Topological graph inverse semigroups

Abstract

To every directed graph E one can associate a graph inverse semigroup G(E), where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph 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) \ {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) \ {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.

Congruences on graph inverse semigroups

“…These inverse semigroups are related to various types of algebras, for instance, C * -algebras, Cohn path algebras, Leavitt path algebras and so on (see [1,2,[8][9][10][11]14]). Graph inverse semigroups have also been studied in their own right in recent years (see [2,5,7,8,12,13]).…”

On locally compact shift-continuous topologies on the α-bicyclic monoid

“…Algebraic and topological properties of the polycyclic monoids were investigated in [19,20,5,6]. The paper [23] is devoted to topological properties of the graph inverse semigroups which are the generalizations of polycyclic monoids. In that paper it was shown that for every finite graph E every locally compact semigroup topology on the graph inverse semigroup over E is discrete, which implies that for every positive integer n every locally compact semigroup topology on the n-polycyclic monoid is discrete.…”

On locally compact semitopological graph inverse semigroups

“…The following Theorem extends Theorem 3 from [23] and Proposition 3.1 from [8] over the case of semitopological graph inverse semigroups. Hence each vertex a is an isolated point in G(E).…”