E-inversive Semigroups And E * - Dense Semigroup And Its Congruence

In this paper,we mainly study certain properties and congruences of E-inversive semigroups and E~*-dense semigroups.It is composed of five sections.The first section is a brief introduction.In the second section,the basic concepts and necessary preliminaries are given.Basic properties of certain class of E-inversive semigroups and E~*-dense semigroups are investigated,respectively.In the third section,it is divided into two parts.In the first part,some properties of E-inversive semigroups and special E-inversive semigroups are obtained.Moreover,some properties about sandwich sets of regular semi-groups are extended to E-inversive semigroups.In the second part,we get some properties of E~*-dense semigroups.The equivalent condition of categorical E~*-dense E-semigroup is given.The fourth section is devoted to the characterizations of congruences on E~*-dense E-semigroups.Firstly,letρbe a O-restricted congruence on S containing D.Then we describe the O-restricted inverse congruence on categorical E~*-dense E-semigroups relying,onρ.Secondly,relying on a given congruence satisfying condition(Ω),we obtain some special congruences,such as the O-restricted inverse congruence on categorical E~*-dense E-semigroups,the O-restricted band congruence on completely categorical E~*-dense E-semigroups, the O-restricted semilattice congruence on completely categorical E~*-dense E-center semigroups.In the fifth section,we give some equivalent characterizations of idempotent-separating congruences on special E-inversive semigroups and E~*-dense semi-groups. The equivalent characterizations of the least group congruence on E-inversive semigroups are given.