Distributivity in congruence lattices of graph inverse semigroups

Abstract

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

11 pages, 3 figures