Topological Graph Inverse Semigroups

TL;DR

This paper explores how to impose topologies on graph inverse semigroups derived from directed graphs, revealing conditions under which these structures are discrete or admit non-discrete topologies, with implications for related algebraic and analytical frameworks.

Contribution

It characterizes topological conditions for graph inverse semigroups, including discreteness and the existence of non-discrete topologies, expanding understanding of their topological algebraic properties.

Findings

In Hausdorff topologies, $G(E)\setminus \{0\}$ must be discrete.

Certain graphs admit $T_1$ topologies with non-discrete $G(E)\setminus \{0\}$.

Descriptions of closures of $G(E)$ in larger topological semigroups.

Abstract

To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, Cuntz-Krieger $C^*$-algebras, and Toeplitz $C^*$-algebras. We investigate topologies that turn $G(E)$ into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, $G(E)\setminus \{0\}$ must be discrete for any directed graph $E$. On the other hand, $G(E)$ need not be discrete in a Hausdorff semigroup topology, and for certain graphs $E$, $G(E)$ admits a $T_1$ semigroup topology in which $G(E)\setminus \{0\}$ is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of $G(E)$ in larger topological semigroups.