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Letbe a directed graph andbe the graph inverse semigroup of. Luo and Wang ( Semimodularity in congruence lattice of graph inverse semigroups , 2021) showed that the congruence latticeof any graph inverse semigroupis 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 graphfor whichis lower semimodular. In the present paper, we show that lower semimodularity, modularity and distributivity in the congruence latticeof any graph inverse semigroupare equivalent.

Distributivity in congruence lattices of graph inverse semigroups