Perfect congruences on bisimple $\omega$-semigroups

TL;DR

This paper characterizes perfect congruences on bisimple omegasemigroups, generalizing previous results on the bicyclic semigroup by leveraging their structural and congruence properties.

Contribution

It provides a complete characterization of perfect congruences on all bisimple omegasemigroups, extending earlier specific cases.

Findings

Complete description of perfect congruences on bisimple omegasemigroups

Generalization of previous results on bicyclic semigroup

Utilization of semigroup structure and congruence descriptions

Abstract

A congruence $\varepsilon$ on a semigroup $S$ is perfect if for any congruence classes $x\varepsilon$ and $y\varepsilon$ their product as subsets of $S$ coincides (as a set) with the congruence class $(xy)\varepsilon$. Perfect congruences on the bicyclic semigroup were found in \cite{key7}. Using the structure of bisimple $\omega$-semigroups determined in \cite{key25} and the description of congruences on these semigroups found in \cite{key20} and \cite{key1}, we obtain a complete characterization of perfect congruences on all bisimple $\omega$-semigroups, substantially generalizing the above mentioned result of \cite{key7}.