Strong R-Unipotent Congruences And Strongly Inverse Congruences On E-Inversive Semigroups

In this paper, we mainly describe strong R-unipotent congruences and strongly inverse semigroup congruences on E-inversive semigroups. It is com-posed of four sections.The first section is an introduction.The second section is preliminaries.In the third section, we describe strong R-unipotent congruences on E-inversive semigroups. We obtain that each strong R-unipotent congruence on E-inversive semigroups is uniquely determined its kernel and hyper-trace. Furthermore, strong R-unipotent congruences on an E-inversive semigroup are described by means of strong R-unipotent congruence pairs for the E-inversive semigroup. It is obtained that the partial order set of all strong R-unipotent congruences on an E-inversive semigroup S is mutually inverse order isomor-phic to the partial order set of all strong R-unipotent congruence pairs for S.In the four section, we describe strongly inverse congruences and group congruences on E-inversive semigroups. We obtain that strongly inverse con-gruences on an E-inversive semigroup are described by means of strongly in-verse congruence pairs for the E-inversive semigroup. Moreover, it is obtained that the partial order set of all strongly inverse congruences on an E-inversive semigroup S is mutually inverse order isomorphic to the partial order set of all strongly inverse congruence pairs for S. We also describe group congruences on E-inversive semigroups in this section. We present here that group congruences on E-inversive semigroups are uniquely described by their kernels.