  Let {\Gamma} be a directed graph and Inv({\Gamma}) be the graph inverse
semigroup of {\Gamma}. Luo and Wang [7] showed that the congruence lattice
C(Inv({\Gamma})) of any graph inverse semigroup Inv({\Gamma}) is upper
semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri,
Mesyan and Mitchell characterized the directed graph {\Gamma} for which
C(Inv({\Gamma})) is lower semimodular [2]. In the present paper, we show that
the lower semimodularity, modularity and distributivity in the congruence
lattice C(Inv({\Gamma})) of any graph inverse semigroup Inv({\Gamma}) are
equivalent.
