operator (] which are not difficult to verify.(2) Definition 4 (see [10]).Let  be an ordered semiring and 0 ̸ =  ⊆ .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 .
We call  an ordered ideal if it is both left ordered ideal and right ordered ideal of .
Example 5 (see [10]).Let [0, 1] be the unit interval of real numbers.Define binary operations ⊕ and ⊙ on [0, 1] by letting (ii) (] is a right ordered ideal of ; (iii) (Σ] 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  (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  ∈ , An element  of an ordered semiring  is said to be an identity if  =  =  for all  ∈ .If  has an identity, then we denote 1 as the identity of .

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) (Σ] ∩ (Σ] ⊆ ; (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,  ⊆ (] ⊆ (] ∩ (] ⊆ (Σ] ∩ (Σ] ⊆ .Let  be a nonempty subset of an ordered semiring .We denote () the smallest ordered quasi-ideal of  containing .If  has an identity, then it is easy to check that () = (Σ] ∩ (Σ] for any  ⊆ .In particular case, we have () = (] ∩ (] for any  ∈ . Let Q() be the set of all ordered quasi-ideals of an ordered semiring .Using Lemma 12, we define the operations ∧ and ∨ on Q() by letting  1 ,  2 ∈ Q(), Then we obtain the following theorem.
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 The converse of Theorem 16 is not true as Example 2.1 page 8 in [2] given by A. H. Clifford.
Corollary 17.Let  be an ordered semiring.Then the following statements hold.
Proof.(i) By Lemma 6, we have (Σ] and (Σ] a left and a right ordered ideal of , respectively.Then by Theorem 16, we have that (Σ] ∩ (Σ] 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 .

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

Theorem 23. Every ordered quasi-ideal of an ordered semiring
is an ordered bi-ideal of .
The converse of Theorem 23 is not generally true as the following example.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 (Σ] ∩ (Σ] = {, } ̸ ⊆ .
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 .
Theorem 26.Let  be an ordered semiring.Then the following statements are equivalent: (i)  is regular; (ii) (Σ] =  ∩  for every right ordered ideal  and left ordered ideal  of ; (iii)  = (Σ] for each ordered bi-ideal  of ; (iv)  = (Σ] for each ordered quasi-ideal  of .

International Journal of Mathematics and Mathematical Sciences
Proof.(i) ⇒ (ii): assume that  is regular and let  and  be a right ordered ideal and a left ordered ideal of , respectively.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 .
Conversely, assume that ∩∩ ⊆ (] for every ordered bi-ideal , every ordered ideal , and every left ordered ideal  of .Then we obtain ∩ = ∩∩ ⊆ (] ⊆ (] ⊆ (Σ] for every right ordered ideal  and left ordered ideal  of .On the other hand, we have (Σ] ⊆ ∩.Hence, (Σ] = ∩.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 duosemiring.
It is not difficult to check that all of them are ordered ideals of .This shows that  is an ordered duo-semiring.
Proof.(i) ⇒ (ii) and (ii) ⇒ (iii) are obvious.(iii) ⇒ (i): 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 (Σ 1  2 ] =  1 ∩  2 for each two ordered quasi-ideals  1 and  2 of .
Proof.Assume that  is a regular ordered semiring.Let  1 and  2 be ordered quasi-ideals of .By Theorem 21,  1 and  2 can be written in the forms for some  1 ,