Abstract

The main aim of this research is to introduce Left Clifford Semi-rings. Using some basic properties of regular semi-rings we shall investigate several properties of Left 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 Clifford Semiring. Also, this research will investigate the distributive congruences of regular semi-rings.

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 [27] 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 [912]. 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 would also be regular if the semi-group is regular. Bourne showed that if there exists such that 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 is called left semi-group of Clifford if . A semi-group 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 [1618]. 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 [2026] 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 [2730]. This research presentes left -semifields and left Clifford semi-rings as abstraction of -semifields and -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 Clifford semi-rings as a simplification of Clifford semi-ring. Being motivated from his work we have introduced Left Clifford semi-rings and also described the basic properties of left -Clifford semi-rings and the equivalent chaacterizations we have committed. We will prove that a semi-group will be a Left Clifford semi-group iff the semi-ring of all subsets of will be Left Clifford semi-ring. We will also characterize the structure of Left Clifford semi-rings. A semi-ring will be left 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

Let then it is said to be semi-ring if it satisfies the following conditions: is a semi-group. is a semi-group. Both (left and right) distributive laws hold i.e. and . Thus; is semi-ring which is denoted by (, , ).

Definition 1. An additive subgroupof a ringthenknown as ideal ofif it satisfies the following properties:
. Ifis left and right ideal ofthenis said to be ideal of ring. A semi-groupwill be known as regular semi-group ifsuch that. Considerbe a semi-ring then for eachsuch thatif and only ifbe a regular semi-ring. Letbe a semi-ring and letis a left ideal ofif the following conditions satisfied:(1) (2) for and .The right ideal ofis defined dually. Ifis left as well as right ideal ofthen I will be called as Ideal of semi-ring.

Definition 2. Letbe a semi-ring andwill be-closure ofdefined as:Suppose that be a semi-ring and be a h-closure of then will be known as -set if .
Let be a semi-ring and is a left ideal of then willl be known as left ideal if:
. The right ideal can be defined dually. Let then is called Band if following axioms satisfied: (i) is semi-group. (ii) each element of is idempotent. .
From [31] we let then will be semilattice if is commutative band. is often a semi-ring the additive reduct is a semilattice and this class of all such semi-ring is denoted by . A semi-ring (,+,.) without zero will be known as (left, right) simple if it has no proper (left, right) -ideal. A semi-ring ( ,+,.)with zero will be known as simple (left, right) if are its only ideals (left, right).

Definition 3. A regular semi-groupwill be called Clifford semi-group ifi.e. idempotent ofcommute with all elements of; whereis set of all idempotents of andIs centre of. Let be a semi-group thenwill be called left Clifford semi-group ifis regular semi-group as well as [18].

Definition 4. A semi-ringwill be known asregular ifsuch thatfor [30]. Ifis a semilattice then we can say that a semi-ringis known asregular if and only if for eachsuch thatfor. Letbe a semi-group and be the hypersemiring of; where,“ and ” is defined as: and thenis a semi-ring whose additive reduct is a semi-lattice that is knwon as hyper semi-ring of semi-group. Letbe a semi-ring and letthen .. will be known asinverse ofifandfor some. Ifthen “” isinverse of. Therefore in anregular semi-ring every component has ainverse. The set ofinverses ofinis denoted by.
We will inagurated Green’s relation on semi-rings in [27], as introduced by Sen and Bhuniya in the following way: for any ; if and only if ; if and only if ; if and only if and . These are the equivalence relations of additive congruences on , where represents the multiplicative right and is just multiplicative left and also is just ideal congruence on . For an element of a semi-ring will be known as idempotent if for some . represents the set of all idempotent. For all , but . Semiring in which for some [30]. The class of semi-rings in constrained by extra character is diversity of . Some sub-classes of are decipated in [29]. Due to their structure we can call them almost-idempotent-semirings. An regular semi-ring in will be knowns as Left semifield if for all and there exists such that for some . It follows that a semi-ring is a left semifield iff for all and there exists such that for some . Right semifield can be defined dually. A semi-ring will be known as semifield if it will be both left and right semifield.

Lemma 1. Letbe a semi-ring inand.(1)If there exists and for some such that then,(2)If and for some and then,(3)If there exists such that then,(4)If and for some and then,(5)If and then,(6)If for some then,(7)If and then,

Proof. (1)As Now to prove Hence desired result is obtained.(2)To prove TakingSimilarly to prove. Taking(3)To prove TakingBy using and as .TakingBy using and as . So we have .(4)To prove . TakingSimilarly it is easy to prove that(5)Since and . As adding on both sides we get(6)SinceNow; for(7)As Also; Adding we have: say.

Definition 5. Aregular semi-ringwill be known as leftClifford semi-ring if for allsuch thatfor some.otherwise; By (Lemma 1); It shows thatis LeftClifford semi-ring iff such thatforsome.

Lemma 2. Forbe a semi-ring inthen;(1)forfor somethen the following conditions are identical:there aresuch that(A2) there aresuch that(A3) there issuch that(2)iffor someare such that and for somethen there issuch that(3)iffor someare such that and for somethen there issuch that and

Proof. (1)As and are obvious.Now; for , suppose thatForNow fori.e.ThusHence has been proved.(2)Put For , takingNow,By and we have . Hence Proved.(3)Put ForNow forHence proved.

Theorem 1. The hypersemiringisregular if and only ifis regular semi-group.

Proof. Suppose that is regular. We have to prove that is regular semi-group. Now let then and then also there is s.t. for some i.e. such that for some . Thus is a regular semi-group.
Conversely, suppose that be a regular semi-group and let then such that . We will choose one such “ ” and denote it by then such that which implies that . Thus is regular semi-ring.

3. Characterizations of Left Clifford Semiring

This section is comprised of some characterizations of Left Clifford Semiring.

Proposition 1. Letbe a semi-ring then the semi-ringis the leftsemifieldis a left-group.

Proof. Suppose that be a left-group. We need to show that is left semifield. Since is a left-group then is regular and left-simple-semigroup then is regular semi-ring. Now let and then there exist . Also let . Since is left simple so such that and for some . Let then we have for some . Thus is left semifield.
Conversely, suppose that be a left semifield then regularity of shows that is a regular semi-group. Let then so such that where and so . Hence such that is left-simple-semigroup. Thus is a left-group.

Proposition 2. Letbe a leftsemifield withandifthen.

Proof. Let then such that for some . Also such that for some then for some </