In 1989, Ahsan and Takahashi [2] introduced the notions of pure ideals and purely prime ideals of semigroups. Later, Changphas and Sanborisoot [3] defined the notions of left pure, right pure, left weakly pure and right weakly pure ideals in ordered semigroups and gave some of their characterizations. In 2007, Shabir and Iqbal [10] studied the concepts of pure ideals on semirings and characterized left and right weakly regular semirings using their pure ideals. As a special case of pure ideals of semirings, Jagatap [5] defined left pure k-ideals and right pure k-ideals of Γ-semirings and use them to characterize left and right weakly k-regular Γ-semirings, respectively. In 2017, Senarat and Pibaljommee [9] characterized left and right weakly ordered k-regular semirings using left pure and right pure ordered k-ideals, respectively.

An ordered semiring defined by Gan and Jiang [4] is a generalization of a semiring; indeed, it is a semiring together with a partially ordered relation connected by the compatibility property. In this work, we present the notions of left pure, right pure, quasi-pure and bi-pure ordered ideals of ordered semirings, investigate some of their properties and characterize regular, left weakly regular and right weakly regular ordered semirings by their pure ordered ideals. Furthermore, we introduce the notions of left and right weakly pure ordered ideals and use them to characterize fully idempotent ordered semirings.

An ordered semiring [4] (S,+, ·,≤) is a semiring (S,+, ·) together with a binary relation ≤ on S such that (S,≤) is a poset satisfying the following condition; if a ≤ b, then a + c ≤ b + c, c + a ≤ c + b, ac ≤ bc and ca ≤ cb for all a, b, c ∈ S. An element 0 of an ordered semiring S is called an absorbing zero [1] if 0 + x = x = x + 0 and 0x = 0 = x0 for all x ∈ S. Throughout this work, we always assume that S is an additively commutative ordered semiring with an absorbing zero 0, i.e., a+b = b+a for all a, b ∈ S.

For nonempty subsets A and B of S, we denote that

In a particular case of a ∈ S, we write ∑a and (a] instead of ∑{a} and ({a}], respectively. If I = ∅︀, we set ∑i∈Iai=0 for all a_i ∈ S.

For the basic properties of the finite sums ∑ and the operator ( ] of nonempty subsets of ordered semirings, we refer to [7, 8].

A nonempty subset A of an ordered semiring S such that A + A ⊆ A is called a left (resp. right) ordered ideal if SA ⊆ A (resp. AS ⊆ A) and A = (A]. If A is both a left and a right ordered ideal of S, then A is called an ordered ideal [4] of S. A nonempty subset Q of S such that Q+Q ⊆ Q is called an ordered quasi-ideal [7] of S if (∑QS] ∩ (∑SQ] ⊆ Q. A subsemiring B of S (i.e., B + B ⊆ B and B² ⊆ B) is called an ordered bi-ideal [7] if BSB ⊆ B.

For an element a of an ordered semiring S, we denote L(a), R(a), J(a), Q(a) and B(a) to be the left ordered ideal, right ordered ideal, ordered ideal, ordered quasi-ideal and ordered bi-ideal of S generated by a, respectively. We recall their constructions which occur in [7] as follows.

Lemma 2.1

Let ∅︀ ≠ A ⊆ S. Then

• (i) L(a) = (∑a + Sa];

• (ii) R(a) = (∑a + aS];

• (iii) J(a) = (∑a + Sa + aS +∑SaS];

• (iv)Q(a) = (∑a + ((aS] ∩ (Sa])];

• (v) B(a) = (∑a+∑a² + aSa].

An ordered semiring S is called regular [6] if a ∈ (aSa] for all a ∈ S (i.e., for each a ∈ S, a ≤ axa for some x ∈ S). If for all a ∈ S, a ∈ (∑SaSa] (resp. a ∈ (∑aSaS]), then S is called left weakly regular (resp. right weakly regular) [5].

In this section, we present the notions of left pure, right pure, quasi-pure, bipure, left weakly pure and right weakly pure ordered ideals of ordered semirings and use them to characterize regular, left weakly regular, right weakly regular and fully idempotent ordered semirings.

Definition 3.1

An ordered ideal A of an ordered semiring S is called left pure (resp. right pure) if x ∈ (Ax] (resp. x ∈ (xA]) for all x ∈ A.

Theorem 3.2

Let A be an ordered ideal of an ordered semiring S. Then the following statements hold:

• (i) A is left pure if and only if A ∩ L = (AL] for every left ordered ideal L of S;

• (ii) A is right pure if and only if R ∩ A = (RA] for every right ordered ideal R of S.

Definition 3.3

An ordered ideal A of an ordered semiring S is called quasi-pure if x ∈ (xA] ∩ (Ax] for all x ∈ A.

The following remark is directly obtained by Definition 3.1 and 3.3.

Remark 3.4

Every quasi-pure ordered ideal of an ordered semiring is both left pure and right pure.

Now, we give an example of a left pure ordered ideal of an ordered semiring which is not quasi-pure.

Example 3.5

Let S = {a, b, c, d, e}. Define binary operations + and · on S by the following tables;

and

Define a binary relation ≤ of S by ≤ =: {(a, a), (b, b), (c, c), (d, d), (e, e), (a, b), (a, c)}.

Figure 1. .

Then (S,+, ·,≤) is an ordered semiring. We have that A = {a, b, c, e} is a left pure ordered ideal of S. However, A is not right pure because c ∉ {a, e} = (cA]. So, c ∉ (cA] ∩ (Ac] and thus A is not quasi-pure.

As a duality of Example 3.5, we now give an example of a right pure ordered ideal of an ordered semiring which is not quasi-pure.

Example 3.6

Let S = {a, b, c, d, e} together with the operation + and the relation ≤ defined in Example 3.5. Define a binary operation · on S by the following table;

Then (S,+, ·,≤) is an ordered semiring. We have that A = {a, b, c, e} is a right pure ordered ideal of S. However, A is not left pure because c ∉ {a, e} = (Ac]. So, c ∉ (cA] ∩ (Ac] and thus A is not quasi-pure.

As a consequence of Example 3.5 and 3.6, the concepts of left pure and right pure ordered ideals of ordered semirings are independent.

Theorem 3.7

An ordered ideal A of an ordered semiring S is quasi-pure if and only if A ∩ Q = (QA] ∩ (AQ] for every ordered quasi-ideal Q of S.

Definition 3.8

An ordered ideal A of an ordered semiring S is called bi-pure if x ∈ (xAx] for all x ∈ A.

The following remark is directly obtained by Definition 3.3 and 3.8.

Remark 3.9

Every bi-pure ordered ideal of an ordered semiring is quasi-pure.

Now, we give an example of a quasi-pure ordered ideal of an ordered semiring which is not bi-pure.

Example 3.10

Let S = {a, b, c, d, e} together with the operation + and the relation ≤ defined in Example 3.5. Define a binary operation · on S by the following table;

Then (S,+, ·,≤) is an ordered semiring. We have that A = {a, b, c, e} is a quasi-pure ordered ideal of S. However, A is not bi-pure because c ∉ {a, e} = (cAc].

Theorem 3.11

An ordered ideal A of an ordered semiring S is bi-pure if and only if A ∩ B = (BAB] for every ordered bi-ideal B of S.

Lemma 3.12

Let S be an ordered semiring. Then the following statements hold:

• (i) if a ∈ (∑a²+aSa+Sa²+∑SaSa] for any a ∈ S, then S is left weakly regular;

• (ii) if a ∈ (∑a² + aSa + a²S + ∑aSaS] for any a ∈ S, then S is right weakly regular.

Theorem 3.13

An ordered semiring S is left (resp. right) weakly regular if and only if every ordered ideal of S is left (resp. right) pure.

Corollary 3.14

An ordered semiring S is both left and right weakly regular if and only if every ordered ideal of S is quasi-pure.

We note that an ordered ideal of an ordered semiring is bi-pure if and only if it is a regular subsemiring. Accordingly, we obtain the following remark.

Remark 3.15

An ordered semiring S is regular if and only if every ordered ideal of S is bi-pure.

Definition 3.16

An ordered ideal A of S is called left weakly pure (resp. right weakly pure) if A ∩ I = (∑AI] (resp. I ∩ A = (∑IA]) for every ordered ideal I of S.

Remark 3.17

Every left (resp. right) pure ordered ideal of an ordered semiring is left (resp. right) weakly pure.

Now, we give an example of a left weakly pure ordered ideal of an ordered semiring which is not left pure.

Example 3.18

Let S = {a, b, c, e}. Define two binary operations + and · on S by the following tables:

and

Define a binary relation ≤ on S by ≤:= {(a, a), (b, b), (c, c), (e, e), (a, c), (b, c)}.

Figure 2. .

Then (S,+, ·,≤) is an ordered semiring with only two ordered ideals S and {e}. Since S ∩ {e} = {e} = (∑S{e}] and S ∩ S = S = (∑SS], S itself is a left weakly pure ordered ideal. However, S is not a left pure ordered ideal because b ∉ {a, e} = (Sb].

As a duality of Example 3.18, we now give an example of a right weakly pure ordered ideal of an ordered semiring which is not right pure.

Example 3.19

Let S = {a, b, c, e}. Define a binary operation · on S by the following table:

Then (S,+, ·,≤) is an ordered semiring together with the operation + and the relation ≤ defined in Example 3.18. We have that S has only two ordered ideals S and {e}. Since {e} ∩ S = {e} = (∑{e}S] and S ∩ S = S = (∑SS], S itself is a right weakly pure ordered ideal. However, S is not a right pure ordered ideal because b ∉ {a, e} = (bS].

Definition 3.20

An ordered semiring S is called fully idempotent if I = (∑I²] for every ordered ideal I of S.

Lemma 3.21

Let S be an ordered semiring. Then the following statements are equivalent:

• (i) S is fully idempotent;

• (ii) a ∈ (∑SaSaS] for all a ∈ S;

• (iii) A ⊆ (∑SASAS] for all ∅︀ ≠ A ⊆ S.

Corollary 3.22

Let S be an ordered semiring. If

for all a ∈ S, then S is fully idempotent.

Theorem 3.23

Let S be an ordered semiring. Then the following statements hold:

• (i) if S is fully idempotent, then every ordered ideal of S is both left and right weakly pure;

• (ii) if every ordered ideal of S is left weakly pure (right weakly pure), then S is fully idempotent.