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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2020; 60(3): 455-465

Published online September 30, 2020

### Purities of Ordered Ideals of Ordered Semirings

Pakorn Palakawong na Ayutthaya, Bundit Pibaljommee*

Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
e-mail : pakorn1702@gmail.com
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Centre of Excellence in Mathematics CHE, Bangkok, 10400, Thailand
e-mail : banpib@kku.ac.th

Received: October 30, 2019; Revised: April 2, 2020; Accepted: April 21, 2020

### Abstract

We introduce the concepts of the left purity, right purity, quasi-purity, bipurity, left weak purity and right weak purity of ordered ideals of ordered semirings and use them to characterize regular ordered semirings, left weakly regular ordered semirings, right weakly regular ordered semirings and fully idempotent ordered semirings.

Keywords: pure ordered ideal, ordered semiring, regular ordered semiring

### 1. Introduction

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.

### 2. Preliminaries

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 ab, then a + cb + c, c + ac + b, acbc and cacb for all a, b, cS. 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 xS. 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, bS.

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

$A+B ={a+b∣a∈A,b∈B},AB ={ab∣a∈A,b∈B},ΣA ={∑i∈Iai∣ai∈A and I is a finite subset of ℕ} and(A] ={x∈S∣x≤a for some a∈A}.$

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

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 + AA is called a left (resp. right) ordered ideal if SAA (resp. ASA) 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+QQ is called an ordered quasi-ideal [7] of S if (∑QS] ∩ (∑SQ] ⊆ Q. A subsemiring B of S (i.e., B + BB and B2B) is called an ordered bi-ideal [7] if BSBB.

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 ∅︀ ≠ AS. 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 aS (i.e., for each aS, aaxa for some xS). If for all aS, a ∈ (∑SaSa] (resp. a ∈ (∑aSaS]), then S is called left weakly regular (resp. right weakly regular) [5].

### 3. Main Results

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 xA.

### 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 AL = (AL] for every left ordered ideal L of S;

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

Proof

(i) Assume that A is left pure. Let L be a left ordered ideal of S. If xAL, then x ∈ (Ax] ⊆ (AL]. So, AL ⊆ (AL]. Clearly, (AL] ⊆ AL. Hence, AL = (AL].

Conversely, let xA. Using assumption and Lemma 2.1, we get

$x∈A∩L(x)=(AL(x)]=(A(Σx+Sx]]⊆((ΣAx+ASx]]⊆(Ax+Ax]⊆(Ax].$

Hence, A is a left pure ordered ideal of S.

(ii) It can be proved similarly.

### Definition 3.3

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

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;

+ a b c d e
a a a a a a
b a a a a b
c a a a a c
d a a a a d
e a b c d e

and

· a b c d e
a a a a a e
b a b c a e
c a a a a e
d a a a a e
e e e e e e

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;

· a b c d e
a a a a a e
b a b a a e
c a c a a e
d a a a a e
e e e e e e

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.

Proof

Assume that A is quasi-pure. Let Q be an ordered quasi-ideal of S. If xAQ, then x ∈ (xA] ∩ (Ax] ⊆ (QA] ∩ (AQ]. So, AQ ⊆ (QA] ∩ (AQ]. Clearly, (QA] ∩ (AQ] ⊆ AQ. Hence, AQ = (QA] ∩ (AQ].

Conversely, let xA. Using assumption and Lemma 2.1, we get

$x∈A∩Q(x)=(Q(x)A]∩(AQ(x)]=((Σx+((xS]∩(Sx])]A]∩(A(Σx+((xS]∩(Sx])]]⊆((Σx+(xS]]A]∩(A(Σx+(Sx]]]⊆((Σx+xS]A]∩(A(Σx+Sx]]⊆(ΣxA+xSA]∩(ΣAx+ASx]⊆(xA+xA]∩(Ax+Ax]⊆(xA]∩(Ax].$

Hence, A is a quasi-pure ordered ideal of S.

### Definition 3.8

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

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;

· a b c d e
a a a a a e
b a b c a e
c a c a a e
d a a a a e
e e e e e e

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 AB = (BAB] for every ordered bi-ideal B of S.

Proof

Assume that A is bi-pure. Let B be an ordered bi-ideal of S. If xAB, then x ∈ (xAx] ⊆ (BAB]. So, AB ⊆ (BAB]. Clearly, (BAB] ⊆ AB. Hence, AB = (BAB].

Conversely, let xA. Using assumption and Lemma 2.1, we get

$x∈A∩B(x)=(B(x)AB(x)]=((Σx+Σx2+xSx]A(Σx+Σx2+xSx]]⊆(ΣxAx]=(xAx].$

Hence, A is a bi-pure ordered ideal of S.

Now, we characterize regular, left weakly regular and right weakly regular semirings using left pure, right pure, quasi-pure and bi-pure ordered ideals.

### Lemma 3.12

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

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

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

Proof

(i) Let aS. Assume that

$a∈(Σa2+aSa+Sa2+ΣSaSa]$$⊆(Sa+ΣSaSa].$

Using (3.2), we obtain that

$a2=aa∈(Sa+ΣSaSa](Sa+ΣSaSa)⊆(ΣSaSa].$

Using (3.2) again, we obtain that

$aSa⊆(Sa+ΣSaSa]S(Sa+ΣSaSa]⊆(ΣSaSa].$

Using (3.3), we obtain that

$Sa2⊆S(ΣSaSa]⊆(ΣSSaSa]⊆(ΣSaSa].$

Using (3.1), (3.3), (3.4) and (3.5), we obtain that

$a∈(Σa2+aSa+Sa2+ΣSaSa]⊆(Σ(ΣSaSa]+(ΣSaSa]+(ΣSaSa]+ΣSaSa]⊆((ΣSaSa]+(ΣSaSa]+(ΣSaSa]+(ΣSaSa]]⊆((ΣSaSa+ΣSaSa+ΣSaSa+ΣSaSa]]=(ΣSaSa].$

Therefore, S is left weakly regular.

(ii) It can be proved in a similar way of (i).

### 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.

Proof

Assume that S is left weakly regular. Let I be an ordered ideal of S and let xI. By assumption, x ∈ (∑SxSx] ⊆ (∑SISx] ⊆ (∑Ix] = (Ix]. Hence, I is left pure.

Conversely, let aS. By assumption, we obtain that J(a) is left pure. Using Lemma 2.1 and Theorem 3.2(i), we obtain that

$a∈J(a)∩L(a)=(J(a)L(a)]=((Σa+aS+Sa+ΣSaS](Σa+Sa]]⊆(Σa2+aSa+Sa2+ΣSaSa].$

By Lemma 3.12(i), we get that S is left weakly regular.

As a consequence of Theorem 3.13 and the fact that every quasi-pure ordered ideal is both left pure and right pure, we directly obtain the following corollary.

### 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.

Proof

Assume that S is regular. Let I be an ordered ideal of S and let xI. By the regularity of S, we have that x ∈ (xSx] ⊆ (xSxSx] ⊆ (xSISx] ⊆ (xSIx] ⊆ (xIx]. Hence, I is bi-pure.

The converse is obvious since S itself is a bi-pure ordered ideal and so S is regular.

Now, we introduce the notions of left and right weakly pure ordered ideals of ordered semirings and then we characterize fully idempotent ordered semirings using their left and right weakly pure ordered ideals.

### Definition 3.16

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

+ a b c e
a a b c a
b b b c b
c c c c c
e a b c e

and

· a b c e
a a a c e
b a a c e
c a a c e
e e e e e

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 SS = 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:

· a b c e
a a a a e
b a a a e
c c c c e
e e e e e

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 SS = 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 aS;

• (iii) A ⊆ (∑SASAS] for all ∅︀ ≠ AS.

Proof

(i) ⇒ (ii): Assume that S is fully idempotent and let aS. Using Lemma 2.1, we obtain that

$a∈J(a)=(ΣJ(a)2]=(Σ(Σa+Sa+aS+ΣSaS](Σa+Sa+aS+ΣSaS]]⊆(Σ(Σa2+aSa+a2S+ΣaSaS+Sa2+ΣSaSa+ΣSa2S+ΣSaSaS]]⊆(Σa2+aSa+a2S+ΣaSaS+Sa2+ΣSaSa+ΣSa2S+ΣSaSaS].$

Using (3.6), we obtain

$a∈(aS+ΣSas+ΣSaSaS],$$a∈(Sa+ΣSaS+ΣSaSaS].$

Using (3.7) and (3.8), we obtain

$a2=aa∈(aS+ΣSaS+ΣSaSaS](Sa+ΣSaS+ΣSaSaS]⊆(aSa+ΣaSaS+ΣSaSa+ΣSaSaS].$

Using (3.7) and (3.8), we obtain

$aSa⊆(Sa+ΣSaS+ΣSaSaS]S(aS+ΣSaS+ΣSaSaS)⊆(ΣSaSaS].$

Using (3.7), we obtain

$ΣSaSa⊆Σ(SaS(aS+ΣSaS+ΣSaSaS])⊆(ΣSaSaS].$

Using (3.8), we obtain

$ΣaSaS⊆Σ((Sa+ΣSaS+ΣSaSaS]SaS)⊆(ΣSaSaS].$

Using (3.9), (3.10), (3.11) and (3.12), we obtain

$a2∈(aSa+ΣaSaS+ΣSaSa+ΣSaSaS]⊆((ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]]⊆((ΣSaSaS+ΣSaSaS+ΣSaSaS+ΣSaSaS]]=(ΣSaSaS].$

Using (3.13), we obtain

$a2S⊆(ΣSaSaS)S⊆(ΣSaSaS],$$Sa2⊆S(ΣSaSaS]⊆(ΣSaSaS],$$Sa2S⊆S(ΣSaSaS]S⊆(ΣSaSaS].$

Using (3.6), (3.13), (3.10),(3.14), (3.12), (3.15), (3.11) and (3.16), we obtain

$a∈(Σa2+aSa+a2S+ΣaSaS+Sa2+ΣSaSa+ΣSa2S+ΣSaSaS]⊆((ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+ΣSaSaS]⊆((ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]+(ΣSaSaS]]⊆(ΣSaSaS].$

(ii) ⇒ (iii): It is obvious.

(iii) ⇒ (i): Assume that (iii) holds and let I be an ordered ideal of S. Clearly, (∑I2] ⊆ (∑I] = I. On the other hand, I ⊆ (∑SISIS] ⊆ (∑ISI] ⊆ (∑I2]. So, I = (∑I2]. Therefore, S is fully idempotent.

By the proof of Lemma 3.21, we can directly obtain the following corollary.

### Corollary 3.22

Let S be an ordered semiring. If

$a∈(Σa2+aSa+a2S+ΣaSaS+Sa2+ΣSaSa+ΣSa2S+ΣSaSaS]$

for all aS, 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.

Proof

(i) Assume that S is fully idempotent. Let A and I be any ordered ideals of S. By assumption and Lemma 3.21, it turns out that if xAI, then

$x∈(ΣSxSxS]⊆(ΣSASIS]⊆(ΣASI]⊆(ΣAI] andx∈(ΣSxSxS]⊆(ΣSISAS]⊆(ΣISA]⊆(ΣIA].$

So, AI ⊆ (∑AI] and AI ⊆ (∑IA]. Clearly, (∑AI] ⊆ AI and (∑IA] ⊆ AI. Hence, AI = (∑AI] = (∑IA] and thus A is both left and right weakly pure.

(ii) Assume that every ordered ideal of S is left weakly pure. Let aS. Then J(a) is left weakly pure. It follows that J(a) = (∑J(a)J(a)]. By Lemma 2.1, we obtain that

$a∈J(a)=(ΣJ(a)J(a)]=(Σ(Σa+Sa+aS+ΣSaS](Σa+Sa+aS+ΣSaS]]=(Σ(Σa2+aSa+a2S+ΣaSaS+Sa2+ΣSaSa+ΣSa2S+ΣSaSaS]]=(Σa2+aSa+a2S+ΣaSaS+Sa2+ΣSaSa+ΣSa2S+ΣSaSaS].$

By Corollary 3.22, we obtain that S is fully idempotent.

It can be proved analogously if every ordered ideal of S is right weakly pure.

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