Abstract
Let be a directed graph and
be the graph inverse semigroup of
. Luo and Wang (Semimodularity in congruence lattice of graph inverse semigroups, 2021) showed that the congruence lattice
of any graph inverse semigroup
is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell (Properties of congruence lattices of graph inverse semigroups, 2022) characterized the directed graph
for which
is lower semimodular. In the present paper, we show that lower semimodularity, modularity and distributivity in the congruence lattice
of any graph inverse semigroup
are equivalent.