Article
Kyungpook Mathematical Journal 2020; 60(3): 455465
Published online September 30, 2020
Copyright © Kyungpook Mathematical Journal.
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
email : 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
email : banpib@kku.ac.th
Received: October 30, 2019; Revised: April 2, 2020; Accepted: April 21, 2020
We introduce the concepts of the left purity, right purity, quasipurity, 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
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
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, quasipure and bipure 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
For nonempty subsets
In a particular case of
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
For an element
Lemma 2.1

(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 +∑a ^{2} +aSa ].
An ordered semiring
In this section, we present the notions of left pure, right pure, quasipure, 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
Theorem 3.2

(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.
(i) Assume that
Conversely, let
Hence,
(ii) It can be proved similarly.
Definition 3.3
An ordered ideal
The following remark is directly obtained by Definition 3.1 and 3.3.
Remark 3.4
Every quasipure 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 quasipure.
Example 3.5
Let
and
·  

Define a binary relation ≤ of
Then (
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 quasipure.
Example 3.6
Let
·  

Then (
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
Assume that
Conversely, let
Hence,
Definition 3.8
An ordered ideal
The following remark is directly obtained by Definition 3.3 and 3.8.
Remark 3.9
Every bipure ordered ideal of an ordered semiring is quasipure.
Now, we give an example of a quasipure ordered ideal of an ordered semiring which is not bipure.
Example 3.10
Let
·  

Then (
Theorem 3.11
Assume that
Conversely, let
Hence,
Now, we characterize regular, left weakly regular and right weakly regular semirings using left pure, right pure, quasipure and bipure ordered ideals.
Lemma 3.12

(i)
if a ∈ (∑a ^{2}+aSa +Sa ^{2}+∑SaSa ]for any a ∈S, then S is left weakly regular; 
(ii)
if a ∈ (∑a ^{2} +aSa +a ^{2}S + ∑aSaS ]for any a ∈S, then S is right weakly regular.
(i) Let
Using (
Using (
Using (
Using (
Therefore,
(ii) It can be proved in a similar way of (
Theorem 3.13
Assume that
Conversely, let
By Lemma 3.12(
As a consequence of Theorem 3.13 and the fact that every quasipure ordered ideal is both left pure and right pure, we directly obtain the following corollary.
Corollary 3.14
We note that an ordered ideal of an ordered semiring is bipure if and only if it is a regular subsemiring. Accordingly, we obtain the following remark.
Remark 3.15
An ordered semiring
Assume that
The converse is obvious since
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
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
and
·  

Define a binary relation ≤ on
Then (
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
·  

Then (
Definition 3.20
An ordered semiring
Lemma 3.21

(i)
S is fully idempotent; 
(ii)
a ∈ (∑SaSaS ]for all a ∈S; 
(iii)
A ⊆ (∑SASAS ]for all ∅︀ ≠A ⊆S.
(i) ⇒ (ii): Assume that
Using (
Using (
Using (
Using (
Using (
Using (
Using (
Using (
(ii) ⇒ (iii): It is obvious.
(iii) ⇒ (i): Assume that (
By the proof of Lemma 3.21, we can directly obtain the following corollary.
Corollary 3.22
Theorem 3.23

(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.
(i) Assume that
So,
(ii) Assume that every ordered ideal of
By Corollary 3.22, we obtain that
It can be proved analogously if every ordered ideal of
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