Congruence-free Restriction Semigroup

Restriction semigroups are common generalizations of ample semigroups and inverse semigroups.In this paper,we mainly investigate restriction semigroups with certain congruence properties.The paper is divided into five sections.Firstly,we provide some preliminaries of ample semigroup and restriction semigroup.We get some results of congruence of restriction semigroups.In Section 4 we give some characterizations of restriction semigroups each of whose proper(2,1,1)-congruences are reduced,so called H-reduced restriction semigroups.And we research congruence-free restriction semigroups.As its applications,then we establish the construction theorem of congruencefree restriction semigroups.Finally,in Section 5 we have added a few additions to congruence-free restriction semigroups.These results extend and enrich the related results of inverse semigroups.