Semigroups which Have a minimum primitive inverse congruence

Abstract

We investigate primitive inverse semigroups and Brandt semigroups as analogues of groups for semigroups with zero. We give a description of the minimum primitive inverse congruence on a categorical E *-dense E-semigroup. We show that a categorical semigroup S with a primitive inverse congruence has a minimum such congruence if $$\tilde D\left( S \right)$$ is *-dense in S. Here $$\tilde D\left( S \right)$$ is the least full, weakly self-conjugate, *-unitary and *-reflexive subsemigroup of S. Our results are analogous to those of Fountain, Pin and Weil for general semigroups.