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.
Categories: math.GR