Graph inverse semigroups and Leavitt path algebras

TL;DR

This paper explores the structure and relationships of inverse semigroups derived from directed graphs, introducing Leavitt inverse semigroups, and establishing connections with Leavitt path algebras and graph $C^*$-algebras.

Contribution

It introduces Leavitt inverse semigroups, provides structural characterizations, and establishes isomorphism conditions linking graph inverse semigroups and Leavitt path algebras.

Findings

Characterization of universal groups of local submonoids

Conditions for homomorphic images to be graph inverse semigroups

Classification of graphs with isomorphic Leavitt inverse semigroups

Abstract

We study two classes of inverse semigroups built from directed graphs, namely graph inverse semigroups and a new class of semigroups that we refer to as Leavitt inverse semigroups. These semigroups are closely related to graph $C^*$-algebras and Leavitt path algebras. We provide a topological characterization of the universal groups of the local submonoids of these inverse semigroups. We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs. We describe the structure of graphs that admit a directed cover or directed immersion into a circle and we provide structural information about graph inverse semigroups of finite graphs that admit a directed cover onto a bouquet of circles. We also find necessary and sufficient conditions for a homomorphic image of a graph inverse semigroup to be another graph inverse semigroup. We find a presentation for the Leavitt inverse semigroup of a graph in terms of generators and relations. We describe the structure of the Leavitt inverse semigroup and the Leavitt path algebra of a graph that admits a directed immersion into a circle. We show that two graphs that have isomorphic Leavitt inverse semigroups have isomorphic Leavitt path algebras and we classify graphs that have isomorphic Leavitt inverse semigroups. As a consequence, we show that Leavitt path algebras are $0$-retracts of certain matrix algebras.