Characterizations of ordered h-regular semirings by ordered h-ideals

Abstract

The objective of this paper is to study the ordered h-regular semirings by the properties of their ordered h-ideals. It is proved that each h-regular ordered semiring is an ordered h-regular semiring but the converse does not follow. Important theorems relating to basic properties of the operator clousre and h-regular semirings are given. It is also proved that each regular ordered semiring is an ordered h-regular semiring but the converse does not hold. The classifications of the left and the right ordered h-regular semirings and the left and the right ordered h-weakly regular semirings are also presented.

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Characterizations of ordered h-regular semirings by ordered h-ideals
Article · July 2020
DOI: 10.3934/math.2020370
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AIMS Mathematics, 5(6): 5768–5790.
DOI: 10.3934/math.2020370
Received: 24 March 2020
Accepted: 06 July 2020
Published: 10 July 2020
Research article
Characterizations of ordered h-regular semirings by ordered h-ideals
Rukhshanda Anjum1, Saad Ullah1, Yu-Ming Chu2,3,∗, Mohammad Munir4, Nasreen Kausar5,∗,
and Seifedine Kadry6
1Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan
2Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
3Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering,
Changsha University of Science & Technology, Changsha 410114, P. R. China
4Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
5Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan
6Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University,
Lebanon
*Correspondence: Email: chuyuming@zjhu.edu.cn, kausar.nasreen57@gmail.com; Tel:
+865722322189; Fax: +865722321163.
Abstract: The objective of this paper is to study the ordered h-regular semirings by the properties of
their ordered h-ideals. It is proved that each h-regular ordered semiring is an ordered h-regular semiring
but the converse does not follow. Important theorems relating to basic properties of the operator clousre
and h-regular semirings are given. It is also proved that each regular ordered semiring is an ordered
h-regular semiring but the converse does not hold. The classifications of the left and the right ordered
h-regular semirings and the left and the right ordered h-weakly regular semirings are also presented.
Keywords: semiring; ordered semiring; ordered h-regular; ordered h-ideal
Mathematics Subject Classification: 16Y99, 16Y60
1. Introduction
Von Neumann gave the idea of regularity in rings in 1935 [1] and showed that if the semigroup,
(S,·) is regular, then the ring (S,+,·) is also regular [1]. In 1951, Bourne showed if ∀x∈Sthere
exist a,b∈Ssuch that x+xax =xbx, then semiring (S,+,·) is also regular [2]. In structure theory
of semirings, ideals play a vital role [3]. In [4], Xueling Ma and Jianming Zhan used the concept
of h-ideals. They used the basic and main concept of h-ideals to prove many properties and results.
Similarly, Jianming Zhan et al., in [5] also used h-ideals in their researches. This class of h-ideals has

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been used in many researches by different researchers. Ideals of semirings used in the structure theory
play an important role in many aspects. Some properties of ideals are discussed in [6–8]. Gan and
Jiang [9] studied the ordered semirings containing 0. Han and others in [10] discussed also the ordered
semirings. Iizuka [11] introduced a new type of ideals namely h-ideals. In [12–14] they used h-ideals
for many purposes related to their researches.
Main and basic concepts related to ordered semirings are given by Gan and Jiang [9]. The authors
also derived some ideas related to minimal ideal, maximal ideal, ordered ideal of an ordered semiring
and simple ordered semirings. Han, Kim and Neggers [10] also worked on semirings by partial ordered
set. Munir and Shafiq [19] characterized the regular semirings through m-ideals. Satyt Patchakhieo
and Bundit Pibalijommee [15] gave the basic definition of ordered semirings and left and right ordered
ideal of the ordered semirings. They used two definitions in their properties and applications to prove
their result.
Keeping in view the different characterizations of the regular semirings by the properties of the
h-ideals, we were motivated to characterize the ordered h-regualr semirings by the properties of their
ordered h-ideals. For this purpose, this paper represents ordered h-regular semirings along with their
ordered h-ideals. In Section 2, we give some basic definitions which will be used in our further course
of work. In Section 3, we characterize the ordered h-ideals semirings by their ordered h-ideals. In
Section 4, we characterize the ordered h-regular semirings, and in Section 5, the characterization of
the ordered h-weakly regular semirings is given. The conclusion of the paper is presented in the final
Section 6.
2. Preliminaries
Definition 1. A non-empty set S together with two binary operations +and ·satisfying the following
properties:
(C1)(S,+)is a semigroup,
(C2)(S,·)is a semigroup,
(C3)Distributive laws hold in S, that is
t1·(t2+t3)=t1·t2+t1·t3,
and
(t1+t2)·t3=t1·t3+t2·t3for all t1,t2,t3∈S,
is called a semiring, which is denoted by (S,+,·).
Definition 2. (S,+,·)is additively commutative ifffor all x1,y1∈S,x1+y1=y1+x1. S is
multiplicatively commutative ifffor all x1,y1∈S,x1·y1=y1·x1.(S,+,·)is called a commutative
semiring iffit is both additively commutative and multiplicatively commutative. Suppose (S,+,·)is a
semiring, if ∀a∈S;a+0=a=0+a and a ·0=0=0·a, then 0∈S is called absorbing zero in S.
Definition 3. [15] Let E ,∅and (S,+,·)is a semiring, E ⊆S , is a left ideal or right ideal if these
properties are satisfied:
(I1)t1+t2∈E for all t1,t2∈E.
(I2)S E ⊆E or ES ⊆E.
If E is left ideal and right ideal of S, then E is an ideal of S .
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Definition 4. [15] Suppose (S,≤)is a partially ordered set satisfying the following properties:
(T1)(S,+,·)is semiring,
(T2)if x1≤x2, then x1+e≤x2+e,
(T3)if x1≤x2, then x1e≤x2e and ex1≤ex2,
for all x1,x2,e∈S , then, (S,+,·,≤)is an ordered semiring.
Definition 5. [15] Suppose (S,+,·,≤)is an ordered semiring. Let E ,∅, F ,∅be subsets of S , then
we denote (E]={g∈S|g≤r for some r ∈E}and EF ={gh|g∈E,h∈F}.
We can write (S,+,·,≤)as S .
Definition 6. [15] Suppose S is an ordered semiring, E ,∅and E ⊆S satisfies the following
properties:
(T1)E is left ideal or right ideal of S;
(T2)if g ≤w for some w ∈E, then g ∈E.
Then E is a left ordered ideal or right ordered ideal.
If E is both left ordered ideal and right ordered ideal of S, then E is ordered ideal of S.
Definition 7. Suppose S is an ordered semiring, if x1∈S , there exist t ∈S such that x1≤x1tx1, then
S is called a regular ordered semiring.
3. Ordered h-ideals semirings
In this section, we characterize the ordered h-ideals semirings by their ordered h-ideals.
Definition 8. Suppose E is a nonempty subset of an ordered semiring S , then E is a left ordered h-ideal
of S if the following properties are satisfied:
(1) E is a left ordered ideal of S ,
(2) if e +x1+t=x2+t for some x1,x2∈E,t∈E, then e ∈E.
Similarly, we define the right ordered h-ideal.
If Eis both a left ordered h-ideal and a right ordered h-ideal of S, then Eis said to be an ordered
h-ideal of S.
Definition 9. Suppose E ,∅, E ⊆S and S is an ordered semiring, then the h-closure of E, denoted by
E, is defined by
E={g∈S,there exist x1,x2∈E,g+x1+h≤x2+h,h∈E}.
Definition 10. Suppose S is an ordered semiring. If for every x1∈S , there exist e,h,c∈S such that
x1+x1ex1+c≤x1hx1+c, . Then S is called h-regular ordered semiring.
Definition 11. Suppose S is an ordered semiring then x1∈S is said to an ordered h-regular if x1∈
(x1S x1]. If each element of S is ordered h-regular, then S is said to be an ordered h-regular semiring.
It is easy to see that each h-regular ordered semiring is an ordered h-regular semiring but converse
does not hold. We see this by the following example.
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Example 1. Suppose S ={t1,t2,t3}. Define binary operations ·and +on S as:
+t1t2t3
t1t1t1t1
t2t1t2t3
t3t1t3t3
and
·t1t2t3
t1t2t2t2
t2t2t2t2
t3t2t2t2
We define order relation ≤on S as follows :
≤=t1,t1,t2,t2,t3,t3,t1,t2,t1,t3,t2,t3.
Then (S,+,·,≤)is an ordered semiring. Furthermore, forall a ∈S(1) a+t1+c≤t2+c, c ∈S(2)
t1,t2∈(aS a]i.e. t1≤asa , t2≤asa, for some asa ∈aS a. Hence S is an ordered h-regular semiring.
On the other hand t3+t3at3+t2≤t3ct3+t2has no solution, so S is not an h-regular ordered semiring.
Lemma 1. Suppose S is an ordered semiring and E ⊆S and F ⊆S , where E and F are nonempty, then
(1) (E]⊆(E].
(2) If E ⊆F, then E ⊆F.
(3) (E]F⊆(EF]and E(F]⊆(EF].
Proof. (1) Let g∈(E]. Then there exists h∈Esuch that g≤h.Since h∈E, then there exist r1,r2∈E
such that h+r1+k≤r2+k,k∈E.It follows that g+r1+k≤h+r1+k≤r2+k.Since E⊆(E],
r1,r2∈(E],k∈(E],g∈(E],i.e. (E]⊆(E].
(2) Consider E⊆F. Let g∈E.Then, there exist r1,r2∈Esuch that g+r1+k≤r2+k,k∈E.By
the assumption, we get r1,r2,k∈F.This implies g∈F, so E⊆F.
(3) Let g∈(E]and w∈F. So, there exist p,q∈(E]such that g+p+s≤q+s,s∈(E]. So,
gw +pw +sw ≤qw +sw. Since p,q,s∈(E],p≤r1and q≤r/
1and s≤r//
1,for some r1,r/
1,r//
1∈E,
so pw ≤r1w∈EF and qw ≤r/
1w∈EF and sw ≤r//
1w∈EF.This implies that gw ∈(EF]. So
(E]F⊆(EF]. Similarly we get E(F]⊆(E F].
Lemma 2. [15] Suppose E ⊆S , where E is nonempty and S is an ordered semiring. If E is closed
under addition, then so are (E],(E].
Now we will use further throughout the section N(set of all positive integers). Let S be ordered
semiring, E,∅and E⊆S, suppose Pf inite Ebe set of all finite sum of elements of E, and for x∈S,
let Nx ={nx|n∈N}.
Lemma 3. Suppose E and F are nonempty subsets of an ordered semiring S , with E +E⊆E and
F+F⊆F. Then
(1) E⊆(E]⊆E⊆(E],
(2) (E]=(E], if E is left ordered h-ideal (or right ordered h-ideal) of S,
(3) E+F⊆E+F⊆E+F,
(4) (E]+(F]⊆(E]+(F]⊆(E+F],
(5) E F ⊆(E] (F],
(6) If E and F are two left ordered h-ideal and right ordered h-ideal of S , respectively then (E] (F]⊆
Pf inite E Fi.
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Proof. (1) We see that E⊆(E].
Let g∈(E], so by definition of “( ]”, there exists r∈Esuch that
g≤r
g+r+r≤r+r+r.
This implies that g∈E=⇒(E]⊆E.
Since E⊆(E],
=⇒E⊆(E].
(2) Let Eis left ordered h-ideal (or right ordered h-ideal) of S.
By(i), (E]⊆(E]; =⇒(E]⊆(E].
Let g∈(E], then by definition of h-closure, there exist h,k∈(E]such that
g+h+s≤k+s,s∈(E].
Since h,k,s∈(E],then by definition of h-closure, there exist r1,r2,r3,r4,r5,r6∈(E],such that
h+r1+s1≤r2+s1,s1∈(E]
k+r3+s2≤r4+s2,s2∈(E]
s+r5+s3≤r6+s3,s3∈(E]
=⇒
g+h+s+r1+s1+r3+s2+r5+s3≤k+s+r1+s1+r3+s2+r5+s3
≤r4+s2+s+r1+s1+r5+s3
≤r4+s2+r6+s3+r1+s1
=r1+r4+r6+s1+s2+s3
=⇒
g+(h+s+r1+r3+r5)+(s1+s2+s3)≤(r1+r4+r6)+(s1+s2+s3).
Since s1,s2,s3∈(E], then by definition of “( ]”, there exist s/
1,s/
2,s/
3∈Esuch that s1≤s/
1and
s2≤s/
2and s3≤s/
3.
=⇒s1+s2+s3≤s/
1+s/
2+s/
3
As Eis a left ordered h-ideal (or right ordered h-ideal) of S, so Eis a left ordered ideal (or right
ordered ideal) of S.
Then by definition of left ordered ideal or right ordered ideal, we get
s1+s2+s3∈E.
Now, since r1,r4,r6∈(E],so by using definition of “( ]”, there exist r/
1,r/
4,r/
6∈E,such that r1≤r/
1
and r4≤r/
4and r6≤r/
6
=⇒r1+r4+r6≤r/
1+r/
4+r/
6.
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Then by definition of left ordered ideal or right ordered ideal, we get
r1+r4+r6∈E.
Then by definition of left ordered ideal or right ordered ideal, we get r1,r2,r5,r6,s1,s3∈E
h+s+(r1+r5+s1+s3)≤(r2+r6+s1+s3)
g+(h+s+r1+r3+r5)+(s1+s2+s3)≤(r1+r4+r6)+(s1+s2+s3)
where s1+s2+s3∈E,then by definition of h-closure, we get g∈E,then by (1), we get g∈E⊆
(E]=⇒g∈(E], =⇒(E]⊆(E]
=⇒(E]=(E]
(3) From (1), we have E⊆Eand F⊆F
=⇒E+F⊆E+F.
Now we show E+F⊆E+F.Suppose g∈E+F,so there exists h∈Eand k∈Fsuch that
g=h+k.
Since h∈E,k∈F,so by using definition of h-closure, there exist r,r/∈Eand w,w/∈Fsuch that
h+r+s1≤r/+s1,s1∈E,
and
k+w+s2≤w/+s2,s2∈F.
=⇒
g+r+s1+w+s2=h+k+r+s1+w+s2
g+(r+w)+(s1+s2)≤r/+s1+w/+s2
=r/+w/+s1+s2
g+(r+w)+(s1+s2)≤r/+w/+(s1+s2)
As (s1+s2)∈E+F,then by definition of h-closure, we get g∈E+F,
=⇒E+F⊆E+F
(4) Let g∈(E]+(F],then there exists p∈(E],q∈(F],such that g=p+q.
Now,
g+(p+q)+h=(p+q)+(p+q)+h=⇒g+(p+q)+h=(p+p)+(q+q)+h.
Since p+p∈(E] and q+q∈(F],then by definition of h-closure, we get g∈(E]+(F],
=⇒(E]+(F]⊆(E]+(F].
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Suppose g∈(E]+(F], so there exists p∈(E],q∈(F],such that g=p+q.
Since p∈(E] and q∈(F], so by using definition of h-closure, there exist r,r/∈(E]and w,w/∈(F]
such that
p+r+s1≤r/+s1,s1∈(E],
and
q+w+s2≤w/+s2,s2∈(F].
g+r+w+s1+s2=p+q+r+w+s1+s2
g+r+w+s1+s2≤r/+w/+s1+s2.
Since (r+w),r/+w/∈(E+F],
g+(r+w)+(s1+s2)≤r/+w/+(s1+s2).
This implies g∈(E+F],=⇒
(E]+(F]⊆(E+F]
(E]+(F]⊆(E+F]
(E]+(F]⊆(E+F].
(5) By Lemma 1,we get
E F ⊆(E] (F].
(6) Let E,Fare two left ordered h-ideal or right ordered h-ideal of S. We will prove that (E]
(F]⊆Pf i j EFi.
For this, let g∈(E](F]then g=hk,as h∈(E],k∈(F],then by definition of h-closure, there exist
p,p/∈(E]and q,q/∈(F]such that
h+p+s1≤p/+s1,s1∈(E],
and
k+q+s2≤q/+s2,s2∈(F].
As
hk +pk +s1k≤p/k+s1k.
Also
pk +pq +ps2≤pq/+ps2and p/k+p/q+p/s2≤p/q/+p/s2.
As
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g=hk
g+pk +pq +p/q+s1k+p/s2+ps2=hk +pk +pq +p/q+s1k+p/s2+ps2
≤p/k+s1k+pq +p/q+p/s2+ps2
≤p/q/+p/s2+s1k+pq +ps2
=pq +p/q/+s1k+p/s2+ps2.
Since Eand Fare left ordered ideal and right ordered ideal of Srespectively, therefore,
pk +pq +p/q∈X
f inite
EF,pq +p/q/∈X
f inite
EF,
and
s1k+p/s2+ps2∈X
f inite
EF.
This implies that
g+pk +pq +p/q+s1k+p/s2+ps2≤pq +p/q/+s1k+p/s2+ps2.
So
g∈X
f inite
EF ⊆






X
f inite
EF







=⇒g∈






X
f inite
EF







.
=⇒(E](F]⊆






X
f inite
EF







.

Example 2. (i) Every regular ordered semiring is an ordered h-regular semiring.
(ii) Consider the semiring (N,+,·,≤), where Nis the set of natural numbers. We define the relation
on Nby gh⇔g≥hfor all g,h∈N. Then, (N,)is a partially ordered set, furthermore
(N,+,·,)is an ordered semiring. Since g+ghg +sghg +sfor all g,h∈N,s∈N,(N,+,·,)is
an ordered h-regular semiring. Moreover, since 2∈N,22h2=4hfor all h∈N,(N,+,·,)is not
a regular ordered semiring. In addition, we get (2N]is an ordered ideal of (N,+,·,)which is not
an h-ideal, for the reason that 2+4+1=3+4as 1<(2N].
Theorem 1. Suppose S is an ordered semiring and E be left ideal or right ideal or ideal, then
conditions given below are equivalent:
(1) E is left ordered h-ideal or right ordered h-ideal or ordered h-ideal of S ;
(2) Let g ∈S,g+r1+h≤r2+h for some r1,r2∈E, h ∈E then g ∈E;
(3) E=E.
Proof. (1)=⇒(2) Suppose Eis a left ordered h-ideal. Suppose g∈Ssuch that g+r1+h≤r2+hfor
some r1,r2∈E,h∈Ethen by using definition of left ordered h-ideal, we get g∈E.
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(2)=⇒(3) Suppose (2) is true. Consider g∈E, then there exist r1,r2∈Esuch that g+r1+h≤r2+h,
h∈E.By condition (2), we get g∈E. So, E⊆E. Since E⊆E, therefore E=E.
(3)=⇒(1) Assume that E=E. Let g∈Sbe such that g+r1+h≤r2+hfor some r1,r2∈E,
h∈E. Then g∈E.Since E=E, so g∈E=E.Thus g∈E.Since g+r1+h≤r2+hfor some
r1,r2∈E,h∈E, then g∈E, so by the definition of left ordered h-ideal or right ordered h-ideal or
ordered h-ideal, we get Eis left ordered h-ideal or right ordered h-ideal or ordered h-ideal of S.
Theorem 2. Suppose S is an ordered semiring, then:
(1) Intersection of any family of left ordered h-ideals of S is a left ordered h-ideal.
(2) Intersection of any family of right ordered h-ideals of S is a right ordered h-ideal.
(3) Intersection of any family of ordered h-ideals of S is an ordered h-ideal.
Proof. (1) Suppose Enis a left ordered h-ideal of Sfor all n∈J, as ∩n∈JEn,∅. Since Enis a left
ordered h-ideal, we get Enis a left ordered ideal for all n∈J. Then ∩n∈JEnis left ordered ideal.
Consider there exist g∈Sand r1,r2∈ ∩n∈JEn,h∈ ∩n∈JEnis such that g+r1+h=r2+h.Since
∩n∈JEn⊆Enfor all n∈J, we get, r1,r2,h∈En. Since Enis a left ordered h-ideal and r1,r2∈En,
g+r1+h=r2+h,h∈Enfor all n∈J,so by using definition of left ordered h-ideal, we get g∈Enfor
all n∈J. So g∈ ∩n∈JEn. Therefore, r1,r2∈ ∩n∈JEn,g+r1+h=r2+h,h∈ ∩n∈JEn. Then g∈ ∩n∈JEn.
By definition of left ordered h-ideal, we get ∩n∈JEnis a left ordered h-idealof S.
(2) Suppose that Enis a right ordered h-ideal of Sfor all n∈J, as ∩n∈JEn,∅. Since Enis right ordered
h-ideal, we get, Enis right ordered ideal for all n∈J. Then ∩n∈JEnis right ordered ideal. Consider that
there exist g∈Sand r1,r2∈ ∩n∈JEn,h∈ ∩n∈JEnsuch that g+r1+h=r2+h. Since ∩n∈JEn⊆Enfor
all n∈J, we have r1,r2,h∈En. Since Enis a right ordered h-ideal and r1,r2∈En,g+r1+h=r2+h,
h∈Enfor all n∈J. So by using the definition of right ordered h-ideal, we get g∈Enfor all n∈J. So
g∈ ∩n∈JEn. Therefore, r1,r2∈ ∩n∈JEn,g+r1+h=r2+h,h∈ ∩n∈JEn. Then g∈ ∩n∈JEn. By definition
of right ordered h-ideal, we have ∩n∈JEnis a right ordered h-idealof S.
(3) From (1) and (2), we get ∩n∈JEnis a left and right ordered h-ideal of S. Therefore, ∩n∈JEnis an
ordered h-ideal of S. Hence proved. 
Remark 1. (1) The sum of two left ordered h-ideals or right ordered h-ideals is a left ordered h-ideal
or right ordered h-ideal.
(2) The sum of two left ordered ideals or right ordered ideals is a left ordered ideal or right ordered
ideal.
(3) The sum of two left ideals or right ideals is a left ideal or right ideal.
Theorem 3. Suppose S is an ordered semiring and E ,∅,F,∅,E⊆S,F⊆S.
(1) Consider E,F be two left ordered h-ideals, then (E+F]is smallest left ordered h-ideal containing
E∪F.
(2) Consider E,F be two right ordered h-ideals, then (E+F]is smallest right ordered h-ideal
containing E ∪F.
(3) Consider E,F be two ordered h-ideals, then (E+F]is smallest ordered h-ideal containing E ∪F .
Proof. (1) Suppose E,Fare two left ordered h-ideal of S. Suppose g,h∈(E+F],s∈S.
By the definition of h-closure, there exist r,r/,w,w/∈(E+F]such that
g+r+f1≤r/+f1,f1∈(E+F],
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and
h+w+f2≤w/+f2,f2∈(E+F].
Hence
g+h+r+w+f1+f2≤r/+w/+f1+f2
and
sg +sr +s f1≤sr/+s f1
As (s f1)∈(E+F], so by using the definition of h-closure, we get (g+h)∈(E+F]and (sg)∈
(E+F].
This implies
(E+F]⊆(E+F]
Let g∈(E+F],then by definition of h-closure, there exist i,v∈(E+F]such that
g+i+f≤v+f,f∈(E+F].
Since i,v,f∈(E+F], then by definition of h-closure, there exist r,r/,w,w/,d,d/∈(E+F],such
that
i+r+f1≤r/+f1,f1∈(E+F]
v+w+f2≤w/+f2,f2∈(E+F]
and
f+d+f3≤d/+f3,f3∈(E].
Now,
g+i+f+r+w+d+f1+f2+f3≤v+f+r+w+d+f1+f2+f3
≤w/+f2+f+r+d+f1+f3
=f+d+f3+w/+f2+r+f1
≤d/+f3+w/+f2+r+f1
=r+w/+d/+f1+f2+f3.
Since r+w/+d/,(i+f+r+w+d)∈E+Fand (f1+f2+f3)∈E+F, then by definition of
h-closure, we get g∈E+F⊆(E+F]=⇒(E+F]⊆(E+F].
So, we get (E+F]=(E+F].
This shows that (E+F]is a left ordered h-ideal.
Suppose g∈E∪F,then g∈Eor g∈F
As g∈E,then g+(g+w)=(g+g)+w∈E+F, for all w∈F. Thus g∈(E+F]
As g∈F,then (r+g)+g=r+(g+g)∈E+F, for all r∈E. Thus g∈(E+F]
Hence,
E∪F⊆(E+F]
Suppose Lis a left ordered h-ideal containing E∪F.
Then E+F⊆Land hence (E+F]⊆(L]=Limplies that (E+F]⊆L=L
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Therefore, (E+F]is the smallest left ordered h-ideal containing E∪F.
(2) This is similar to (1).
(3) From (1) and (2), we prove that (E+F]is smallest left and right ordered h-ideal containing
E∪F. Therefore, (E+F]is smallest ordered h-ideal containing E∪F.
Theorem 4. Suppose S is an ordered semiring and E ,∅, E ⊆S . Then these properties hold.
(1) Consider E a left ideal, then (E]is the smallest left ordered h-ideal containing E.
(2) Consider E a right ideal, then (E]is the smallest right ordered h-ideal containing E.
(3) Consdier E an ideal, then (E]is the smallest ordered h-ideal containing E.
Proof. Suppose Eis a left ideal. We know that (E] is closed with respect to the operation of addition.
Suppose g∈(E],and k∈E,then by using definition of h-closure, there exist r,w∈(E]such that
g+r+h≤w+h,h∈(E].
Hence
kg +kr +kh ≤kw +kh.
So by using definition of “( ]”, we have kh ∈(E].Since (kr),(kw)∈(E],kg +(kr)+(kh)≤
(kw)+(kh),(kh)∈(E].
Then by definition of h-clousre, we get kg ∈(E].Therefore,(E]is a left ordered h-ideal.
We know that (E]is a left ordered h-ideal containing E.
Suppose Qis a left ordered h-ideal containing E. So (E]⊆(Q]=Q. Then, (E]⊆Q=Q.
Therefore, (E] is the smallest left ordered h-ideal containing E.
(2) This is similar to (1).
(3) From (1) and (2), we prove that (E]is the smallest left and right ordered h-ideal containing E.
Therefore, (E]is the smallest ordered h-ideal containing E.
Corollary 1. Suppose S is an ordered semiring, let ∅,E⊆S . We denote the smallest left ordered h-
ideal containing E by Lh(E), the smallest right ordered h-ideal containing E by Rh(E), and the smallest
ordered h-ideal of S containing E by Mh(E). Then, the following results follows:
(1) Lh(E)=(Pf inite E+Pf inite S E],
(2) Rh(E)=(Pf inite E+Pf inite ES ],
(3) Mh(E)=(Pf inite E+Pf inite S E +Pf inite ES +Pf inite S ES ].
Proof. We want to prove Pf inite E+Pf inite S E is a left ideal.
For this,
(I1)Let a,b∈Pf inite E+Pf inite S E.Then
a+b∈X
f inite
E+X
f inite
S E
(I2)Let a∈Pf inite E+Pf inite S E,r∈S
=⇒
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5779
ra ∈r






X
f inite
E+X
f inite
S E







=X
f inite
rE +X
f inite
rS E
⊆X
f inite
E+X
f inite
S E
=⇒ra ∈X
f inite
E+X
f inite
S E
Therefore, Pf inite E+Pf inite S E is a left ideal. By Theorem 4,we get
Lh(E)=(X
f inite
E+X
f inite
S E]
We see that the proofs of (2) and (3)are similar to that of (1).
Corollary 2. Suppose S is an ordered semiring, let r ∈S . Then
(1) Lh(r)=(Nr +S r];
(2) Rh(r)=(Nr +rS ];
(3) Mh(r)=(Nr +S r +rS +Pf inite S rS ],where N =Set of natural numbers.
4. Ordered h-regular semirings
We now give the characterization of ordered h-regular semirings by their ordered h-ideals.
Theorem 5. Consider S is an ordered semiring. Then S is an ordered h-regular iffE∩F=(EF], for
all right ordered h-ideals E, left ordered h-ideals F of S .
Proof. Suppose Sis an ordered h-regular semiring and Eis right ordered h-ideal, F is left ordered
h-ideal of S. Then, we have EF ⊆Eand EF ⊆F. Thus, (EF]⊆(E]=Eand (EF]⊆(F]=F.
This implies (EF]⊆E=Eand (E F]⊆F=F.Thus (EF]⊆E∩F
Let p∈E∩F.As Sis an ordered h-regular, there exist h,k∈(pS p,such that
p+h+o≤k+o,o∈(pS p.
Since h,k,o∈(pS p, then by definition of “( ]”, there exist s,j,j1∈Ssuch that
h≤ps p,k≤p j p,o≤p j1p.
Since Eis a right ordered h-ideal, Fis a left ordered h-ideal, we have psp,p jp,p j1p∈EF.Since
h≤ps p ∈EF,k≤p jp ∈E F,o≤p j1p∈EF , so by using definition of “( ] ”, we have h,k,o∈(EF]
so p∈(EF]=⇒E∩F⊆(E F]
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=⇒E∩F=(EF]
Conversely, Consider E∩F=(E F]for all right ordered h-ideals Eof S, left ordered h-ideals Fof
S. Suppose d∈S,then by above Corollary 2 we get
Lh(d)=(Nd +S d] and Rh(d)=(Nd +dS ].
By assumption, (Rh(d)∩Lh(d)=(Rh(d)Lh(d)]).Now we show that Rh(d)Lh(d)⊆(S d]∩(dS ]
Let p∈Rh(d) and h∈Lh(d).
Since p∈Rh(d), then by the definition of h-closure,
There existi,i/∈(Nd +dS ],such that p+i+o1≤i/+o1,o1∈(Nd +dS ].
Since h∈Lh(d), then by the definition of h-closure,
There exist v,v/∈(Nd +S d],such that h+v+o2≤v/+o2,o2∈(Nd +S d],
ph +ih +o1h≤i/h+o1h.
Since h,k,o∈(pS p, then by definition of “( ]”, there exist s,j,j1∈Ssuch that
i≤ed +ds,i/≤f d +d j,o1≤gd +d j1
It follows that
ih ≤edh +dsh =d(eh +sh)∈dS
i/h≤f dh +d jh =d(f h +jh)∈dS
o1h≤gdh +d j1h=d(gh +j1h)∈dS .
Then by definition of “( ]”, we get ih,i/h,o1h∈(dS ]. Since ih,i/h∈(dS ],
ph +ih +o1h≤i/h+o1h,o1h∈(dS ].
Then by definition of h-closure, we get ph ∈(dS ].So,
Rh(d)Lh(d)⊆(dS ]
Similarly, we can show that
Rh(d)Lh(d)⊆(S d]
Therefore,
Rh(d)Lh(d)⊆(S d]∩(dS ].
Since (dS ] is a right ordered h-ideal, (S d] is a left ordered h-ideal and by assumption, we have,
(dS ]∩(S d]⊆(dS ]∩(S d]i.
Now we will show that (dS ](S d]⊆(dS d].For this let m∈(dS ] and k∈(S d],then by definition of
h-closure, there exist l,l1∈(dS ] and q,q1∈(S d],such that
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5781
m+l+o3≤l1+o3,o3∈(dS ]
and
k+q+o4≤q1+o4,o4∈(S d]
From above equations we get ,
mk +lk +o3k≤l1k+o3k
lk +lq +lo4≤lq1+lo4
l1k+l1q+l1o4≤l1q1+l1o4
o3k+o3q+o3o4≤o3q1+o3o4.
Since S is a multiplicatively commutative ordered semiring,
mk +(lk +lq +l1q)+(o3k+lo4+l1o4)≤l1k+o3k+lq +l1q+lo4+l1o4
≤l1q1+l1o4+o3k+lq +lo4
=(l1q1+lq)+(o3k+lo4+l1o4).
Since l,l1,o3∈(dS ],q,q1,o4∈(S d],then by definition of “( ]”, there exist s1,s2,s3,l/,l//,l/// ∈S,
such that
l≤ds1,l1≤ds2,o3≤ds3and q≤l/d,q1≤l// d,o4≤l///d.
Hence, we obtained,
lk +lq +l1q≤l1q+lo4+lq1+lo4
≤ds2l/d+ds1l///d+ds1l//d+d s1l/// d∈dS d
=⇒
lk +lq +l1q∈(dS d]
and
o3k+lo4+l1o4≤o3q+o3o4+lo4+l1o4+o3q1+o3o4
≤ds3l/d+ds3l///d+ds1l///d+d s2l/// d+ds3l//d+ds3l///d∈dS d.
Then by definition of “( ]”,
o3k+lo4+l1o4∈(dS d]and l1q1+lq ≤d s2l// d+ds1l/d∈dS d
=⇒
l1q1+lq ∈(dS d].
So, mk ∈(dS d]. Hence (dS ](S d]⊆(dS d].
Hence
(dS ]∩(S d]⊆(dS ]∩(S d]i⊆(dS d]=(dS d].
Thus,
Rh(d)∩Lh(d)=(Rh(d)Lh(d)]⊆(dS ]∩(S d]i⊆(dS d]=(dS d]
It turns out, d∈(dS d].Hence, Sis an ordered h-regular. 
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Corollary 3. Suppose S is a commutative ordered semiring. Then S is an ordered h-regular ifffor
each ordered h-ideal E of S , E =(E2].
Proof. Suppose Sis an ordered h-regular. Consider Eis an ordered h-ideal of S. Obviously we have
E=E∩E=(E2].
Conversely, Suppose that for each ordered h-ideal E of S, E=(E2].Consider r∈S. As Sis a
commutatively multiplication ordered semiring, we get
Mh(r)=Lh(r)=Rh(r).
So,
r∈Mh(r)=(Mh(r)Mh(r)]
=(Rh(r)Lh(r)]
=(Nr +rS ](Nr +S r]i
⊆






X
f inite
(Nr +rS ](Nr +S r]







⊆(rS ]i=(rS ].
Since Sis commutative i.e., (rS ]=(S r], and (rS ] is an ordered h-ideal, therefore
r∈(rS ]=((rS ]2=(rS ](rS ]i
=(rS ](S r]i
⊆







(X
f inite
rS S r]







⊆(rS r]i=(rS r].
Thus, Sis an ordered h-regular. 
Definition 12. Suppose S is an ordered semiring, let r ∈S . Suppose r ∈(S r2],then r is said to be left
ordered h-regular. Suppose r ∈(r2S],then r is called right ordered h-regular. Suppose each element
of S is left or right ordered h-regular. Then, ordered semiring S is said to be a left or right ordered
h-regular.
Theorem 6. Suppose S is left ordered h-regular semiring. Then
(1) for each left ordered h-ideal E of S, (E2]=E;
(2) Q∩E=(QE], for each left ordered h-ideal E and each ordered h-ideal Q of S .
Proof. (1) Suppose Eis left ordered h-ideal of S. Then, we get (E2]⊆(E]=E.
Suppose r∈E. As Sis a left ordered h-regular, so r∈(S r2].
Since (S r2]⊆(S E2]⊆(E2],=⇒r∈(E2]. Hence, E⊆(E2]=⇒(E2]=E.
(2) Let Eis left ordered h-ideal and Qis ordered h-ideal of S. Then, we get, (QE]⊆(Q]=Qand
(QE]⊆(E]=E.Hence, (QE]⊆Q∩E. Let a∈Q∩E. As Sis left ordered h-regular,
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5783
a∈(S a2]⊆(S QE]⊆(QE].
=⇒a∈(QE]. Hence, Q∩E⊆(QE].Thus,
=⇒Q∩E=(QE].

Theorem 7. Suppose S is a right ordered h-regular semiring. Then
(1) for each right ordered h-ideal E of S, (E2]=E;
(2) E∩Q=(EQ], for each right ordered h-ideal E and each ordered h-ideal Q of S .
Theorem 8. Suppose S is an ordered semiring,then the conditions given below are equivalent:
(1) for each left ordered h-ideal E, F of S , E ∩F=(EF].
(2) for each left ordered h-ideal E and each ordered h-ideal Q of S , E ∩Q=(E Q].
(3) S is left ordered h-regular and Rh(E)⊆Lh(E)for all ∅,E⊆S
Proof. (1) ⇒(2) Let Eis left ordered h-ideal of Sand Fis ordered h-ideal of S. Then E∩F=(EF],
Fbeing left ordered h-ideal of S, we get E∩F=(EF].
(2) ⇒(3) Let ∅,E⊆S. By assumption, we get Lh(E)=Lh(E)∩S=(Lh(E)S]. We have
Rh(E)=(X
f inite
E+X
f inite
ES ]
⊆(X
f inite
Lh(E)+X
f inite
Lh(E)S]
=






X
f inite
(Lh(E)S]+X
f inite
(Lh(E)S]S







⊆






X
f inite
(Lh(E)S]+X
f inite
(Lh(E)S S ]







=






X
f inite
(Lh(E)S]







=X
f inite
Lh(E)
=Lh(E).
Moreover, we show that Lh(E)=Mh(Lh(E)). Since Rh(E)⊆Lh(E), we get, Lh(E)⊆Rh(Lh(E)) ⊆
Lh(Lh(E)) =Lh(E).
Hence, Lh(E)=Rh(Lh(E)). It follows that Lh(E)=Mh(Lh(E))
Let p∈S.From assumption, we get
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5784
p∈Lh(p)∩Mh(p)=(Lh(p)Mh(p)]
=(Lh(p)Mh(Lh(p))]
=(Lh(p)Lh(p)]
⊆(N p2+X
f inite
S p2+X
f inite
pS p +X
f inite
S pS p]
⊆(N p2+X
f inite
S p2+X
f inite
Rh(p)p+X
f inite
S Rh(p)p]
⊆(N p2+X
f inite
S p2+X
f inite
Lh(p)p+X
f inite
S Lh(p)p]
⊆(N p2+X
f inite
S p2+X
f inite
Lh(p)p+X
f inite
Lh(p)p]
=(N p2+X
f inite
S p2+X
f inite
Lh(p)p]
=(N p2+S p2+Lh(p)p]
=N p2+S p2+(N p +S p]pi
⊆N p2+S p2+(N p2+S p2]i
=(N p2+S p2]i
=(N p2+S p2]
=⇒
p∈(N p2+S p2]
Since p∈(N p2+S p2], then by definition of h-closure, there exist v,v/∈(N p2+S p2],such that
p+v+t1≤v/+t1,where t1∈(N p2+S p2].
Since v,v/,t1∈(N p2+S p2],then by definition of “( ]”,there exist e,f,g∈Nand s,r,r1∈Ssuch
that
v≤ep2+sp2,v/≤f p2+r p2,t1≤gp2+r1p2
In a similar way, we obtain
p2∈(N p4+S p4].
Then by definition of h-closure, there exist u,u/∈(N p4+S p4],such that
p2+u+t2≤u/+t2,t2∈(N p4+S p4].
Since u,u/,t2∈(N p4+S p4],then by definition of “( ]”, there exist e/,f/,g/∈Nand s/,r/,r/
1∈S
such that
u≤e/p4+s/p4,u/≤f/p4+r/p4,t2≤g/p4+r/
1p4.
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5785
From p2+u+t2≤u/+t2we get
ep2+eu +et2≤eu/+et2.
Now we have
v+eu +et2≤ep2+sp2+eu +et2
≤eu/+et2+sp2
≤e f /p4+er/p4+et2+sp2
v+eu +et2+f u +f t2≤e f /p4+er/p4+et2+s p2+f u +f t2
≤e f /p4+er/p4+sp2+f(e/p4+s/p4)+et2+f t2
≤e f /p4+er/p4+sp2+f e/p4+f s/p4+e(gp2+r1p2)+f(gp2+r1p2)
≤e f /p4+er/p4+sp2+f e/p4+f s/p4+egp2+er1p2+f gp2+f r1p2∈S p2.
Then by definition of “( ]”, v+eu +et2+f u +f t2∈S p2i.
Now
v/+f u +f t2≤f p2+r p2+f u +f t2
v/+eu +et2+f u +f t2≤f p2+r p2+eu +et2+sp2+f u +f t2
v/+eu +et2+f u +f t2≤f p2+r p2+e(e/p4+s/p4)+e(g/p4+r/
1p4)+sp2+f(e/p4+s/p4)
+f(g/p4+r/
1p4)
v/+eu +et2+f u +f t2≤f p2+r p2+ee/p4+es/p4+eg/p4+er/
1p4+sp2+f e/p4+f s/p4
+f g/p4+f r/
1p4∈S p2.
Then by definition of “( ]”,
=⇒v/+eu +et2+f u +f t2∈S p2i.
Now,
et2+f t2≤e(g/p4+r/
1p4)+f(g/p4+r/
1p4)
=eg/p4+er/
1p4+f g/p4+f r/
1p4∈S p2.
Then by definition of “( ]”,
et2+f t2∈S p2i.
Now
p+(v+eu +et2+f u +f t2)+(et2+f t2)≤v/+eu +et2+f u +f t2+(et2+f t2)
=⇒p∈S p2.
Hence, Sis a left ordered h-regular.
(3) ⇒(1)
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Let E,Fare left ordered h-ideals of S, then we get (EF]⊆(F]=FWe see that E⊆Rh(E)⊆
Lh(E)=E. Hence, Eis an ordered h-ideal. Thus (EF ]⊆(E]=E. So, (EF ]⊆E∩F
Suppose p∈E∩F. By assumption, we get p∈(S p2].Since (S p2]⊆(S EF]⊆(EF],p∈(EF]. It
turns out that E∩F⊆(EF].
Therefore,
E∩F=(EF]

Theorem 9. Suppose S is an ordered semiring. Then the conditions given below are equivalent:
(1) for each right ordered h-ideal E, F of S , E ∩F=(EF].
(2) for each right ordered h-ideal E of S , each ordered h-ideal Q of S , Q ∩E=(QE].
(3) S is right ordered h-regular, Lh(E)⊆Rh(E)for all ∅,E⊆S.
5. Ordered h-weakly regular semirings
Definition 13. Suppose S is an ordered semiring, let r ∈S . Suppose r ∈(Pf init e(S r)2], then r is said
to be a left ordered h-weakly regular. Suppose r ∈(Pf inite (rS )2], then r is said to be a right ordered
h-weakly regular. Suppose each element in S is left or right ordered h-weakly regular, then ordered
semiring S is said to be left or right ordered h-weakly regular.
Theorem 10. Suppose S is an ordered semiring, then the conditions given below are equivalent:
(1) S is a left ordered h-weakly regular.
(2) for each left ordered h-ideal E of S , (Pfinite E2]=E.
(3) for each left ordered h-ideal E of S and each ordered h-ideal Q of S , Q ∩E=(Pf inite QE].
Proof. (1)⇒(2) Suppose Eis a left ordered h-ideal of S. Then, we get, (Pfinit e E2]⊆(E]=E
Let a∈E. By assumption, we have
a∈(X
f inite
S aS a]⊆(X
f inite
S ES E]⊆(X
f inite
E2].
Hence,
E⊆(X
f inite
E2].
Thus,
(X
f inite
E2]=E
(2)⇒(1) Let r∈S. From assumption, Lemma 3, and Corollary 2. We get,
AIMS Mathematics Volume 5, Issue 6, 5768–5790.

5787
r∈Lh(r)=(X
f inite
Lh(r)2]
=






X
f inite
(Nr +S r](Nr +S r]







⊆






X
f inite
(X
f inite
(Nr +S r)(Nr +S r)]







⊆






X
f inite
(S r]







=(S r]i
=(S r].
Since (S r] is left ordered h-ideal, we get r∈(S r]=Pf inite (S r]2.
By Lemma 2 and Theorem 4, we have







X
f inite
(S r]2







⊆






X
f inite
(X
f inite
S rS r]







=







(X
f inite
S rS r]







=(X
f inite
(S r)2].
Hence,
r∈(X
f inite
(S r)2].
Therefore, Sis a left ordered h-weakly regular.
(2)⇒(3) Suppose Eis a left ordered h-ideal of Sand Qis an ordered h-ideal of S. Then,
(X
f inite
QE]⊆(X
f inite
Q]=Q,
and
(X
f inite
QE]⊆(X
f inite
E]=E.
Hence, (Pf inite QE]⊆Q∩E.
Let a∈Q∩E.By assumption, we get,
AIMS Mathematics Volume 5, Issue 6, 5768–5790.

5788
a∈Lh(a)=(X
f inite
Lh(a)2]
⊆






X
f inite
Mh(a)Lh(a)







⊆(X
f inite
QE].
Hence,
Q∩E⊆(X
f inite
QE].
Thus
Q∩E=(X
f inite
QE]
(3) ⇒(2) Suppose Eis a left ordered h-ideal of S, then, we get (Pf inite E2]⊆(E]=E.
By Lemma 1, 2, Theorem 4 and Corollary 1, we get
E=Mh(E)∩E
=(X
f inite
Mh(E)E]
=






X
f inite
(X
f inite
E+X
f inite
S E +X
f inite
ES +X
f inite
S ES ]E







⊆






X
f inite
(X
f inite
EE +X
f inite
S EE +X
f inite
ES E +X
f inite
S ES E]







⊆






X
f inite
(X
f inite
E2]







=







(X
f inite
E2]







=(X
f inite
E2].
Thus,
(X
f inite
E2]=E.

Theorem 11. Suppose S is an ordered semiring, then the conditions given below are equivalent:
(1) S is a right ordered h-weakly regular.
(2) for each right ordered h-ideal E of S , (Pfinite E2]=E.
(3) for each right ordered h-ideal E of S and each ordered h-ideal Q of S , E ∩Q=(Pf inite E Q].
Proof. Straightforward. 
AIMS Mathematics Volume 5, Issue 6, 5768–5790.

5789
6. Conclusions
Concepts of the ordered h- ideals in semirings, alongside their essential properties, were presented.
The classes of the semirings like ordered h-regular and ordered h-weakly regular semirings were
characterized by the properties of the ordered h-ideals.
The ideas of the ordered h-ideals can be extended to the non associative structures like the ones in
( [16–18, 20–22] ). Moreover, ordered h-ideals can be extended for fuzzification in semiring theory.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (Grant Nos.
11971142, 11871202, 61673169, 11701176, 11626101, 11601485).
Conflict of interest
The authors declare no conflict of interest.
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