Studies On 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:The first chapter, we give some fundamental definitions which will be used in this passage.In the second chapter, we characterize some properties ofρ-quasi adequate semigroups, and conform the existence of the good congruence on it.The main results are given in follow:Definition 2.1.3 S,T are semigroups, p is a congruence on S. A semigroup homomorphismφ:S→T is called a good homomorphism, if for all a, b∈S, we can obtain aφ(?)*bφfrom a(?)ρb and aφ(?)*bφfrom a(?)ρb. A congruence T on semigroup S is called good congruence, if the natural homomorphism S→S/T is a good homo-morphism, that is for all a, b∈S, we can obtain aT(?)*bT from a(?)ρb, and aT(?)*bT from a(?)ρb.Lemma 2.1.4 S isρ-abundant semigroup,ρ,T are congruences on S, then the follow results are equivalent:(1) T is a good congruence;(2) for all a∈S, there existe idempotent e∈Laρ, f∈Raρ, so that for all x, y∈S1(a) (ax, ay)∈T(?)(ex, ey)∈T;(b) (xa, ya)∈T(?)(xf, yf)∈T.Definition 2.2.1 S is a semigroup,p is a congruence on S, A, B are nonempty classes on S, then we let AB={ab|a∈A,b∈B}, and Aρ={aρ|a∈A} is a subclass of semigroup{S/ρ,·). Definition 2.2.2 S is aρ-quasi adequate semigroup,ρis a congruence on S, we definite the equivalence relationδon S:Theorem 2.2.6 S is aρ-quasi adequate semigroup,ρ:δare eongruences on it:thenδis a good congruence on S.Theorem 2.2.9 S is aρ-quasi adequate,thenδis a congruence if and only if for all a,b∈S,aE(aρ*)E(bρ+)bρ(?)E((ab)ρ+)abE((ab)ρ*)ρ.In the third chapter,we obtain some.characterization theorems for perfect(?)ρ-abundant semigroups,and describe such semigroups from different aspects. The main results are given in follow:Lemma 3.1.9 Let S be a strong(?)ρ-abundant semigroup whose idepotents form a normal band E(s),ρis a congruence on S.Define a relationγon S by aγb if and only if a=ebf, where a.b∈S,e,f∈E(bρ*),thenγis a congruellce on S.Theorem 3.1.11 S is a strong(?)ρ-abundant semigroup,ρis a congruence on s,E(S)is a normal band,and(?)(?)ρ,γis the congruence on S aS defined in Lemma 3.1.9.Defined a relationρ' on S/γby aγρ'bγ(?)aρb:thenρ' is a congruence on S/γ.Lemma 3.1.12 S is a strong(?)ρ-abundant semigroup,ρis a congruence on S,and(?)(?)ρ,γis the congruence on S as defined in Lemma 3.1.9. IF for all a,b∈S,a(?)ρb,then aγ(?)ρ/γ(S/γ)bγ.Theorem 3.2.2 S is a strong(?)ρ-abundant semigroup,ρis a congruence on S,and E(S)is a normal band,then the following conditions are equivalent:(1)S is a perfect(?)ρ-abundant semigroup；(2)(ab)ρ*=aρ*bρ*for all a,b∈S.Theorem 3.2.3 S is a strong(?)ρ-abundant semigroup,ρis a congruence on S:and(?)(?)ρ,E(S)is a normal band,then the following conditions are equivalent:(1)S is a perfect(?)ρ-abundant semigroup；(2)S/γis a C-(?)ρ/γ-abundant semigroup,whereγis the congruence on S as defined in Lemma 3.1.9. Theorem 3.2.5 S is a strong (?)ρ-abundant semigroup,ρis a congruence on S, and (?)(?)ρ, then S is a perfect (?)ρ-abundant semigroup if and only if there exist a quasi-strong C-(?)ρ0-abundant semigroupT (whereρ0 is a congruence on T)and a surjective homomorphismφ:S→T satisfying if a(?)ρb then aφ(?)ρ0(T)bφsuch that for all e,f∈E(S), the resrictionφ|esf ofφto eSf is injective.Lemma 3.3.3 S is a strong(?)ρ-abundant semigroup, andρ, (?)ρare congru-ences on S, if E(S) is a semilattice, then the idempotents of S are central, that is S is a C-(?)ρ-abundant semigroup.Theorem 3.3.4 S is a strong (?)ρ-abundant semigroup on which (?)ρis a congruence. Then S is a perfect (?)ρ-abundant semigroup if and only if S is an orthodox locally C-(?)ρ-abundant semigroup.In the fourth chapter, we give a definition of idempotent-connectedρ-adequate semigroups and type A-ρ-ad equate semigroups. And also we give some equal descriptions of them. The main results are given in follow:Definition 4.2.1 S is a semigroup,ρis a left congruence on S, S is called idempotent-connected(IC) when for each element a∈S and for some aρ*∈Laρ∩E(S), aρ+∈Raρ∩E(S), there is a bijectionα:→, satisfying xa=a(xa) for any x∈. And there is a bijectionβ:→, satisfying ay=[yβ)a for any y∈. Now we callα,βare connected bijection, and aρ-abundant semigroup which satisfies idempotent-connected condition is called idempotent-connectedρ-abundant semigroup.Definition 4.2.5 S. isρ-adequate semigroup,ρis a left congruence on S, then the bijectionα,βdefined in Definition 4.2.1 are insomorphisms.Theorem 4.2.8 S is aρ-adequate semigroup,ρis a left congruence on S, then S is a type A-ρ-adequate semigroup if and only if for any a∈S, are inverse insomorphisms.Theorem 4.2.9 S isρ-adequate semigroup,ρis a left congruence on S, then S is idempotent-connected if and only if S is a type A-ρ-adequate semigroup.