Completely simple congruences on E-inversive semigroups

Abstract

We study completely simple congruences on an arbitrary [Formula: see text]-inversive semigroup [Formula: see text]. In particular, we show that every such congruence [Formula: see text] on [Formula: see text] is uniquely determined by its kernel and trace, and that the trace of [Formula: see text] is a congruence on the biordered set [Formula: see text]. Moreover, we investigate the complete lattice of all completely simple congruences on [Formula: see text] and show that the trace relation is a complete congruence on this lattice. We also construct a family of completely simple congruences on [Formula: see text].