| In this dissertation,we mainly discuss some bi-semirings,some property theorems and structure theorems are given,the main idea is using the distributive lattice of bi-semirings to study the structure and property of the bi-semirings.The ddissertation is divided into three chapters,the main results axe given in follow:In the fist chapter,We study the bi-semirings whose(S,+)semigroup are semilattices、(S,·)semigroup are inverse semigroups and(S,*)semigroup are semilattices.In order to prove the bi-semirings S is a so-called distributive lattice,we construct the continuation between two partial orders which satisfies certainconditions.Let S∈(S|+)l∩(C|·)∩(S|*)l,In order to prove Green-H relation H on(S,·)semigroup of S to be a bi-semiring congruence,we construct three partial order relations on S.The main results are given in follow:Lemma 1.2.1 Let S ∈(S|+)l∩(I|·)∩(S|*)l,then the idempotent set E(S)of multiplication of S is a subbi-semirings of S.Lemma 1.2.2 Let S ∈(S|+)l∩(I|·)∩(S|*)l,if ≥+ and ≥*are continuations of ≤.,and for any a,b ∈ S,satisfies(a + b)-1 = a-1 + b-1;(a*b)-1=a-1*b-1,then ≥+=≤.=≥*,where ≥+ and ≥*respectively are the inverse of der of ≤+ and then ≥+=≤.=≥*,where ≥+and ≥*respectively are the inverse order of ≤+ andLemma 1.2.4 Let S ∈ (S|+)l ∩(I|·)∩(S|*)l,≥+=≤.=≥*,then S is a distributive lattice.Theorem 1.2.5 Let S ∈ (S|+)l ∩(I|·)∩(S|*)l,then the following statements are equivalent:(1)≥+ and ≥*are continuations of ≤.,and satisfies(a,b ∈ S)(a + b)-1=a-1 +(a*b)-1 = a-1*b-1;(2)≥+=≤.=≥*;(3)S is a distributive lattice.Lemma 1.3.1 Let S ∈ (S|+)l ∩(C|·)∩(S|*)l,then Q1,Q2 are subsemigroups of(S,·).Lemma 1.3.2 Let S ∈ (S|+)l ∩(C|·)∩(S|*)l,thenH is a congruence on S(?)(a + b)0 = a0 + b0,(a*b)0 = a0*b0,(?)a,b E S.Lemma 1.3.3 Let S ∈ (S|+)l ∩(C|·)∩(S|*)l,≤0 is a binary relation on S,which is defined as a ≤0b(?)a0 ≤b0,ba-1 ∈ Q1,(?)a,b ∈ S,then(S,·,≤0)is a partial orders Clifford semigroup.Lemma 1.3.4 Let S ∈ (S|+)l ∩(C|·)∩(S|*)l,≤0 is a binary relation on S,which is defined as a≤0b(?)a0 ≤ b0,ba-1 ∈ Q2,(?)a,b ∈ S.then(S,·,≤0)is a partial orders Clifford semigroup.Lemma 1.3.5 Let S ∈ (S|+)l ∩(C|·)∩(S|*)l,(?)e,f ∈ E(S),satisfies e+f = ef = e*f,and H is a bi-semirings congruence on S,then ≤+=≤0,≤*=≤0.Theorem 1.3.6 Let S ∈ (S|+)l ∩(C|·)∩(S|*)l,H is a bi-semirings congruence on S,then Q1 = Q2 = E(S)(?)≤.=≤0=≤0.In the second chapter:we study semilattice congruences and the smallest semilattice congruences on commutative distributive bi-semirings on distributive bi-semirings.The main results are given in follow:Lemma 2.1.3 Let S is a distributive bi-semiring,define a relation η:aηb(?)((?)e ∈ E(S))a + e =(a + e)(b + e)(a + e),b + e =(b + e)(a + e)(b+ e).then η is a congruence on S,and(S/η,·)is a semilattice.Lemma 2.1.4 Let S is a distributive bi-semiring,on S/r7,define a relationρ:aηρbη(?)((?)∈+E(S)(ae))η＝=(ae)η＋+(be))η＋+(ae)η,(be))η＝=(be)η+(ae)η +(be)η.then p is a congruence on S/η,and((S/η)/ρ,+)is a semilattice.Lemma 2.1.5 Let S is a distributive bi-semiring,define a relation λ on S/ηaηλbη(?)((?)e ∈*E(S))(ae)η =(ae)η*(be)η*(ae)η,(be)η =(be)η*(ae)η*(be)η.then A is a congruence on S/η,and((S/η)/λ,*)is a semilattice.Theorem 2.1.6 Let S is a,distributive bi-semiring,define σ:a,b ∈ S,aσb(?)aη(p ∨ λ)bη.then σ is a semilattice congruence on bi-semiring SLemma 2.2.2 Let S is a commutative distributive bi-semiring,define a relation η on S:a,b ∈ S,aηb=(?)3m,n ∈ Z+,x,y∈ S satisfies am=bx,bn = ay,then η is a congruence on bi-semiring S,and a smallest semilattice congruence on(S,·).Lemma 2.2.3 Let S is a commutative distributive bi-semiring,define a relation p on S/η:aηρbη(?)k,l∈Z+,uη,vη∈S/η,k(aη)=(bη)+(uη),l(bη)=(aη)+(vη),then p is a congruence on S/η,and a smallest semilattice congruence on(S/η,+).Lemma 2.2.4 Let S is a commutative distributive bi-semirings,define a relation λ on S/η:aηλbη(?)m,n ∈ Z+,sη,tη∈S/η,(aη)*m =(bη)*(sη),(bη)*n = {aη)*(tη),then λ is a congruence on S/η,and a smallest semilattice congruence on(S/η,*)Theorem 2.2.5 Let S is a commutative distributive bi-semirings,define a relation σ on S:a,b ∈ S,aσb(?)aη(ρ∨λ)bη,then σ is a smallest semilattice congruence on S.In the third chapter:we mainly discuss the relation of ideals of bi-semirings with multiplication zero elements and congruences.The main results are given in follow:Lemma 3.1 Let a≤b,c≤d,then a + c≤b + d,a*c≤b*d.Lemma 3.2 Let p is a congruence of S,the partial order relation correctly induced by multiplication“·”on S/p:aρ(?)bρ(?)(ab,b)∈p.Lemma 3.4 Let ρ∈ C(S),then e⌒ is a ideal,which writed for I(ρ).Lemma 3.5(x,y)∈ρ(I)(?)a,b∈I,satisfies ax=by.Theorem 3.6 Let I ∈ T(S),then I(ρ(I))= I.Theorem 3.8 Let I ∈ T(S),then ρ(I)∈ RC(S).Theorem 3.9 Let ρ ∈ RC(S),then ρ(I(ρ))= ρ.Theorem 3.10 There is a one-to-one correspondence between T(S)and RC(S). |
