Graph inverse semigroups: Their characterization and completion

Graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of C⁎C⁎-algebras. In this paper, we characterize such semigroups and show how they may be completed, under suitable conditions, to form what we call the Cuntz–Krieger semigroup of the graph. This semigroup is proved to be the ample semigroup of a topological groupoid associated with the graph, and the semigroup analogue of the Leavitt path algebra of the graph.