Semidirect Products And Congruences Of Some Quasi-Regular Semigroups

Combination and separation of semigroups is one of the important fields in the research about semigroups, through which more properties of semigroups can be revealed. We now have many ways and means in the study on combination and separation of semigroups, and semidirect product has showed its advantages in the studing combination of semigroups. Thus, it is necessary and important to pay attention to study on semidirect product. By describing semidirect products and structures and congruences, we can manage to obtain some structures and congruences on this kind of semigroup.It is under the condition that semigroups have identity elements that most of researches on semidirect products. In[4],[6],[3],[5],the authors have given us descriptions of semidirect products and wreath products of monoids to be orthodox, quasi-regaular, weak Cliff or dian quasi-regular,strongly π-inverse semigroups. ; bothsides in [5], [6],the authors determine structures and the least group congruences. These results make us have a more explicit conception about semidirect products and structures and congruences on them. But the results above all base on the fact that semigroups have identity elements. Thus it is relatively has certain confinement.In order to break the confinement, in this paper we manage to study semidirect products of semigroups regardless of identity elements.That is, we get rid of the especially important condition that semigroups have identity elements. Unlike having identity elements, we describe the properties of semidirect products in virtue of S and Te(= , the subsemigroup of T) ,not in virtue of S and T. By finding the relation between E(S ×a T) and E(S),E(T),the relation between Reg(S ×a T) andR(S),R(T), we describe the relations between congruences on the semidirect product and congruences on semigroups. In addition, by constructing homomorphisms we obtain structures on semidirect products same or similar to semigroups. By this means, we get the descriptions of semidirect products to be strongly π-inverse semigroup,Clifford quasi-regular semigroup, strongly π-E-unitary inverse semigroup and E-unitary inverse semigroup. For strongly π-E-unitary inverse semigroup, we define it in the third chapter. The purpose of this paper is to discuss semidirect products of some quasi-regular semigroups which are strongly π-inverse semigroup,Clifford quasi-regular semigroup, strongly π-E-unitary inverse semigroup and E-unitary inverse semigroup. We not only give the necessary and sufficient conditions for the semidirect products of 5 and T to be these semigroups but also discuss structures on these semidirects or the relations between congruences on these semidirect products and semigroups. We define strongly π-E-unitary inverse semigroup in the third chapter. In this paper we give necessary and sufficient conditions for the semidirect products of 5 and T to be strongly π-inverse semigroup,Cilfford quasi-regular semigroup,strongly π-E- unitary inverse semigroup and E-unitary inverse semigroup in the four chapters. Furthermore, in the second chapter semidirect product of 5 and Te is discussed. We have the result that it is also Clifford quasi-regular semigroup. Besides semidirect product of S and Te is semilattice of quasi-groups. In the fourth chapter the paper discusses the following relation between the least group congruences on the E-unitary inverse semigroup and the semidirect product: Moreover we describe the following structure of the semidirect product of two E-unitary semigroups by McAlistertriple:.These results make the area of research about semidirect product not to be confined to monoids. Thus the application of semidirect product as an instrument of studing semigroups is extended.