# Characterizations of Left H-Clifford Semirings by Their H-Ideals

Authors: Rukhshanda Anjum; Zeeshan Saleem Mufti; Dilshad Alghazzawi; Nasser M. Al-Zidi; Muhammad Rizwan Ullah Khalid
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The main aim of this research is to introduce Lefth− Clifford Semi-rings. Using some basic properties of h− regular semi-rings we shall investigate several properties of Left h− Clifford semi-rings and their characterizations. We will also establish that a semi-group Q will be a Left Clifford Semi-group iff the semi-group P (Q) of all subsets of Q is a Left h− Clifford Semiring. Also, this research will investigate the distributive congruences of regular semi-rings.


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## Body

## 1. Introduction

Principal thought of semi-group and monoid is given by Wallis [1]. Wallis communicated that a set which fulfills the associative law under some binary operations is called semi-group and a set which is semi-group with identity is called monoid. A few analysts additionally work on semi-group hypothesis like [2–7] talks about the real concept of group and commutative group, as well as other major points. Several scholars have expounded about semi-ring in their studies. In their analysis, they had checked at different aspects of semi-ring. Vandiver [8] discusses semi-ring and clarifies several main concepts. The algebraic system was invented by Vandiver. He imagined that an algebraic structure is made up of a nonempty set that can be manipulated using the binary operations + and ⋅. Semirings play a significant role in geometry, but they also play a role in pure mathematics. Several ideas and results relating rings and semi-rings have been introduced by various researchers like [9–12]. In Applied mathematics and information sciences semi-rings have been set up for tackling the various issues. Semirings with commutative addition and zero element are also very necessary in theoretical computer science. The concept of regular semi-rings was introduced by Von Neumann regularity [13] and Bourne regularity [14]. Von Neumann showed that the ring M,⋅ would also be regular if the semi-group is regular. Bourne showed that if ∀m∈M there exists w′,v′∈M such that m+mw′mt=mv′m then the semi-ring will be regular. Due to their rich structure of Clifford semi-ring, it is obvious to look for classes of regular semi-groups close to Clifford semigroups. Zhu et al. [15] presented the idea of left Clifford semi-groups which sum up Clifford semi-groups. A regular semi-group M is called left semi-group of Clifford if mM⊂Mm∀m∈M. A semi-group M will be a left Clifford semi-group, if and only if it will be a semilattice of left groups. Since its presentation, semi-group section shows their advantage to this new class of semi-groups for large and clear interpretation of structure [16–18]. According to more abstraction of left Clifford semigroups, Shum, Guo and Zhu have presented left C-rpp semi-groups [19]. We additionally suggest to visit [20–26] as an application of fuzzy and intutionistic fuzzy structures in decision making.During the study of semi-rings, the fascinating component is the examination of how the structured characteristics of two reducts are powered by distributive laws to collaborate. Bhuniyaa and Sen carry on their work on the k-regular semi-rings in which the two reducts cooperate more than the normal communication constrained by the distributive laws [27–30]. This research presentes left h-semifields and left h− Clifford semi-rings as abstraction of h-semifields and h-Clifford semi-rings, respectively. Distinct equivalant characterizations and distributive lattice decompositions have been done likewise. It was in 1937 A. H. Clifford (1908–1992) and G. B. Preston, first describe the relation between representation of group and those of a normal subgroup in order to introduce Clifford semigroups. Bhuniya [28] has initiated left k− Clifford semi-rings as a simplification of k− Clifford semi-ring. Being motivated from his work we have introduced Left h− Clifford semi-rings and also described the basic properties of left h-Clifford semi-rings and the equivalent chaacterizations we have committed. We will prove that a semi-group Q will be a Left Clifford semi-group iff the semi-ring PQ of all subsets of Q will be Left h− Clifford semi-ring. We will also characterize the structure of Left h− Clifford semi-rings. A semi-ring M will be left h− Clifford semi-ring iff it will be distributive lattice of the left h-semifields.In first section of this article we will present its introduction and litrature review then in section two basic definitions will be depicted. The third and forth sections are the main research and in the last we will conclude our research.


## 2. Preliminaries

LetM≠∅ then it is said to be semi-ring if it satisfies the following conditions:D1M,+ is a semi-group.D2M,⋅ is a semi-group.D3 Both (left and right) distributive laws hold i.e. m1⋅m2+m3=m1⋅m2+m1⋅m3 and m1+m2⋅m3=m1⋅m3+m2⋅m3;∀m1,m2,m3∈M. Thus; M is semi-ring which is denoted by (M, +, ⋅).Definition 1.
An additive subgroupEof a ringSthenEknown as ideal ofSif it satisfies the following properties:
aE⊆E&Eb⊆E∀a,b∈E. IfEis left and right ideal ofSthenEis said to be ideal of ringS. A semi-groupF′will be known as regular semi-group if∀f′∈F′∃p∈F′such thatf′pf′=f′. ConsiderM,+,.be a semi-ring then for eachm∈M∃a∈Msuch thatmam=mif and only ifMbe a regular semi-ring. LetM,+,.be a semi-ring and let∅≠I⊆Mis a left ideal ofMif the following conditions satisfied:(1)
i1+i2∈I∀i1,i2∈I (2) m.i∈Mform∈M and i∈I.The right ideal ofMis defined dually. IfIis left as well as right ideal ofMthen I will be called as Ideal of semi-ringM.Definition 2.
LetMbe a semi-ring and∅≠W⊆Mwill beh-closure ofWdefined as:(1)W¯=m∈M:m+w1+v=w2+vfor somew1,w2∈W&v∈M.
Suppose thatM,+,. be a semi-ring and W¯ be a h-closure of M then M will be known as h-set if W¯=W.
LetM,+,. be a semi-ring and ∅≠W⊆M is a left ideal of M then W willl be known as left h− ideal if:
W¯=m∈M:m+w1+v=w2+vfor somew1,w2∈W&v∈M=W.. The right h− ideal can be defined dually. Let ∅≠W⊆M then W is called Band if following axioms satisfied: (i) W is semi-group. (ii) each element of W is idempotent. ∀w∈W⇒w2=w.
From [31] we let ∅≠W⊆M then W will be semilattice if W is commutative band. M is often a semi-ring the additive reduct is a semilattice and this class of all such semi-ring is denoted by ML+. A semi-ring (M,+,.) without zero will be known as (left, right) h− simple if it has no proper (left, right) h-ideal. A semi-ring (M ,+,.)with zero will be known as h−0− simple (left, right) if 0&M are its only h− ideals (left, right).Definition 3.
A regular semi-groupMwill be called Clifford semi-group ifEM⊆CMi.e. idempotent ofMcommute with all elements ofM; whereEMis set of all idempotents ofM and(2)CM=c∈M:mc=cm∀m∈M.
Is centre ofM. Let Mbe a semi-group thenMwill be called left Clifford semi-group ifMis regular semi-group as well aswM⊆Mw∀w∈M [18].Definition 4.
A semi-ringMwill be known ash−regular if∀m∈M∃w′,v′∈Msuch thatm+mw′m+p=mv′m+pforp∈M [30]. IfM,+is a semilattice then we can say that a semi-ringM∈ML+is known ash−regular if and only if for eachm∈M∃w′∈Msuch thatm+mw′m+p=mw′m+pforp∈M. LetQbe a semi-group and PQbe the hypersemiring ofQ; where,“+” and “⋅” is defined as: D+F=D∪F and DF=df:d∈D,f∈F∀D,F∈PQthenPQ,+,⋅is a semi-ring whose additive reduct is a semi-lattice that is knwon as hyper semi-ring of semi-groupQ. LetMbe a semi-ring and letm∈Mthen .. will be known ash−inverse ofmifm+mnm+f=mnm+fandn+nmn+f′=nmn+f′for somef,f′∈M. Ifm+mxm+f=mxm+fthen “xmx” ish−inverse ofm. Therefore in anh−regular semi-ring every component has ah−inverse. The set ofh−inverses ofminMis denoted byWhm.
We will inagurated Green’s relation on semi-rings in [27], as introduced by Sen and Bhuniya in the following way: for any u,t∈M; uℒ¯t if and only if Mu¯=Mt¯; uℛ¯t if and only if uM¯=tM¯; uJ¯t if and only if MuM¯=MtM¯ and ℋ¯=ℒ¯∩ℛ¯. These are the equivalence relations of additive congruences on M, where ℒ¯ represents the multiplicative right and ℛ¯ is just multiplicative left and also J¯ is just ideal congruence on M. For an element e of a semi-ring M will be known as h− idempotent if e+e2+f=e2+f for some f∈M. EhM represents the set of all h− idempotent. For all e1,e2∈EhM⇒e1+e2+f∈EhM, but e1⋅e2+f∉EhM. Semiring M in which e1e2+f=e2e1+f∀e1,e2∈EhM for some f∈M [30]. The class Aℐ of semi-rings in ML+ constrained by extra character e+e2≈e2 is diversity of ML+. Some sub-classes of Aℐ are decipated in [29]. Due to their structure we can call them almost-idempotent-semirings. An h− regular semi-ring M in ML+ will be knowns as Left h− semifield if for all u∈M and v∈M′=M−0 there exists w1,w2∈M such that u+w1v+f=w2v+f for some f∈M. It follows that a semi-ring M is a left k− semifield iff for all u∈M and v∈M′ there exists w∈M such that u+wv+f=wv+f for some f∈M. Right h− semifield can be defined dually. A semi-ring M will be known as h− semifield if it will be both left and right h− semifield.Lemma 1.
LetMbe a semi-ring inML+anda′,b′,c′,d′∈M.(1)
If thereexistsx1′,x2′,y1′,y2′∈M and for some f∈M such that a′+x1′b′y1′+f=x2′c′y2′+f then,(3)a′+x1′+x2′b′+c′y1′+y2′+f=x1′+x2′b′+c′y1′+y2′+f.(2)
Ifa′+x′c′x′+f=x′c′x′+fandb′+y′d′y′+f=y′d′y′+f for some x′,y′∈M and f∈M then,(4)a′+x′+y′c′+d′x′+y′+f=x′+y′c′+d′x′+y′+fb′+x′+y′c′+d′x′+y′+f=x′+y′c′+d′x′+y′+f.(3)
Ifthere exists x1′,x2′∈M such that a′+x1′b′+f=x2′c′+f then,(5)a′+x1′+x2′b′+c′+f=x1′+x2′b′+c′+f.(4)
Ifa′+x′c′+f=x′c′+f and b′+y′d′+f=y′d′+f for some x′,y′∈M and f∈M then,(6)a′+x′+y′c′+d′+f=x′+y′c′+d′+fb′+x′+y′c′+d′+f=x′+y′c′+d′+f.(5)
Ifa′+b′+f=b′+f and b′+c′+f=c′+f then,(7)a′+c′+f=c′+f.(6)
Ifa′+b′c′+f′=b′c′+f′ for some f′,x′,y′∈M then,(8)b′+x′+f=x′+f⇒a′+x′c′+f′=x′c′+f′c′+y′+f=y′+f⇒a′+b′y′+f′=b′y′+f′.(7)
Ifa′+b′+f=b′+f and c′+d′+f=d′+f then,(9)a′c′+b′d′+f′=b′d′+f′.Proof.
(1)
Asa′+x1′b′y1′+f=x2′c′y2′+f⟶iNow to provea′+x1′+x2′b′+c′y1′+y2′+f=x1′+x2′b′+c′y1′+y2′+f(10)L.H.S=a′+x1′+x2′b′+c′y1′+y2′+f=a′+x1′b′y1′+x2′b′y1′+x1′c′y1′+x2′c′y1′+x2′b′y2′+x2′b′y2′+x1′c′y2′+x2′c′y2′+f=x2′c′y2′+f+x1′b′y1′+x2′b′y1′+x1′c′y1′+x2′c′y1′+x2′b′y2′+x2′b′y2′+x1′c′y2′R.H.S.=x1′+x2′b′+c′y1′+y2′+f=x1′b′y1′+x2′b′y1′+x1′c′y1′+x2′c′y1′+x2′b′y2′+x2′b′y2′+x1′c′y2′+x2′c′y2′+f=x1′b′y1′+x2′b′y1′+x1′c′y1′+x2′c′y1′+x2′b′y2′+x2′b′y2′+x1′c′y2′+a′+x1′b′y1′+f=a′+x1′b′y1′+x2′b′y1′+x1′c′y1′+x2′c′y1′+x2′b′y2′+x2′b′y2′+x1′c′y2′+f∵AsM∈ML+x1′b′y1′+x1′b′y1′=x1′b′y1′=x2′c′y2′+f+x1′b′y1′+x2′b′y1′+x1′c′y1′+x2′c′y1′+x2′b′y2′+x2′b′y2′+x1′c′y2′.Hence desired result is obtained.(2)
To provea′+x′+y′c′+d′x′+y′+f=x′+y′c′+d′x′+y′+fTaking(11)L.H.S.=a′+x′+y′c′+d′x′+y′+f=a′+x′c′x′+y′c′x′+x′d′x′+y′d′x′+x′c′y′+y′c′y′+x′d′y′+y′d′y′+fsincea′+x′c′x′+f=x′c′x′+f=x′c′x′+y′c′x′+x′d′x′+y′d′x′+x′c′y′+y′c′y′+x′d′y′+y′d′y′+f=x′+y′c′+d′x′+y′+f=R.H.S.Similarly to prove.b′+x′+y′c′+d′x′+y′+f=x′+y′c′+d′x′+y′+f.Taking(12)L.H.S.=b′+x′+y′c′+d′x′+y′+f=b′+x′c′x′+y′c′x′+x′d′x′+y′d′x′+x′c′y′+y′c′y′+x′d′y′+y′d′y′+fSinceb′+y′d′y′+f=y′d′y′+f=y′d′y′+f+x′c′x′+y′c′x′+x′d′x′+y′d′x′+x′c′y′+y′c′y′+x′d′y′=x′+y′c′+d′x′+y′+f=R.H.S.(3)
To provea′+x1′+x2′b′+c′+f=x1′+x2′b′+c′+fTaking(13)L.H.S.=a′+x1′+x2′b′+c′+f=a′+x1′b′+x2′b′+x1′c′+x2′c′+f=x2′c′+f+x1′c′+x2′c′+x2′c′=x2′c′+x2′b′+x1′c′+f.By usinga′+x1′b′+f=x2′c′+f and x2′c′+x2′c′=x2′c′ as M∈ML+.Taking(14)R.H.S.=x1′+x2′b′+c′+f=x1′b′+x2′b′+x1′c′+x2′c′+f=a′+x1′b′+f+x1′b′+x2′b′+x1′c′=a′+x1′b′+x2′b′+x1′c′+fAsa′+x1′b′+f=x2′c′+f=x2′c′+x2′b′+x1′c′+f.By usinga′+x1′b′+f=x2′c′+f and x1′b′+x1′b′=x1′b′ as M∈ML+. So we have L.H.S=R.H.S.(4)
To provea′+x′+y′c′+d′+f=x′+y′c′+d′+f. Taking(15)L.H.S.=a′+x′+y′c′+d′+fa′+x′c′+y′c′+x′d′+y′d′+f=x′c′+f+y′c′+x′d′+y′d′x′+y′c′+d′+f=R.H.S.a′+x′c′+f=x′c′+f.Similarly it is easy to prove that(16)b′+x′+y′c′+d′+f=x′+y′c′+d′+f.(5)
Sincea′+b′+f=b′+f and b′+c′+f=c′+f. As a′+b′+f=b′+f adding c′ on both sides we get(17)a′+b′+c′+f=b′+c′+fa′+c′+f=c′+f∵b′+c′+f=c′+f.(6)
Since(18)b′+x′+f=x′+fb′+x′+fc′=x′+fc′b′c′+x′c′+fc′=x′c′+fc′a+b′c′+x′c′+fc′=a+x′c′+fc′a+b′c′+x′c′+f′=a+x′c′+f′sayfc′=f′b′c′+x′c′+fc′=a+x′c′+f′∵a′+b′c′+f′=b′c′+f′·b′+x′+fc′=a+x′c′+f′x′+fc′=a+x′c′+f′x′c′+fc′=a+x′c′+f′x′c′+f′=a+x′c′+f′∵fc′=f′.Now; for(19)c′+y′+f=y′+fc′+y′+fb′=y′+fb′b′c′+b′y′+b′f=y′b′+b′fa′+b′c′+b′y′+b′f=a′+y′b′+b′fa′+b′y′+b′f+b′c′=a′+y′b′+b′fb′y′+b′f+b′c′=a′+y′b′+b′fsaybf′=f′b′y′+f+c′=a′+y′b′+b′f∵a′+b′y′+f′=b′y′+f′b′y′+f=a′+y′b′+b′f∵c′+y′+f=y′+fb′y′+b′f=a′+y′b′+b′fb′y′+f′=a′+y′b′+f′.(7)
Asa′+b′+f=b′+f⇒a′+b′+fc′=b′+fc′⇒a′c′+b′c′+fc′=b′c′+fc′⟶iAlso;c′+d′+f=d′+f⇒b′c′+d′+f=b′d′+f⇒b′c′+b′d′+b′f=b′d′+b′f⟶iiAddingi&ii we have: a′c′+b′c′+fc′+b′c′+b′d′+b′f=b′c′+fc′+b′d′+b′fb′c′+fc′+b′d′+b′f=b′d′+b′f+fc′∵byi&ii⇒b′c′+b′d′+b′+c′f=b′d′+b′+c′fb′c′+b′d′+f′=b′d′+f′∵say. b′+c′f=f′Definition 5.
Ah−regular semi-ringMwill be known as lefth−Clifford semi-ring if for allu,v∈M∃w1,w2∈Msuch thatuv+w1u+f=w2u+ffor somef∈M.otherwise; By (Lemma 1); It shows thatMis Lefth−Clifford semi-ring iff∀u,v∈M∃w∈Msuch thatu+wv+f=wv+fforsomef∈M.Lemma 2.
ForMbe a semi-ring inML+then;(1)
fora′,b′∈M&for somef∈Mthen the following conditions are identical:A1there aresi′,ti′∈Msuch thatb′+s1′a′t1′+f=s2′a′t2′+f(A2) there ares′,t′∈Msuch thatb′+s′a′t′+f=s′a′t′+f(A3) there isx′∈Msuch thatb′+x′a′x′+f=x′a′x′+f(2)
ifa′,b′,c′∈Mfor somef∈Mare such thatb′+x′a′x′+f=x′a′x′+f and c′+y′a′y′+f=y′a′y′+ffor somex′,y′∈Mthen there isz′∈Msuch thatb′+z′a′z′+f=z′a′z′+f=c′+z′a′z′+f(3)
ifa′,b′,c′∈Mfor somef∈Mare such thatc′+x′a′x′+f=x′a′x′+f and c′+y′a′y′+f=y′a′y′+ffor somex′,y′∈Mthen there isz′∈Msuch thatc′+z′a′z′+f=z′a′z′+f and c′+z′b′z′+f=z′b′z′+fProof.
(1)
AsA3⇒A2 and A2⇒A1 are obvious.Now; forA1⇒A3, suppose that(20)x′=s1′+s2′+t1′+t2′.For(21)s1′a′t1′+x′a′x′+f=s1′a′t1′+s1′+s2′+t1′+t2′a′s1′+s2′+t1′+t2′+f=s1′a′t1′+s1′a′s1′+s2′a′s1′+t1′a′s1′+t2′a′s1′+s1′a′s2′+s2′a′s2′+t1′a′s2′+t2′a′s2′+s1′a′t1′+s2′a′t1′+t1′a′t1′+t2′a′t1′+s1′a′t2′+s2′a′t2′+t1′a′t2′+t2′a′t2′+fass1′a′t1′+s1′a′t1′=s1′a′t1′∵M,+is semillatice=s1′a′s1′+s2′a′s1′+t1′a′s1′+t2′a′s1′+s1′a′s2′+s2′a′s2′+t1′a′s2′+t2′a′s2′+s1′a′t1′+s2′a′t1′+t1′a′t1′+t2′a′t1′+s1′a′t2′+s2′a′t2′+t1′a′t2′+t2′a′t2′+fs1′a′t1′+x′a′x′+f=x′a′x′+f.Now for(22)s2′a′t2′+x′a′x′+f=s2′a′t2′+s1′+s2′+t1′+t2′a′s1′+s2′+t1′+t2′+f=s2′a′t2′+s1′a′s1′+s2′a′s1′+t1′a′s1′+t2′a′s1′+s1′a′s2′+s2′a′s2′+t1′a′s2′+t2′a′s2′+s1′a′t1′+s2′a′t1′+t1′a′t1′+t2′a′t1′+s1′a′t2′+s2′a′t2′+t1′a′t2′+t2′a′t2′+fass2′a′t2′+s2′a′t2′=s2′a′t2′∵M,+is semillatice.=s1′a′s1′+s2′a′s1′+t1′a′s1′+t2′a′s1′+s1′a′s2′+s2′a′s2′+t1′a′s2′+t2′a′s2′+s1′a′t1′+s2′a′t1′+t1′a′t1′+t2′a′t1′+s1′a′t2′+s2′a′t2′+t1′a′t2′+t2′a′t2′+fs2′a′t2′+x′a′x′+f=x′a′x′+f,i.e.(23)s1′a′t1′+x′a′x′+f=x′a′x′+f=s2′a′t2′+x′a′x′+f.Thus(24)b′+s1′a′t1′+x′a′x′+f=x′a′x′+fb′+x′a′x′+f=x′a′x′+f.HenceA1⇒A3 has been proved.(2)
Putz′=x′+y′Forb′+z′a′z′+f=z′a′z′+f , taking(25)L.H.S.=b′+z′a′z′+f=b′+x′+y′a′x′+y′+f=b′+x′a′x′+x′a′y′+y′a′x′+y′a′y′+f=x′a′x′+x′a′y′+y′a′x′+y′a′y′+f∵b′+x′a′x′+f=x′a′x′+f=z′a′z′+f⟶i.Now,(26)R.H.S.=c′+z′a′z′+f=c′+x′+y′a′x′+y′+f=c′+x′a′x′+x′a′y′+y′a′x′+y′a′y′+f=x′a′x′+x′a′y′+y′a′x′+y′a′y′+f∵c′+y′a′y′+f=y′a′y′+f=z′a′z′+f⟶ii.Byi and ii we have b′+z′a′z′+f=z′a′z′+f=c′+z′a′z′+f. Hence Proved.(3)
Putz′=x′+y′For(27)c′+z′a′z′+f=c′+x′+y′a′x′+y′+fSincec′+x′a′x′+f=c′+x′a′x′+x′a′y′+y′a′x′+y′a′y′+f=x′a′x′+fwe have,c′+z′a′z′+f=x′a′x′+x′a′y′+y′a′x′+y′a′y′+f=z′a′z′+f.
Now for(28)c′+z′b′z′+f=c′+x′+y′b′x′+y′+f=c′+x′b′x′+x′b′y′+y′b′x′+y′b′y′+f∵c′+y′a′y′+f=y′a′y′+fso,c′+z′b′z′+f=x′b′x′+x′b′y′+y′b′x′+y′b′y′+f=z′a′z′+f.
Hence proved.Theorem 1.
The hypersemiringPQ,+,⋅ish−regular if and only ifQis regular semi-group.Proof.
Suppose thatPQ,+,⋅ is h− regular. We have to prove that Q is regular semi-group. Now let g∈Q then G=g∈PQ and f∈Q then F=f∈PQ also there is W∈PQ s.t. G+GWG+F=GWG+F for some F∈PQ i.e. G+F⊂GWG+F⇒∃g∈G such that g+f=gwg+f for some f∈F. Thus G,⋅ is a regular semi-group.
Conversely, suppose thatQ,⋅ be a regular semi-group and let G∈PQ then ∀g∈G∃w∈Q such that g=gwg. We will choose one such “ w” and denote it by wg then W=wg:g∈G∈PQ such that G+F⊂GWG+F which implies that G+GWG+F=GWG+F. Thus PQ,+,⋅ is h− regular semi-ring.


## 3. Characterizations of Lefth− Clifford Semiring

This section is comprised of some characterizations of Lefth− Clifford Semiring.Proposition 1.
LetQbe a semi-ring then the semi-ringPQis the lefth−semifieldiffQis a left-group.Proof.
Suppose thatQ be a left-group. We need to show that PQ is left h− semifield. Since Q is a left-group then Q is regular and left-simple-semigroup then PQ is h− regular semi-ring. Now let G,H∈PQ and H≠∅ then there exist h∈H. Also let G=g1,g2,g3,…,gn,n∈ℕ. Since Q is left simple so ∃q1,q2,q3,…,qn∈Q such that gi+f=qih+f∀i=1,2,3…n and for some f∈Q. Let R=q1,q2,q3,…,qn then we have G+F⊂RH+F⇒G+RH+F=RH+F for some F∈PQ. Thus PQ is left h− semifield.
Conversely, suppose thatPQ be a left h− semifield then h− regularity of PQ shows that Q is a regular semi-group. Let g,h∈Q then G=g,H=h∈PQ so ∃R∈PQ such that G+RH+F=RH+F where F∈PQ and so G+F⊂RH+F. Hence ∃q∈R such that g+f=qh+f⇒Q is left-simple-semigroup. Thus Q is a left-group.Proposition 2.
LetMbe a lefth−semifield with″0″andm1,m2∈Mifm1,m2≠0thenm1⋅m2≠0.Proof.
Let0≠m1,m2∈M then ∃w∈M such that m1+wm2+f=wm2+f for some f∈M. Also ∃v∈M such that w+vm1+f=vm1+f for some f∈M then m1+vm1m2+f′=vm1m2+f′ for some f′∈M⇒m1⋅m2≠0 otherwise m1 would be zero.Theorem 2.
A semi-ringMwith “0″ is a lefth−semifield iffMish−regular and lefth−0−simple.Proof.
Suppose thatM is a left h− semifield. Let S≠0 be a left h− ideal and let s1≠0 be an element of S. Let s2∈M then ∃m∈M such.that s2+ms1+f=ms1+f for some f∈M. Since S is left h− ideal so ms1∈S and so s2∈S. Thus M=S⇒M is a left h−0− simple.
Conversely, suppose thatM be an h− regular and left h−0− simple. Let s1∈M and s2∈M′, since M is h− regular so s2+f∈Ms2¯+f⇒Ms2¯+f=M+f for some f∈M then s1+f∈Ms2¯+f and hence ∃m1,m2∈M such that s1+m1s2+f=m2s2+f for some f∈M. Thus M is left h− semifield.Theorem 3.
LetQbe a semi-group thenPQis a lefth−Clifford semi-ringiffQis a Left Clifford semi-group.Proof.
Letg,h∈Q then g, h∈PQ this implies that ∃W∈PQ s.t. g⋅h+Wg+f=Wg+f for some f∈PQ, gh∈Wg. Hence there exists w∈W such that gh+f=wg+f for some f∈Q. Thus gQ⊂Qg and so Q is a left Clifford semi-group.
Conversely, letG,H∈PQ and let g∈G and h∈H. Since Q is left Clifford semi-group so gQ⊂Qg⇒ there exist w∈Q such that gh+f=wg+f for some f∈Q. For each g∈G and h∈H we choose one such “ w” and denote it by “ wg,h”. Since both G and H are finite so W=wg,h:g∈G,h∈H∈PQ. Also gh∈WG∀g∈G and h∈H. Hence GH+F⊂WG+F⇒GH+WG+F=WG+F where F∈PQ. Thus PQ is a left h− Clifford semi-ring.Theorem 4.
LetMbe a semi-ring then the following conditions are equivalent:(1)
M is left h− Clifford.(2)
M is h− regular and iM¯⊆Mi¯ for all i∈EhM.(3)
M is h− regular and mM¯⊆Mm¯ for all m∈M.(4)
ℛ¯⊆ℒ¯.(5)
All lefth− ideals are two sided and E1∩E2=E1E2¯ for any two h− ideals E1,E2 of M.(6)
K1∩K2=K1K2¯ for any two h− ideals K1,K2 of M.Proof.
1⇒2. This is trivial.
2⇒3. Let m,n∈M then ∃u,v∈M such that(29)m+mum+f=mum+fandmn+mnvmn+f=mnvmn+f.And hence(30)m+mwm+f=mwm+fandmn+mnwmn+f=mnwmn+fwherew=u+v∈M.
thenwm∈EhM and 2 implies that there exist t∈M such that wmn+twm+f=twm+f then;(31)mn+mntwm+f=mntwm+fbyLemma1.
HencemM¯⊆Mm¯.
3⇒4 Let m,n∈M such that mℛ¯n then ∃u∈M such that(32)m+nu+f=nu+fandn+mn+f=mu+f.
Again by3∃g,h∈M such that(33)nu+gn+f+gn+fandmn+hm+f=hm+f.implies that(34)m+gn+f=gn+fandn+hm+f=hm+f.
Thus,mℛ¯n and so ℛ¯⊆ℒ¯.
4⇒5 Let K be a left h− ideal of M. Consider k∈K and m∈M. let h∈M be such that(35)k+khk+f=khk+f.then(36)km+mk+f=mk+fandk+khk+f=khk+f.
This implies that there isu=m+h∈M s.t.(37)km+mu+f=mu+fandk+kuk+f=kuk+f.Then kℛ¯ku implies that kℒ¯ku.
Hence there existw∈M s.t. ku+wk+f=wk+f thus, km+wk+f=wk+f i.e. km∈Mk¯⊆K hence K is two-sided h− ideal of M.
LetE1, E2 be two sided h− ideal of M then E1.E2⊆E1, E2 implies that E1.E2⊆E1∩E2 and so E1E2¯⊆E1∩E2. Now let k∈E1∩E2 since M is a h− regular semi-ring ∃u∈M s.t. k+kuk+f=kuk+f then k∈E1⇒ku∈E1 whence kuk∈E1.E2∴k∈E1.E2¯ and so E1∩E2⊆E1E2¯.
Thus,E1∩E2=E1E2¯5⇒6 Trivial.
6⇒1 Let m,n∈M since M is h− regular semi-ring there exist u∈M s.t.(38)mn+mn.u.mn+f=mn.u.mn+f.Which implies that(39)mn+mn.u.mn.u.mn+f=mn.u.mn.u.mn+f.Then(40)mn+mnumnumn+f=mnumnumn+f⇒mn∈Mm¯.Mn¯¯⊆Mm¯∩Mn¯⇒mn∈Mm¯.Mn¯¯⊆Mm¯∩Mn¯.
Somn∈Mm¯.
Hence there arew1,w2∈M such that mn+w1m+f=w2m+f. Thus M is left h− Clifford semi-ring.Theorem 5.
LetMbe ah−regular semi-ring then the following conditions are equivalent:(i)
M is left h− Clifford semi-ring(ii)
for allm∈M and i∈EhM∃w∈M such that im+iwi+f=iwi+f for some f∈M(iii)
for allm∈M and i∈EhM∃w∈M such that mi+wi+f=wi+f for some f∈M(iv)
for allm∈M and m′∈Whm∃w∈M such that mm′+wm+f=wm+f for some f∈MProof.
i⇒ii Let m∈M and i∈EhM. Since M is a left h− Clifford semi-ring so iM¯=Mi¯ which implies that there exisst b∈M such that im+bi+f=bi+f. Again h− regularity of M implies that there is x∈M such that(41)im+imxim+f=imxim+f.
Which implies that(42)im+imxbi+f=imxbi+f.
ii⇒iii Let m∈M and i∈EhM then the regularity of M together with by lemma1 implies that ∃u∈M such that(43)mi+mi.u.mi+f=mi.u.mi+fandm+mum+f=mum+f.
Sinceum∈EhM∃v∈M such that(44)umi+umvum+f=umvum+f.Implies that(45)mi+miumvum+f=miumvum+f.
iii⇒iv Let m∈M and m′∈Whm then(46)m′+m′mm′+f=m′mm′+f.
And so(47)mm′+mm′mm′+f=mm′mm′+f.Which implis that(48)m′+m′xm′+f=m′xm′+fandmm′+mxm′+f=mxm′+f.Where x=m+mm′ then xm′∈EhM.
Hence∃v∈M such that mxm′+vm+f=vm+f thus we get mm′+vm+f=vm+fby lemma1iv⇒i Let m,n∈M then ∃x∈M such that. m+mxm+f=mxm+f
Also,mn+mn=mn hence(49)m+mbm+f=mbm+fandmn+mb+f=mb+f.where b=x+n then bmb∈Whm and so there exist w∈M such that(50)mbmb+wm+f=wm+f.Then mn+mb+f=mb+f implies that(51)mn+mbmb+f=mbmb+f.
Which again implies thatmn+wm+f=wm+f.
ThusM is a h− Clifford semi-ring.Theorem 6.
A semi-ringMis lefth−Clifford if and only if for alli,j∈EhMthere existsk∈EhMsuch thatij+ki+f=ki+ffor somef∈M.Proof.
LetM be a left h− Clifford semi-rings and i,j∈EhM then ∃m∈M such that ij+mi+f=mi+f for f∈M again, i+i2+f=i2+f implies that i+ii+i2+f=ii+i2+f i.e. i+i3+f=i3+f for f∈M
Then we haveij+ni+f=ni+f and i+ini+f=ini+f where n=m+i&f∈M then, k=ni∈EhM and ij+ni+f=ni+f which implies that ij+ni2+f=ni2+f for f∈M , ij+ki+f=ki+f for f∈M.
Conversely, suppose thatr,s∈M then ∃x,y∈M s.t.(52)r+rxr+f=rxr+fands+sys+f=sys+f.
Hence(53)rs+rxrsys+f=rxrsys+f.
Again we havez∈M s.t.(54)r+rzr+f=rzr+fs+szs+f=szs+fandrs+zrzs+f=zrzs+fforf∈M.
Theni=zr, j=zs∈EhM⇒∃k∈EhM s.t. ij+ki+f=ki+frs+ij+f=ij+f⇒rs+kzr+f=kzr+f for f∈M.
Thus,M is a left h− Clifford semi-ring.


## 4. Structure of Lefth− Clifford Semiring

In this section we will work on the structure ofh− Clifford semi-ring. The "ξ−class" resolved by the least Distributive Lattice Congruence “ ξ” on M are left h− semifields for a Left h− Clifford semi-ring M. Hence each left h− Clifford semi-ring will be distributive lattice of left h− semifields.Definition 6.
LetMbe a semi-ring and∅≠A⊆Mthen lefth−centralizer ofAis defined by(55)CtA=v∈M:there existw∈Msuch thatuv+vwu+f=vwu+fforf∈M&∀u∈A.Theorem 7.
A semi-ringMis a lefth−Clifford semi-ring if and only ifMish−regular andi∈CtiMfor alli∈EhM.Proof.
LetM is left h− Clifford semi-ring then ∃w∈m s.t. imi+wim+f=wim+f then(56)i+i2+f=i2+f⇒imi+i2mi+f=i2mi+f⇒imi+iwim+f=iwim+f.
Thusi∈CtiM.
Conversely, suppose thati∈CtiM∀i∈EhM. Consider m,n∈M then similarly we can get z∈M s.t.(57)m+mzm+f=mzm+f,n+nzn+f=nzn+fandmn+mzmzn+f=mzmzn+f.
Thenmz, zn∈EhM and so i=mz+zn∈EhM. Since EhM is a subsemilattice of the additive reduct M,+. Now i∈CtiM implies that there exist w∈M such that(58)imi+iwim+f=iwim+f.Then we have(59)mn+mzmzn+f=mzmzn+fmn+mz+znmmz+zn+f=mz+znmmz+zn+fmn+imi+f=imi+fmn+iwim+f=iwim+f.
Thus,M is a left h− Clifford semi-ring.Definition 7.
Suppose thatCis a class of semi-rings we will call the semi-rings inCasC−semi-rings. A semi-ringMwill be known as a distributive lattice ofC−semi-rings if∃congruenceρonMsuch thatM/ρis a distributive lattice and eachρclass is a semi-ring inC.Theorem 8.
LetMbe ah−regular semi-ring thenMis a Lefth−Clifford semi-ringiffℒ¯=J¯is the least distributive lattice congruences onM.Proof.
Suppose thatM is a left h− Clifford semi-ring then ℛ¯⊆ℒ¯. ⇒ℒ¯°ℛ¯⊆ℒ¯ and so J¯⊆ℒ¯ [27]. Again in a h− regular semi-ring ℒ¯⊆J¯. Therefore J¯=ℒ¯.
Letm,n∈M be such that mℒ¯n and p∈M then ∃u∈M s.t.(60)m+un+f=un+fn+um+f=um+ffor some f∈M.
SinceM is a left h− Clifford semi-ring so ∃h∈M s.t. pu+hp+f=hp+f for some f∈M, then we get(61)pm+pun+f=pun+f⇒pn+hpn+f=hpn+f.
and similarly(62)pn+hpm+f=hpm+f.
Which implies thatpmℒ¯pn.Thus ℒ¯ is a congruence on M,+,⋅.
Now letm∈M then ∃u∈M such that(63)m+mum+f=mum+f.
SinceM is a left h− Clifford semi-ring so there exists w∈M such that(64)mu+wm+f=wm+f.
Then we havem+wm2+f=wm2+f and hence mℒ¯m2.
Letm,n∈M then ∃u∈M s.t.(65)mn+mn.u.mn+f=mn.u.mn+f.
SinceM is a left h− Clifford semi-ring so there exists g∈M such that mu+gm+f=gm+f and hence h∈M such that(66)nug+hn+f=hn+fforf∈M,
then we have(67)mn+mn.u.mn+f=mn.u.mn+f⇒mn+mn.u.gm+f=mn.u.gm+f⇒mn+m.hn.gm+f=m.hn.m+f⇒mn+wnm+f=wnm+fwherew=mh.
Similarly;∃j∈M such that mn+jmn+f=jmn+f; for f∈M thus; mnℒ¯nm.
Now, letm,n∈M then ∃m,w∈M s.t.(68)m+mum+f=mum+fandmn+wm+f=wm+f,forf∈M,
then we have(69)m+mum+f=mum+f⇒m+mn+mum+wm+f=mum+wm+f⇒m+mn+um+wm+f=um+wm+f,
againm+mum+f=mum+f implies that. m+mum+mn+f=mum+mn+f
Hence,m+mnℒ¯m. Thus ℒ¯ is a distributive lattice congruence on M.
Now let “σ” be a distributive lattice congruence on M. Let m,n∈M be s.t. mℒ¯n then ∃u∈M s.t(70)m+um+f=un+fandn+um+f=um+f.
Then,(71)mσm+munσm2+mun+f=mm+un+f=munσnum+f=n2+numσn+f.
Shows thatmσn.
Thusℒ¯ is the least distributive lattice congruence on M.
For the converse, letm,n∈M then mnℒ¯nm implies that ∃u∈M such that mn+unm+f=unm+f; for f∈M.
This shows thatM is left h− Clifford semi-ring.Theorem 9.
LetMbeh−regular semi-ring thenMis a distributive lattice of lefth−SemifieldsiffMis lefth−Clifford semi-ring.Proof.
Suppose thatM is distributive lattice S of left h− semifields Pα′:α′∈S. Let m,n∈M then ∃α′,β′∈S such that m∈Pα′ and n∈Pβ′ hence, m+mn+f∈Pα′+Pα′Pβ′⊆Pα′+α′β′=Pα′ since, Pα′ is a left h− semifield there exist u∈Pα′ s.t.(72)m+mn+um+f=um+fforf∈M.
Implies that(73)mn+mn+um+f=um+fforf∈M.
Since,M,+ is semilattice i.e mn+um+f=um+f. Thus, M is a left h− Clifford semi-ring.
Conversely, suppose thatM be a left h− Clifford semi-ring then ℒ¯ is a distributive lattice congruence on M. So, each ℒ¯− class is a subsemiring of M. Let T be a ℒ¯− class and m,n∈T then n2∈T and hence mℒ¯n2 which implies that there exist g∈M such that(74)m+gn2+f=gn2+f.
Also there existh∈M such that(75)n+nhn+f=nhn+f.
Then there existu=g+h∈M such that m+un2+f=un2+f and u+nun+f=unu+f Take w=un then we have(76)n+nw+f=nw+fandw+un+f=un+fforf∈M.
Which shows thatwℒ¯n i.e. w∈T and such that m+wn+f=wn+f hence T is a left h− semifield. Thus each ℒ¯− class is a left h− semifield. Therefore M is a distributive lattice ℳ/ℒ¯ of left h− semifields.


## 5. Conclusion

This research represents the structure and characterizations of Lefth− Clifford Semirings. This article also explains some proofs relevant to the specific properties of h− regular semi-rings. Several properties of left h− Clifford semi-rings and the equivalent characterizations have been proved in this research. We have shown that a semi-group Q is left Clifford semi-group iff the semi-ring PQ of all subsets of Q is Left h− Clifford semi-ring. Lastly, we worked on the structure of Left h− Clifford semi-ring. We prove that a semi-ring M is left h− Clifford semi-ring iff it is distributive lattice of left h− semifield. In future a similar work can be done for other algebraic structures and one can apply these concepts in decision making problems.



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