Study On The Properties And Structure Of (?)-Abundant Semigroups

In this paper, we mainly study LU-abundant semigroups , and some properties and some structure theorems of such semigroups are given . The main idea is to describe structures and propertiees of generalized regular semigroups by generalized Green relations in generalized regular semigroups . There are three chapters , the main contest are given in follow:In the first chapter, we give the introductions and preliminaries.In the second chapter, we used Green =U relations to define LU-abundant semi-groups,strong LU abundant semigroups,C-LU- abundant semigroups, perfect LU-abundant semigroups, to discuss their fundamental properties.The main results are given in follow:Theorem 2.2.1 Let S be a strong LU - abundant semigroup and U is a normal band.We defined relation ? :a?b if and onlt if a = ubv, (u, v ? U(b)).Then ?is a congruence on S.Theorem 2.2.3 Let S is a strong LU- abundant semigroup and U is a normal band.Then the following statements are equivalent:(1) Sis a preferct LU- abundant semigroup;(2) (?)a, b ? S, (ab) = ab;(3) S/9 is a C-LU abundant semigroup.Theorem 2.2.4 Let S be a strong LU - abundant semigroup,LU is a congru-ence on S. If U is a semilattice, then S is a C-LU abundant semigroup.Theorem 2.2.5 Let S be a strong LU-abundant semigroup,LU is a congruence on S.Then S is a preferct LU - abundant semigroup if and only if S is an orthordoxloally C-LU- abundant semigroup.Theorem 2.2.6 Let S is a strong LU- abundant semigroup, then S is a preferct LU - abundant semigroup if and only if S is a semigroup and excit C-LU-abundant semigroup T and a epimorphism ? : S -?T of keep the relation LC ,and satisfied (?)u,v ? U,?|uSv, is a monomorphism and U? (?)C U(T).Theorem 2.3.2 Let S be a semigroup.Then the following statements are equivalent:(1) S is a preferct LU - abundant semigroup;(2) LU is a congruence on S,and S is a spined product of the C - LU -abundant semigroup M = [Y; M?;??,?]and a narmal band B = [Y;U?;??,?} and U??Y{1?} ×Ua is a sub-semigroup;(3) LU is a congruence on S,and S is a strong semilattice of planks [Y;S?;??,?]and U?a?Y{1?} × U?a is a sub-semigroup.In the third chapter: we give the description of the structure of orthodox superLU -abundant semigroup. The main result is given in follow:Theorem 3.2.1 Let S is a semigroup. Then the following statements are equivalent:(1) S is a strong LU -abundant semigroup and U ? I× {1T}×? is a rectanglar band;(2) S is a orthodox super LU - abundant semigroup and U = I× {1T}×? is a rectanglar band;(3) S is equivalent to a rectangular unipotent semigroup I× T×?.Theorem 3.2.2 Let S is a semigroup. Then the following statements are equivalent:(1)S(U)is an orthodox super LU - abundant semigroup;(2)S ?[Y;S? ? I?×T?×??](? ? Y),S? is a rectangular unipotent semigroup,U=U??Y U?is a band.And (?)(i, a,? A)?S? , (j,?)? I?× ??,(k, v) ?I?×??[(i,a,?)(j, 1T?,?)]PI?? = [(i,?, ?)(k, 1T?,?)]PI??[(i ,1T?,A)(j,1T?,?)]PI??=[(i,1T?,?)(k,1T?,?)]PI??...