Semidirect Products And Congruences Of Some Semigroups

In this dissertation, we mainly discuss semidirect products of three quasi-regular semigroups, such as left Clifford quasi-regular semigroups,right Clifford quasi-regular semigroups ,left normal quasi-regular semigroups and LR-C semigroups.lt is under the condition that semigroups have no identity elements that we get these results.We describe the closeness of semidirect products of the four semigroups.In the fourth chapter we define left normal quasi-regular semigroups . Unlike the discussion of monoids,we describe the properties of semidirect products in virtue of S and T6,the subsemigroup of T,not in virtue of S and T.In Chapter l,we give the introduction and preliminaries.In Chapter 2,we describe the necessary and sufficient conditions for the semidirect products of S and T to be left Clifford quasi-regular semigroups.In Chapter 3,we describe the necessary and sufficient conditions for the semidirect products of 5 and T to be right Clifford quasi-regular semigroups.Theorems.1 Let 5,T be two semigroups,a : a(s) be a given homomorphism,then the semidirect product S xf, T is a right Clifford quasi- regular semigroup if and only if:In chapter 5,we give the necessary and sufficient conditions of a regular semigroup: LR-C semigroup.Theorems.3 Let 5, T be two semigroups,o :(s) be a given homomorphism,then the semidirect product S xa T is a LR-C semigroup if and only if:(i)for every e € E(S), 5 is a LR-C semigroup and Te is a regular semigroup ,whereThese results make the area of research about semidirect products not to be confied to monoids.Thus the application of semidirect products as an instrument of studing semigroups is extended.