On lattice of congruences on graph inverse semigroups

TL;DR

This paper investigates the structure of the lattice of congruences on graph inverse semigroups, revealing it is upper semimodular but not lower semimodular, based on recent graph-based descriptions.

Contribution

It provides a detailed lattice-theoretic analysis of congruences on graph inverse semigroups, highlighting their semimodularity properties.

Findings

Lattice of congruences is upper semimodular.

Lattice of congruences is not lower semimodular.

Descriptions are based on the underlying graph.

Abstract

Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the lattice of congruences on a graph inverse semigroup is upper semimodular but not lower semimodular.