On locally compact semitopological graph inverse semigroups

Abstract

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph E is strongly connected and has finitely many vertices, then any Hausdorff shift-continuous locally compact topology on the graph inverse semigroup G(E) is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for Hausdorff locally compact shift-continuous topologies on polycyclic monoids P 1 and P λ , respectively.

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

“…For instance, in [23] it was proved that a Hausdorff locally compact semitopological bicyclic semigroup with an adjoined zero C 0 is either compact or discrete. In [6] and [8] this result was extended for polycyclic monoids and graph inverse semigroups over strongly connected graphs with finitely many vertices, respectively. Similar dichotomy also holds for other generalizations of the bicyclic monoid (see [9,24,30]).…”

On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zero

“…A dichotomy for the bicyclic monoid with an adjoined zero C 0 = C (p, q)⊔{0} was proved in [11]: every locally compact semitopological bicyclic monoid C 0 with an adjoined zero is either compact or discrete. The above dichotomy was extended by Bardyla in [5] to locally compact λ-polycyclic semitopological monoids, in [6] to locally compact semitopological graph inverse semigroups in [13] to locally compact semitopological interassociates of the bicyclic monoid with an adjoined zero and are extended in [12] to locally compact semitopological 0-bisimple inverse ω semigroups with compact maximal subgroups.…”

Embedding of graph inverse semigroups into CLP-compact topological semigroups

“…Algebraic theory of graph inverse semigroups is well developed (see [2,8,10,26,27,30,32]). Topological properties of graph inverse semigroups were investigated in [9,11,12,13,33].…”

“…Let G(E) be an arbitrary semitopological GIS. Observe that each non-zero element of G(E) is isolated (see [12,Theorem 4]). For an arbitrary GIS G(E) by τ c we denote the topology on G(E) which is defined as follows: each non-zero element is isolated in (G(E), τ c ) and open neighbourhood base of the point 0 consists of cofinite subsets of G(E) which contain 0.…”