The Structure And Properties About Some Generalized Regular Semigroups

In this paper, we mainly give the definitions of some generalized regular semi-groups,and some properties and some structure theorems of such semigroups are given.The main idea is to describe structures of generalized regular semigroups by generalized Green relations in generalized regular semigroups.There are four chap-ters,the main contest are given in follow:In the first chapter, we give the introductions and preliminaries.In the second chapter, We extent Green**relations from usual semigroups to (n, m)-semigroups,so that the corresponding notions such as Left congruence、Right congruence、Quasi strong wrpp(n, m)-semigroups、strong wrpp(n, m)-semigroups are defined and their fundamental properties are discussed. The main results are given in follow:Theorem2.1.7Let S be a (n, m)-semigroup,then L**is a right congruence, R*is a left congruence.Theorem2.1.9Let (S,[]) be a (n, m)-semigroup, a∈Sm, and e∈Sm is a idempotent, Then the following statements are equivalent:(1)aL**e.(2) a(RnL**)[ae△], and for any x, y∈S+, such that[ax]R[ay](?)[ex]R[ey].Theorem2.1.10Let (S,[]) be a(n,m)-semigroup, for any a、b∈Sm, if aLb, then aL**b.Theorem2.2.6Let S is a Quasi strong wrpp(n, m)-semigroup, Under the condition of def2.2.5and e∈E, then [e△Sme△]is a (n,m)-**sub-semigroup.Theorem2.3.2Let S is a strong wrpp (n,m)-semigroup, then(1)for all a∈S+, b∈Sm,[ab]*=[ab*]* (2)for all a∈Sm,b∈S+,[ab]+=[a+b]+.(3)for all∈Sm,e∈E,[e△a]*=[e△a*],[ae△]+=[ae△].In the third chapter,we mainly discuss the structure of quasi U-abundant semi-groups with weak normal idempotents.Secondly,we obtain some characterizations of such semigroups.Thirdly,we establish the structure of broad semigroups with weak normal idempotents.The main result is given in follow:Theorem3.3.3In the semigroup(SQ,o),(?)(x,s,y),(u,t,v)∈SQ,if Psuch that(C5)令x∈L,y∈R,s∈T,if x∈LSU.,y∈RSU ands=spyxS,则s=pyx then the following results are true:(1)(x,s,y)LU(SQ)(u,t,v)(?)sLUt,y=v(2)(x,s,y)RU(SQ)(u,t,v)(?)sRTt,x=u(3)SQ is a U-abundant semigroup.(4)if SQ is a quasi U-abundant semigroup,(ε1,ε,εr,)is a idempotent wn-of SQ.Theorem3.3.4Let(T;L,R;P)be a SQ system.If P satisfies(C5),then SQ(T;L,R;P)is a quasi U(SQ)-abundant semigroup with a weak normalidempotent-(ε1,ε,εr)；otherwise,quasi U-abundant semigroups with weak normal idempotents can also be made as this.In the fourth chapter:we mainly discuss the structure of U-abundant semi-groups with normal medial idempotents.Secondly,we obtain some characterizations of such semigroups.Thirdly,we establish the structure of broad semigroups with normal medial idempotents.The main result is given in follow:Theorem4.3.2Given U(W)={(e,x,f)∈W|x∈U°,fe=x),then(1)W=W(U,S)is a U-abundant semigroup;(2)U(W)={(e,x,y)∈W|x∈U°}≌U;(3)(u,u,u)is a normal medial idempotent of W(4)(u,u,u)W(u,u,u)≌. Theorem4.3.5Let S is a U-abundant semigroups with normal medial idempotent, U=<U> is a regular semigroup of idempotent generated, then S≌W(U,uSu).