The Structure And Congruence Of Some Bi-Semirings

In this dissertation,we mainly discuss the structures and congruences of the bi-semirings,give the relationship between the congruences of the bi-semirings with multi-plication identity element and their strong ideals,characterize the congruence pairs of thebi-semirings with reversible addition,find out some necessary and sufcient conditions ofchanging the bi-semirings without multiplication identity element into the bi-semiringswith multiplication identity element,and discuss the structures of a quasi distributivelattice of bi-semirings with multiplication identity element.This article is divided intofour chapters:In the first chapter,we give the introductions and preliminaries.In the second chapter,we mainly discuss the structures and congruences of the bi-semirings with multiplication identity element,find out the one-to-one correspondencebetween the congruences of the bi-semirings with multiplication identity element andtheir strong ideals,and characterize the congruence pairs of the bi-semirings with re-versible addition.The main results are given as follows:Lemma 2.1.2 Let R be the strong ideal of S,defineσon S:σ={（x, y）∈S×S|xR = yR}Thenσis the congruence of S§and the kernel ofσis K = R.Theorem 2.1.4 Let （S,+,·, ） be the bi-semirings with multiplication identityelement, （S,·）is a group, s∈S,s = ss + s = s + ss, s = ss s=s ss,Then there is a one-to-one correspondence between the congruences of S and strong ideals of S.In the third chapter,we give the least quasi bi-ring congruence of bi-semirings thattheir multiplication semigroups become inverse semigroups,and the conditions that thesets of their idempotents become strong ideals.The main results are given as follows:Theorem 2.2.3 Let （S,+,·, ） be the bi-semirings with multiplication identityelement,and （S,·） is an inverse group,defineρon S as followsμρ= {（x, y）∈S×S | e, f∈E[·],such that xe = yf},（E[·]= {e∈S|ee = e}）.Thenρis the least quasi bi-ring congruence of S.Theorem 2.2.4 Let （S,+,·, ） be the bi-semirings with multiplication identity element,and the idempotents of （S,·） are commutative, s∈S,1 = s + 1, 1 = 1 + s, 1 =1 s,1 = s 1,and （S, ）,（S,+）satisfied with cancellation law,then the set E[·]= {e∈S|ee =e} is an ideal of S.Moreover,if （S,·） is an inverse group and satisfied with cancellationlaw,then the set E[·]is a strong ideal of S.Theorem 2.3.9 Let （S,+,·, ）be the bi-semirings with reversible addition,ρbe thebi-semirings congruence on S,then （Kerρ, trρ） is a congruence pair of S;Conversely,if（N,τ） is a congruence pair of St,thenρ（N,τ）= {（a, b）∈S×S|（a + a, b + b）∈τ, a + b∈N}is a bi-semirings congruence on S,and Kerρ（N,τ）= N,trρ（N,τ）=τ,ρ（Kerρ,trρ）=ρ.In the fourth chapter§we discuss how to change the bi-semirings without multipli-cation identity element into the bi-semirings with multiplication identity element and thestructures of a quasi distributive lattice of the bi-semirings with multiplication identityelement is given.The main results are given as follows:Theorem 4.1.1 Let（S,+,·, ） is a bi-semiring§1∈/ S,if S satisfies conditions:（1） （?）s∈S {1},1_s = s1 = s;（2） （?）s∈S {1},1 + s = s + 1 = 1（3） （?）s∈S {1},1 s = s 1 = 1Then （S {1},+,·, ）is the bi-semiring with multiplication identity element,if and only ifS satisfies conditions: s, x∈Ss = sx + s = xs + s = s + sx = s + xs = sx s = xs s = s sx = s xsTheorem 4.1.2 Let（S,+,·, ） is a bi-semiring§1∈/ S,if S satisfies conditions:（1） （?）s∈S {1},1_s = s1 = s;（2） （?）s∈S {1},1 + s = s + 1 = s（3） （?）s∈S {1},1 s = s 1 = sThen （S {1},+,·, ）is the bi-semiring with multiplication identity element,if and only ifS satisfies conditions: s, x∈Ssx = x + sx = sx + x = s + sx = sx + s = x sx = sx x = s sx = sx s;s x = s + （s* x） = （s* x） + s = x + （s* x） = （s* x） + x;s + x = s* （s + x） = （s + x） s = x *（s + x） = （s + x）* x.Theorem 4.1.3 Let（S,+,·, ） is a bi-semiring§1∈/ S,if S satisfies conditions:（1） （?）s∈S {1},1_s = s1 = s;（2） （?）s∈S {1},1 + s = 1, s + 1 = s（3） （?）s∈S {1},1 s = 1, s 1 = s Then （S {1},+,·, ） is the bi-semiring with multiplication identity element,if and only ifS satisfies conditions: s, x∈Ss = s + sx = s + xs = s sx = s xs;sx = sx x = sx s = sx + x = sx + s;s+x = （s* x） + s = （s* x） + x;s+x = （s + x） s = （s + x） x.Lemma 4.2.2 Let S = [D; S_α],then （S, +,·, ） is a bi-semiring.Theorem 4.2.3 Let S = [D; S_α], a∈S_α, b∈Sβ, （α,β∈D）,if a·1_αβ= b·1_αβ, thenδ≤αβ, a·1δ= b·1δ（C₄）;ifδ≤αβ, a·1δ= b·1δ, then a·1_αβ= b·1_αβ（C₅）;Defineρon S, a∈S_α, b∈Sβaρb a·1_αβ= b·1_αβ, （ ）;Thenρis a bi-semiring congruence of S,and S is a subdirect product of a distributivelattice D and a bi-semiring S/ρ;Conversely,if there exists the same congruence as （ ） onS = [D; S_α]t,and 1_αβ= 1_α·1_β, （α,β∈D）, then S satisfies （C₄）, （C₅）.Theorem 4.2.4 Let S = [D; S_α],ifα,β∈D, 1_α·1_β= 1_αβ,then S = D; S_α,ψα,β.