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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.
Anagnostopoulou‐Merkouri et al. (Sat,) studied this question.