Some idempotent-separating congruences on a ?-regular semigroups

Abstract

Edwards describes in [4] the maximum idempotent-separating congruence on a eventually regular (equivalently, ? -regular) semigroup which is given by $\mu={(a,b)∈ S? S: \quad\textit{if}\quad x∈ S\quad\textit{is regular then each of}\quad xRxa, xRxb\quad\textit{ implies }\quad xaHxb,\quad\textit{ and each}\quad xLax, xLbx\quad \textit{imples}\quad axHbx}$. In this paper we describe the maximum idempotent-separating congruence and their kernels on some subclasses of ?-regular semigroups. Also, we describe the minimum idempotent-separating r- semiprime congruence and its kernel on an r-semigroup.In this way we obtain a generalization of results of Meakin [10],[11], Feigenbaum [5],[6]and Howie[8].