Topological Graph Inverse 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 $C^{\ast }$-algebras, and Toeplitz $C^{\ast }$-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.

25 pages. The second version contains additional references and improved exposition

References in corpus (3)

• Graph inverse semigroups, groupoids and their C*-algebras
• The Structure of a Graph Inverse Semigroup
• A categorical invariant of flow equivalence of shifts

Cited by in corpus (13)

• On a complete topological inverse polycyclic monoid
• Classifying locally compact semitopological polycyclic monoids
• Topological properties of Taimanov semigroups
• On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers
• On locally compact semitopological $0$-bisimple inverse $\omega$-semigroups
• On the semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ which is generated by the family ${\mathcal{F}}_n$ of finite bounded intervals of $\omega$
• On lattice of congruences on graph inverse semigroups
• On a semitopological polycyclic monoid
• E-disjunctive inverse semigroups
• On locally compact shift-continuous topologies on the $\alpha$-bicyclic monoid
• Closed inverse subsemigroups of graph inverse semigroups
• The monoid of order isomorphisms of principal filters of a power of the positive integers
• Congruences on graph inverse semigroups