Furthermore, there is an order-preserving bijection from the set of all completely regular congruences on an eventually regular semigroup onto the set of all completely regular kernel normal systems for this semigroup.

The results that the strongly ordered congruences are the strongly regular congruences and that the converses are not true are proved.

What subsets of an ordered semigroup S can serve as a congruence class of certain regular congruence on S is still an open problem to be solved.

Introducing the concept of Rees matrix semigroups of matrix type,we prove the equivalence of completely simple matrix semigroups and this kind of Rees matrix semigroups, and characterize the minimal ideal of a topological matrix semigroup as well as the completely regular matrix semigroups.

In the second chapter ,we give the definition of the normal subset of aπ-regular semigroup S , the normal equivalence on E(S) and then we give the description of completely regular congruence pairs of S.

This paper first introduces the concepts of regular P-congruences and partial kernel normal system on S(P),then gives the regular P-congruences on S(P)an abstract characterization by means of the partial kernel normal system and a necessary and sufficient condition for a set B={B_i∶i∈I}of pairwise disjoint subsets of S(P)which is a partial kernel normal system in S(P)with P∩B_i as its C-set.