Inverse graph semigroups were defined by Ash and Hall in 1975. They found necessary and sufficient conditions for the semigroups to be congruence free. In this paper we give a description of congruences on a graph inverse semigroup in terms of the underlying graph. As a consequence, we show that the inverse graph semigroup of a finite graph is congruence Noetherian.

We give a short proof for the associativity of the binary operation defined on a refined system of semigroups indexed by a semilattice. The main result given by Zhang, Shum and Zhang in 2001 on the refined semilattice of semigroups is substantially strengthened.

Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the lattice of congruences on a graph inverse semigroup is upper semimodular but not lower semimodular.

Let τ be an equivalence relation on a semigroup. We introduce τ -congruence-free semigroups, extending the notion of congruence-free semigroups, and classify all completely regular semigroups which are τ -congruence-free, where τ is one of Green's relations H, L and D respectively. Taking τ as H as well as D, this settles two open problems posed by M. Petrich and N.R. Reilly.