Congruence Lattices Of Graph Inverse Semigroups

Based on the description of congruences on general graph inverse semigroups,we use the properties of directed graphs to perform some research on congruence lattices of graph inverse semigroups.First we prove that the set of 0-restricted congruences on a graph inverse semigroup forms a distributive lattice under inclusion relations of sets.Secondly,we show that the congruence lattice of graph inverse semigroup is upper semimodular but not necessarily lower semimodular,and the lattice is modular if and only if it is distributive.Next we prove that the congruence lattice on a finite graph inverse semigroup is M-symmetric.Finally we give several special types of directed graphs such that the congruence lattices on the graph inverse semigroups are distributive.