Studies On Some Regular Semigroups And Some Generalized Regular Semigroups

This passage gives the definitions of some Generalized Regular Semigroups, and the theorems and some property of such semigroups are given. The main results are given in follow:In the first chapter , we give the introductions and preliminaries.In the second chapter , we give the definition of PI-strongly (?)-abundant semigroups and a theorem which discribes the structure of PI-strongly (?)-abundant semigroups.The main results are given in follow:Definition 2.1.1 S is a semigroup.If each (?)-class of S contain an idempotent,then S is called a (?)-abundant semigroup.In particular,if for any a∈S, the set L_a^ρ∩I_a contains a unique element a^ρ, then S is called a strongly (?)-abundant semigroup.Definition 2.1.3 S is a semigroup.If S is strongly (?)-abundant semigroups which satisfies replacement identical conditions,then S is called PI-strongly (?) -abundantsemigroup.Theorem 2.2.1 S is a PI-strongly (?)-abundant semigroup if and only if S is the spined product of some normal band B = [Y; B_α] and an exchange C-(?)-orthogroup T = [Y; T_α] about semilattice Y.In the third chapter ,we give a definition of W-adequate semigroup. and study the keep (?) congruence on W-adequate semigroup.The main results are given in follow:Definition 3.2.1 S is a semigroup.S is called idempotent-connected when for each element a of S and for some a^**∈L_a^**∩E(S), a^-∈R_a^**∩E(S), there is a bijectionα:< a^- >→< a^** >, satisfying xa = a(xα), for any x∈< a^- >.And there is a bijectionβ:< a^** >→< a^- >, satisfying ay = (yβ)a,for any y∈< a^** > . Now,we callα,βare connected bijection ,and a W-abundant semigroup which satisfies idempotent-connected condition is called idempotent-connected W-abundant semigroup.Theorem 3.2.7 S is W-adequate semigroup,then S is W-A semigroup if and only if for any a in S,α_a : a^-E→a^**E,x (?) (xa)^** andβ_a : a^**E→a^-E, y (?) (ay)^- are inverse isomorphisms.Theorem 3.2.8 S is W-adequate semigroup,then S is idempotent-connected if and only if S is W-A semigroup.In the fourth chapter ,we give a definition of Quasi-semiadequate semigroup, semiadequate semigroup and the minimum semiadequate keep (?) congruence on Quasi-semiadequate semigroup.The main results are given in follow:Definition 4.1.1 A semigroup S is called Quasi-semiadequate semigroup when it is semiabundant and its idempotents form a subsemigroup.Definition 4.1.2 A Quasi-semiadequate semigroup S is called semiadequate semigroup when its idempotents form a semilattice.Theorem 4.2.8δ= {(a,b)|E(a')aE(?) = E(b')bE(?)} is in any subsemiadequatecongruence.Theorem 4.2.9 S is Quasi-semiadequate semigroup ,δis a congruence,thenδ.is a keep (?) congruence if and only if aδ∈E(S/δ), then exists x∈E(S),aδ= xδ.Theorem 4.2.10 If 6 is a keep (?) congruence, thenδis the minimum semiadequate keep (?) congruence on S.Theorem 4.2.11δis congruence if and only if (?)a, b G S,aE(?)E(b')b (?) E((ab)')abE(?).Theorem 4.2.13 (?)∩δ=1.In the fifth chapter,we give a definition of (?)-semiabundant semigroup, strongly-(?) -semiabundant semigroup,A-(?)-semiabundant semigroup and a structure theoremof the A-(?)-semiabundant semigroup's translational hull, when (?) is right congruence. The main results are given in follow:Definition 5.1.3 A semigroup S is called A-(?)-semiabundant semigroup, if S is strongly-(?)-semiabundant semigroup,E(S) is semilattice. Theorem 5.2.11 If S is A-(?)-semiabundant semigroup, (?) is right congruence,then the translational hall of S is also A-(?)-semiabundant semigroup.