On The Structures Of Some Generalized Regular Semigroups

In this paper, we mainly study some generalized regular semigroups. The main idea is to describe structures of generalized regular semigroups by generalized Green relations and the set of the idempotents of generalized regular semigroups. There are four chapters, 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 broad (n,m)-semigroups, quasi adequate broad (n,m)-semigroups and adequate broad (n,m)-semigroups are defined and their fundamental properties are discussed. The main results are given in follow:Theorem 2.1.10 Let S be a (n, m)-semigroup, ??LC(S), then ??Lp.Theorem 2.1.12 Let S be a (n,m)-semigroup, ?? LC(S), ??Sm, and e?Sm is a idempotent, then the following statements are equivalent:(1) (?)Lpe.(2) [(?)?](??L?)(?), and for any x,yES+, [(?)]?[(?)](?)[(?)]?[(?)].Theorem 2.2.5 Let S is a quasi strong ?- broad (n,m)-semigroup,??LC(S), e?E, then [(?)?Sm(?)?]is a quasi strong (n,m)-?broad sub-semigroup.Theorem 2.3.2 Let S is a strong broad (n, m)-semigroup,??LC(S), then(1) for all (?)?S+, (?)?Sm, [(?)]?*?[(?)?*]?*.(2) for all (?)?Sm, (?)?S+, [(?)]?+= [(?)?+(?)]?+.Theorem 2.3.3 Let S is a strong broad (n, m)-semigroup, ??LC(S), then for all (?)?Sm, (?)?E, [(?)?(?)]?*= [(?)?(?)?*], [(?)?]?+=[(?)?+(?)?].In the third chapter:we study the structure of good regular L?-orthogroup. Firstly, we give the definition of C-L?-rthogroup, i.e.L?-rthogroup with the set of idempotents being C- bands. Secondly, we give the structure of left regular L?-rthogroup,right regular L?-rthogroup, regular L?-rthogroup and LR-egular L?-rthogroup. At last, we obtain the spined product structure and the ?-product structure of these semigroups. The main result is given in follow:Theorem 3.2.7 Let S is a semigroups, there exist p E ?(S), Then S is a regular L?-orthogroup which satisfies (C1) if and only if S is isomorphic to a spined product Si×T?,? S2 of a left regularL?1- orthogroupS1=[Y; I?×T?]satisfying (C1) and a right regularL?2-orthogroupS2= [Y; T?×??]satisfying (C1), having a common C -L?-abundant semigroup component T with respect to the semigroup homomorphisms ?:(i, x)(?)x,(i, x) ?S1i and ?:(x,?) (?) x, (x,?)?S2, where ?i?LC(Si)(i= 1,2).In the fourth chapter, we mainly give the description of the structure of even-tually C -L?-bundant semigroups. Firstly, we give the definition of C -C?-bundant semigroups. Secondly, The eventually C-L?-abundant semigroups are introduced by using the inflation of semigroup. At last, we mainly give the description of the structure of eventually C -L?-abundant semigroups. The main result is given in follow:Definition 4.2.1 Let S be a eventually C -L?-abundant semigroups, if each C?-classes of S contain a idempotent, and the idempotent is central.Theorem 4.2.7 If S is a eventually C-L?-abundant semigroup, ??LC(S), e?E, then the following statements established:(1) L(?) is a congruence of semilattice on S;(2) Le(?) is a eventually C-L?-abundant semigroup;(3) Sis a semilattice of Lg(?)(g?E).Theorem 4.2.8 If S is a semigroup, ??LC(S), then the following state-ments are equivalent:(1) S is a eventually C-L?-abundant semigroup with only one L(?)- class;(2) S is a inflation of ?-left cancellative monoid;(3) S have a central idempotent, and meet the following ciditions:for allx, y?S2 Stands ? S (sx, sy)??(?)(x,y)??.