Semidirect Products And Structures Of Some Generalized Regular Semigroups

In this dissertation, we mainly discuss semidirect products of some generalized regularsemigroups, such as C - rpp semigroups, right adequate semigroups,C - wrpp semi-groups.left C - rpp semigroups and perfect rpp semigroups. It is under the condition that semigroups have no identity elements that we get the rusults about C - rpp semigroups and right adequate semigroups.In the first chapter , we give the introductions and preliminaries.In the second chapter , we describe the necessary and sufficient conditions for the semidirect products of S and T to be C - rpp semigroups.The main results are given in follow:Theorem 2.1.1 Let S, T be two semigroups,α: S→End(T), s (?)α(s) be a given homomorphism,then the semidirect product S×αT isaC- rpp semigroup if and only if:(i)S and T are C - rpp semigroups;(ii)for every e∈E(S),t∈T,then te = i;for every f∈E(T), s∈S.then fs = f; and for every x∈S, g∈Mt,then g∈Mtx.In the third chapter ,we describe the necessary and sufficient conditions for the semidirect products of S and T to be right adequate semigroups.The main results are given in follow:Theorem 3.1.1 Let S, T be two semigroups,α: S→End(T), s (?)α(s) be a given homomorphism,and for all e∈E(S),α(e) = 1∈End(T),then the semidirect product S×αT is a right adequate semigroup if and only if:(i)S and T are right adequate semigroups;(ii)for every x∈S,f∈E(T), t∈T,if f∈Mt,then fx∈Mtx.Theorem 3.2.1 Let S, T be two semigroups,α: S→End(T), s (?)α(s) be a given homomorphism.then the semidirect product S×αT is a right adequate semigroup if and only if:(i)for every e∈E(S), 5and Te are right adequate semigroups;(ii)if (e, f)∈E(S xa T),then e∈E(S), f = fe∈E(T);(iii)for every e,f∈E(S),u,v∈T,if ueu = u,vfv = v,then (uv)e = (uv)f;(iv)for every e∈E(S), t∈T.,x∈S,if f∈MteTe,then fe∈Mtex;(v)for every s∈S,t∈T.there exists e∈E(T) s.t.se = s,te = t and (e,t)(?)(s,t).In the forth chapter ,we describe the necessary and sufficient conditions for the semidirect products of S and T to be C - wrpp semigroups.The main results are given in follow:Theorem 4.1 Let S, T be two monoids.α: S→End(T),s (?)α(s) be a given homomorphism of monoids,then the semidirect product S×αT is & C - wrpp semigroup if and only if:(i)S and T are C - wrpp semigroups;(ii)for every e∈E(S),t∈T,then te = t;for every f∈E(T), s∈S,then fs = f;(iii)if e∈Ms,then (e, 1)∈M(s,1);if f∈Mt,then for every x∈S,then f∈Mtx.In the fifth chapter ,we describe the necessary and sufficient conditions for the semidirectproducts of S and T to be left C - rpp semigroups .The main results are given in follow:Theorem 5.1 Let S, T be two monoids.α: S→End(T).s (?)α(s) be a given homomorphism of monoids.then the semidirect product S×αT is a left C-rpp semigroup if and only if:(i)S and T are left C - rpp semigroups;(ii)for every e∈E(S), t∈T,then te = t;for every f∈E(T), s∈S,then fsf = fs;(iii)for every t∈T,x∈S,if f∈Mt,then fx∈Mtx;(iv)for every s∈S.every t∈T.there exists unique f∈Mt,s.t. fst = t.In the sixth chapter ,we describe the necessary and sufficient conditions for the semidirect products of S and T to be perfect rpp semigroups.The main results are given in follow: Theorem 6.1 Let S,T be two monoids,α: S→End(T),s (?)α(s) be a given homomorphism of monoids,then the semidirect product S×αT is a perfect rpp semigroup if and only if:(i)S and T are perfect rpp semigroups;(ii)for every t∈T,x∈S.then (tx)+ = (t+)x = (t+)x+,tx+t+ = t:(iii)if (e, f)∈M(s,t),and s.t.es = s, fst = t,then f = t+;(iv)if e1,e2,e3,e4∈E(S), f1 = f1e1f1,f2 = f2e2f2.f3 = f3e3f3,f4 = f4e4f4∈T,thenf1e2e3e4f2e3e4f3e4f4=f1e3e2e4f3e2e4f2e4f4...