Graph inverse semigroups: their characterization and completion

TL;DR

This paper characterizes graph inverse semigroups abstractly and demonstrates how to complete them into Cuntz-Krieger semigroups, linking algebraic structures with topological groupoids and Leavitt path algebras.

Contribution

It provides an abstract characterization of graph inverse semigroups and introduces a method to complete them into Cuntz-Krieger semigroups, connecting algebraic and topological frameworks.

Findings

Characterization of graph inverse semigroups

Construction of Cuntz-Krieger semigroups

Connection to topological groupoids and Leavitt path algebras

Abstract

Graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of C*-algebras. This paper has two main goals: first, to provide an abstract characterization of graph inverse semigroups; and second, to show how they may be completed, under suitable conditions, to form what we call the Cuntz-Krieger semigroup of the graph. This semigroup is the ample semigroup of a topological groupoid associated with the graph, and the semigroup analogue of the Leavitt path algebra of the graph.