Properties of congruence lattices of graph inverse semigroups

Abstract

From any directed graph E one can construct the graph inverse semigroup [Formula: see text], whose elements, roughly speaking, correspond to paths in E. Wang and Luo showed that the congruence lattice [Formula: see text] of [Formula: see text] is upper-semimodular for every graph E, but can fail to be lower-semimodular for some E. We provide a simple characterization of the graphs E for which [Formula: see text] is lower-semimodular. We also describe those E such that [Formula: see text] is atomistic, and characterize the minimal generating sets for [Formula: see text] when E is finite and simple.