Classifying locally compact semitopological polycyclic monoids

Abstract

We present a complete classification of Hausdorff locally compact polycyclic monoids up to a topological isomorphism. A {\em polycyclic monoid} is an inverse monoid with zero, generated by a subset $\Lambda$ such that $x{x}^{-1}=1$ for any $x\in \Lambda$ and $x{y}^{-1}=0$ for any distinct $x,y\in \Lambda$. We prove that any non-discrete Hausdorff locally compact topology with continuous shifts on a polycyclic monoid $M$ coincides with the topology of one-point compactification of the discrete space $M\setminus \{0\}$.

References in corpus (3)

• Topological Graph Inverse Semigroups
• On a complete topological inverse polycyclic monoid
• On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero

Cited by in corpus (8)

• Topological properties of Taimanov semigroups
• On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers
• On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of ${\mathbb{N}}^n$ with adjoined zero
• On the lattice of weak topologies on the bicyclic monoid with adjoined zero
• The monoid of monotone injective partial selfmaps of the poset $({\mathbb{N}}^3,⩽)$ with cofinite domains and images
• On the semigroup ${\text{ID}}_{\infty }$
• On locally compact shift-continuous topologies on the $\alpha$-bicyclic monoid
• The monoid of order isomorphisms of principal filters of a power of the positive integers