Properties of Congruence Lattices of Finite Graph Inverse Semigroups

Given a finite acyclic digraph $E$ one can construct the graph inverse semigroup $G(E)$ of $E$ whose elements correspond to paths in $E$. In this paper we examine the properties of the congruence lattices $L(G(E))$ of graph inverse semigroups for finite acyclic digraphs $E$. More specifically, we characterise the following: the minimal generating set of $L(G(E))$ for any finite graph inverse semigroup $G(E)$ in terms of the digraph $E$; the digraphs $E$ such that the lattice of congruences $L(G(E))$ is lower-semimodular, modular, or distributive; the digraphs $E$ such that $L(G(E))$ is atomistic, geometric, or isomorphic to a power set lattice.