A network of congruences on an inverse semigroup

Abstract:

A congruence $\rho$ on an inverse semigroup $S$ is determined uniquely by its kernel and its trace. Denoting by ${\rho ^{\min }}$ and ${\rho _{\min }}$ the least congruence on $S$ having the same kernel and the same trace as $\rho$, respectively, and denoting by $\omega$ the universal congruence on $S$, we consider the sequence $\omega$, ${\omega ^{\min }}$, ${\omega _{\min }}$, ${({\omega ^{\min }})_{\min }}$, ${({\omega _{\min }})^{\min }} \ldots$. These congruences, together with the intersections of corresponding pairs, form a sublattice of the lattice of all congruences on $S$. We study the properties of these congruences and establish several properties of the quasivarieties of inverse semigroups induced by them.