The Structure And Characterizations Of Quasi Clifford Semirings

In this dissertation,we mainly discuss semirings whose additive semigroups are orhogroups.Under the extension of left Clifford semirings and rectangular Clifford semirings,we gived their definitions,structures and properties.This paper is divided into three chapters.The main results are given as follow:In part of chapter one,we study the semiring whose called quasi Clifford semir-ings.Firstly,we give the definition of quasi Clifford semirings:A semiring S is called a normal Clifford semiring if S is a distributive lattice of a rectangular rings and E+(S)is a regular band.Secondly,we study some properties and structures of quasi-Clifford from the structure of additive Semigroups of semirings.The main results are as follows:Theorem 1.2.1 A semiring S is a quasi Clifford semiring if and only if the additive reduct(S,+)of S is a quasi Clifford semigroups in which each maximal subgroup is abelian,E+(S)(?)E·(S)and S satisfies the following conditions.(1)(?)α∈S,V+(a)+α(?)α(α+V+(a));(2)(?)a,b∈S,V+(ab)+ab(?)(b+ V+(b))a;(3)(?)a,b∈S,V+(a)+a(?)a+ab+V+(ab))+V+(a).Corollary 1.2.1 A semiring S is a normal Clifford semiring if and only if(?)is a distributive lattice congruence on S,every(?)-class is a rectangular semiring and E+(S)is a regular band.Theorem 1.2.2 For some distributive lattice skeleton D,let L=∪α∈D Lαbe the distributive lattice D decomposition of left regular band semiring L into left zero band semirings Lα,∪α∈D Tα be the distributive lattice D decomposition of Clifforg semiring T into rings Tα,R=∪α∈D Rα be the distributive lattice D decomposition of right regular band semiring R into right zero band semirings Rα,Sl=[D;Lα×Tα]be a left Clifford semiring,Sr=[D;Tα×Rα]be a right Clifford semiring,we have:The semiring S is a quasi Clifford semiring if and only if S is isomorphic of the spined product Sl×TSτ of a left Clifford semiring Sl=[D;Lα×Tα]and a right Clifford semiringSr=[D;Tα×Rα]having a common Clifford semiring componnent T=[D;Tα]with respect to the semiring homomorphisms φ:(i,x)(?)x,for(i,x)∈Si and ψ:(x,λ)(?)x,for(x,λ)∈Sr.Chapter 2:In this chapter,we give the definition of Δ-product of semirings and study the A-product of left Clifford semirings and quasi Clifford semirings.The main conclusions are as follows:Deinition 2.2.1 Let D is a distributive,T=∪α∈D Tα is the distributive D of the semiring Tα,I=∪α∈D Iα is the distributive D of the semiring Iα.For anyα∈D,denote Sα=Tα×Iα,then(?)α,β∈D,α≤β,let map:Ψα,β:Sα→(?)(Iβ)a(?)φα,βa satisfing:(P1):If(u,i)∈Sα,i’∈Iα,then Ψα,α(u,i)i’=i.(P2):If(u,i)∈Sα,(v,j)∈Sβ,then(a)ψα,α+β(u,i)ψβ,α+β(v,j)is a constat mapping on Iα+β,rccord thu value as<ψα,α+β(u,i)ψβ,α+β(v,j)>.(b)When α+β≤δ,<ψα,α+β(u,j)ψβ,α+β(v,j)>=k,we have ψα+β,δ(u+v,k)=ψα,δ(u,i)ψβ,δ(v,j).Let(u,i)+(v,j)=(u+v,<,<ψα,α+β(u,i)ψβ,α+β(v,j)>)(1)(u,i)·(v,j)=(uv,ij)(2)where u+v is the addition of u and v in T,uv is the multiplication of u and v in T,ij is the multiplication of i and j in I.It is easy to prove S=∪α∈D Sα is a semiring whith the formula(1)(2),we call this semiring is the Δ-product of the semiring T and I about D and Ψα,β,and sign it as S=TΔD,Ψ.We call {Ψα,β|α,β∈D} is the structure function of Δ-product.Theorem 2.2.1(1)Let T=∪α∈D Tα be the distributive lattice decomposi-tion of Clifford semiring T into ring Tα,I=∪α∈D Iα be he distributive lattice decomposition of left regular band semiring I into left zero band semiring Iα,Then the product of T and I on D and any structure homomorphism Ψα,β的Δ-product is a left Clifford semiring,it takes Sα=Tα×Iα as its subsemiring,and it is the distributive lattice D of Sa.(2)Any left Clifford semiring can be constructed in this way.Theorem 2.2.2 Let T=∪α∈D Tα be the distributive lattice decomposition D of Clifford semiring T into ring Tα,I=∪α∈D Iα hand Λ=∪α∈D Λα are the dis-tributive lattice decomposition D of left、right regular band semiring Iα、Λα into left、right zero band semiring Iα、Λα respectively.For any α,β∈D,α≤β,let map:Φα,β:Sα→(?)1(Iβ)a(?)φα,βa,Ψα,β:S’α→(?)2(Λβ)a(?)ψα,βa.where Φα,β satisfies:(Q1):If(i,x)∈Sα,i’∈Iα,then φα,α(i,x)i’=i.(Q2):if(i,x)∈Sα,(j,y)∈Sβ,then(a)φα,α+β(i,x)φβ,α+β(j,y)is a constant mapping on Iα+β,record the value as<φα,α+β(i,x)φβ,α+β(j,y)>.(b)When α+β≤δ,<φα,α+β(i,x)φβ,α+β(j,y)>=k,we have φα+β,δ(i,x+y)φα,δ(i+x)φβ,δ(j+y).Ψα,β satisfies(Q’1),(Q’2)which is dual with(Q1),(Q2),wherex+y is the addition of x and y in T.Defining operation in S=∪α∈D(Iα×Tα×Λα):(?)(i,x,λ)∈Sα,(j,y,μ)∈Sβ.(i,x,λ)+(j,y,μ)=(<φα,α+β(i+x)φβ,α+β(j+y)>,x+y,<ψα,α+β(x+λ)φβ,α+β(y+μ)>),(1)(i,x,λ)·(j,y,μ)=(ij,xy,λμ).(2)where x+y is the addition of x and y in T,ij is the multiplication of i and j in I,xy is the multiplication of x and y in T,λμ is the multiplication of A and μ in Δ.Then S is a quasi Clifford semiring whith the formula(1)(2),we call S is the Δ-product of the semiring I,A and T with D and structure homomorphism Φα,β,Ψα,β.Conversely,any quasi Clifford semiring can be constructed in this way.Chapter 2:In this chapter,We study the structure and properties of normal Clifford.Firstly,we give the definition of normal Clifford semiring:A semiring S is called a normal Clifford semiring if S is a distributive lattice of a rectangular ring and E+(S)is a normal band.Secondly,we study the structure and properties of quasi Clifford semirings based on the structure of additive Semigroups.Finally,we introduce the strong distributive lattice form of normal Clifford semirings.The main conclusions are as follows:Theorem 3.1.1 A semiring S is a normal Clifford semiring if and only if the additive reduct(S,+)of S is a normal orthogroup in which each maximal subgroup is abelian,E+(S)(?)E’(S)and S satisfies the following conditions.(1)(?)α ∈ S,V+(α)+α(?)α(α+V+(α);(2)(?)α,b ∈ S,V+(αb)+αb(?)(b+V+(b))α;(3)(?)α,b∈S,V+(α)+α(?)α+αb+V+(αb)+V+(α).Corollary 3.1.1 A semiring S is a normal Clifford semiring if and only if(?)is a distributive lattice congruence on S,every(?)-class is a rectangular semiring and E+(S)is a normal band.Theorem 3.1.2 For some distributive lattice skeleton D,let[D,Lα,φα,β]be the strong distributive lattice D decomposition of left normal band semiring L into left zero band semirings Lα,∪α∈D Tα be the distributive lattice D decomposition of Clifforg semiring T into rings Ta,[D,Rα,φα,β]be the strong distributive lattice D decomposition of right normal band semiring R into right zero band semirings Rα,we have:The spined product(?)R=∪α∈D(Lα×Tα×Rα)of left normal band semiring L=[D,Lα,φα,β],Clifford semiring T=∪α∈DTα and right normal band semiring R=[D,Rα,φα,β]with respect to the same distributive skeleton D is a nor-mal Clliford semiring.Coversely,every normal Clifford semiring can be expressed by such a spined product.Theorem 3.2.1 Every strong distributive lattice S=[D,Sα,φα,β]of rectan-gular semiring Sα is a normal Clifford semiring if and only if E+(S)is left unitary in(S,+).