Congruences on graph inverse semigroups

TL;DR

This paper characterizes congruences on graph inverse semigroups using the structure of the underlying graph, showing finite cases are congruence Noetherian, thus advancing understanding of their algebraic properties.

Contribution

It provides a graph-based description of congruences on inverse graph semigroups, extending previous results on congruence conditions.

Findings

Congruences are described in terms of the underlying graph structure.

Finite inverse graph semigroups are congruence Noetherian.

Abstract

Inverse graph semigroups were defined by Ash and Hall in 1975. They found necessary and sufficient conditions for the semigroups to be congruence free. In this paper we give a description of congruences on a graph inverse semigroup in terms of the underlying graph. As a consequence, we show that the inverse graph semigroup of a finite graph is congruence Noetherian.