The Structure And Congruence Of Some Semirings And Bi-Semirings

In this dissertation,we mainly discuss some semirings and bi-semirings.Some property theorems and structural theorems of them arc given.This dissertation is divided into two chapters.The main results are given in follow:In part one of chapter one,we study the semiring whose additive semiring is a semilattice and multiplicative semiring is nil extensions of a rectangular group.According to the subset of semiring,we construct relation on multiplicative semi-group.We obtain sufficient and necessary condition for H^*to be a congruence on semirings(_Reg（S）.+).We give the conditions under which the nil extension of rectangular group can be transformed into the nil extension of rectangular band,and the conditions under which the nil extension of rectangular group can be tra.ns-formed into the nil extension of rectangular band.We extend the properties of the nil extension of rectangular group to the nil extension of rectangular band.The main results are as follows:Lemma 1.2.2 let semiring S be a semiring whose additive semiring is a semi-lattice and multiplicative semiring is nil extensions of a rectangular group.Then H^*is a congruence on S if and only if（a⁰+b⁰）⁰=（a+b）⁰,a⁰b⁰=（ab）⁰.Theorem 1.2.5 let semiring S be a semiring whose additive semiring is a semilattice and mulliplicative somiring is nil extensions of a rectangular group).S satisfies（1）（?）e,f∈E（S）,e+f=ef,（2）H^*is a congruence relation on S;（3）（Reg（（?））,+,·）is a semiring.Then （?）is a congruence relation on semiring（Req（S）,+,·）if and only if≤₊=≤₀.Theorem 1.2.G let semiring S be a semiring whose additive semiring is a semilattice and multiplicative seiniring is nil extensions of a rectangular group.H^*is a congruence on S,then Q₁ |_Reg（S）=Q₂|(_Reg（S）)=E（S）if and only if ≤=≤0=≤O.Corollary 1.2.7 Let S be a semiring whose S is semilattice and S is nil extension of a rectangular band,then H^*is a congruence relation of S if and only if（a⁰+b⁰）⁰=（a+b）⁰,a⁰b⁰=（ab）⁰.Corollary 1.2.8 Let S be a semiring whose（?）is a semilattice and S is nil extension of a rectangular group.If H^*is a congruence relation on S,thenQ₁ is a.subsemigroup of S.Corollary 1.2.9 Let S be a semiring whose（?）is a semilattice and S is nil extension of a rectangular group,then Q₁ |_Reg（S）=Q₂|_Reg（S）=Q₃|_Reg（S）=_?.Corollary 1.2.10 Let S be a semiring whose S is semilattice and S is nil extension of a rectangular group.If H^*is a congruence on S,then ≤_.=≤⁰=≤₀=（≤’）⁰Corollary 1.2.11 let semiring S be a semiring whose additive semiring is a semilattice and multiplicative semiring is nil extension of a rectangular band.S satisfies（1）（?）e,f∈E（S）,e+f=ef;（2）（?）is a congruence relation on S;（3）（E,+,·）is a semiring.Then （?）is a congruence relation on semiring（（?）,+.·）if and only if≤+=≤0.Corollary 1.2.12 let semiring S be a semiring whose additive semiring is a semilattice and multiplicative semiring is nil extensions of a rectangular band.H^*is a congruence relation on S.Then Q₁ |Reg（E）=Q₂ | E)=E（S）if and only if≤_.=≤⁰=≤₀.Secondly,we study the structure of a distributive semiring whose multiplica-tive semigroup is a GV-semigroups.We define GV-distributive semirings,GV-distributive inverse semirings,GV-distributive inverse semirings,complete Archimedean semirings and quasi-banded semirings We study the properties of GV-distributive semirings.GV-distributive inverse semirings,and complete Archimedean semi-groups.A ccording to the properties of GV-semigroups,GV-inverse semigroups and completely Archimedeansemigroups,the properties of GV-distributive semirings,GV-distributive inverse semirings and completely Archimedeansemrings are stud-ied.The main conclusions are as follows:Lemma 1.3.6 Let S be a GV-distributive inerse semiring and E（?）.And for all e,f∈（?）,we have e=e（e+f）,e＋f=（f+e）（e+f）（f+e）.Then （?）is a distributive lattice congruence on a π group semiring.Corollary 1.3.8 Let S be a GV-distributive inverse semiring and E ∈（?）.If （?） is a quasi-band semiring,then E is a distributive lattice.Theorem 1.3.9 Let S be a distributive semiring and E（?）.Then S is a GV-distributive inverse semiring and E is a distributive lattice if and only if（1）S is a distributive lattice of π—group:（2）（?）e.f E E,we have e=e（e+f）,（f+e）=（f+e）（e+f）（f+e）.Theorem 1.3.10 Let S be a dist.ributive semiring and （?）-Then S is a GV-distributive inverse semiring if and only if S is a multiplicative semilattice semiring of a π—group semiring.Theorem 1.3.12 Let S be a distributive semiring,and E（?）.Then S is a GV-distributive inverse semiring and E is a Quasi band semiring if and only if（1）S is a distributive lattice of a complete Archimedean semiring;（2）（?）e,f ∈（?）,we have e=e（e+f）e,e+f=（f+e）（e+f）（f+e）.Chapter 2:In this ehapter,we study the minimum semilattice congruence on idempotent distributive bi-semirings and commutative distributive bi-semiring.The main conclusions are as follows:Lemma 2.2.1 Let S be a idempotent distrilbutive bi-semiring.Define rela-tionship ηon S:aηb（?）a=aba,b=bab.Then η is a congruence on S and it is a minimum semilattice congruence on（S,·）.Lemma 2.2.2 Let S be a idempotent distributive bi-semiring.Define rela-tionship p on S/η:aηρbη（?）aη+bη+aη,bη=bη+aη+bη.Then p is a congruence on S/η.And it is a minimum semilattice congruence on（S/η,+）.Lemma 2.2.3 Let S be a idempotent distributive bi-semiring.Define rela-tionship λ on S/η:aηρbη（?）aη+aη*bη*aη,bη=bη*aη*bη.Then A is a congruence on S/η.And it is a minimum semilattice congruence on（S/η,·）.Theorem 2.2.4 Let S be a idempotent distributive bi-semiring.Define rela-tionship a on S/η（a,b ∈ S）aσb（?）aη（ρ ∨ λ）bη.Then σ is a minimun semilattice congruence on S.Lemma 2.3.1 Let S be a commuta.tive distributive bi-semiring.Define rela-tionship ξ on S:a,b ∈ S.aξb（（?）m.n ∈Z+,x,y ∈ S）ma=b x.nb=a+y,Then ξis a congruence on S.And it is a minimum semilattice congruence on（S,+）Lemma 2.3.2 Let S be a commutative distributive bi-semiring.Define rela-tionship（?）on S/ξ:aξ（?）bξ（?）（（?）k,l ∈ Z+,uξ,vξ∈Sξ）（aξ）k=bξuξ,（bξ）l=aξvξ.Then（?）is a congruence on S/ξAnd it is a minimum semilattice congruence on（S/ξ,·）.Lemma 2.3.3 Let S be a commutative distributive bi-semiring.Define rela-tionship ε on S/ξ:aξεbξ（?）(（?）k,l ∈Z+,uξ,vξ∈Sξ（aξ）*k=bξ*uξ,（bξ）*l,=aξ*vξ,Then ε is a congruence on S/ξ And it is a minimum semilattice congruence on（S/ξ,*）.Theorem 2.3.4 Let S be a commutative distributive bi-semiring.Define rela-tionship θ on S:（a,b ∈ S）aθb（?）aξ（（?）∨ ε）bξ.Then 0 is a minimum semilattice congruence on S.