Some Studies Of Congruences On E-inversive Semigroups

This Ph.D.dissertation consists of six chapters.We mainly study the properties and congruences of E-inversive semigroups.In the first chapter,we introduce and study the regular congruences on an E-inversive semigroup S.The kernel normal systems for an E-inversive semigroup S is introduced.We show that each regular congruence on S is uniquely determined by its kernel normal systems and describe the regular congruences on S in terms of their kernel normal systems.In Chapter 2,we investigate the relationship between the regular congruence and its characteristic trace on an E-inversive semigroup S.We establish the minimum and the maximum regular congruences on S with the same characteristic trace.A description of all regular congruences on S with the same characteristic trace T is investigated.We establish a one to one order-preserving map of the normal andτ-normal subset onto the set of all regular congruences with the same characteristic traceτon S.In Chapter 3,we study the relationship between the regular congruence and its kernel on an E-inversive semigroup S.The n-relation of regular congruences on an E-inversive semigroup is introduced by means of the kernels of the regular congruence. The least and the greatest elements in eachκ-class are established.We also give the connections between the two regular congruences on an E-inversive semigroup.In Chapter 4,we investigate the relationship between the congruences and the equivalence relation(?)((?)).The equivalence relationγon eventually regular semigroups is studies.We give some basic properties about the equivalence relation(?)((?))and discuss the relationship between(?)((?))and(?)((?))by means of homomorphic images.Chapter 5 is devoted to investigate the strong P-congruences on P-inversive semigroups S(P).We describe the minimum strong P-congruences whose hyper-trace co- incide with the hyper-trace of given congruences on S(P).Furthermore,we investigate the semigroup generated by certain operators on the strong P-congruence lattices of P-inversive semigroups.The properties about the subsemigroup of the transformation semigroup on Cp(P)generated by the transformations t:p→pt,T:p→pT, k:p→pk and K:p→pK,p∈Cp(S)are investigated.In Chapter 6,firstly we introduce the concept of fuzzy strong P-congruences on P-inversive semigroups S(P).Secondly we investigate the fuzzy strong P-congruences by means of 'weak P-inverse'.Lastly,the fuzzy strong P-congruences on P-inversive semigroups S(P)are described by means of certain fuzzy strong P-congruence pair (ξ,K),whereξis a certain fuzzy normal equivalence on P and K is a certain fuzzy normal subset of S(P).