Some properties of congruence lattices of path semigroups

Each quiver corresponds to a path semigroup, and such a path semigroup also corresponds to an associative K-algebra over an algebraically closed field K. Let Q be a quiver and S_Q, KQ be its path semigroup, path algebra, respectively. In this paper, we study some properties of the congruence lattice of S_Q. First, we show that there is a one-to-one correspondence between congruences on S and certain algebraic ideals of KQ. Based on such a description, we consider acyclic quivers and show that the congruence latices of such path semigroups are strong upper semimodular but not necessarily lower semimodular. Moreover, we provide some equivalent conditions for the congruence lattices to be modular and distributive.