On a semitopological polycyclic monoid

TL;DR

This paper investigates the algebraic and topological properties of the $ ext{lambda}$-polycyclic monoid, revealing its structural characteristics and limitations of its topologizations, including discreteness and compactness constraints.

Contribution

It extends known algebraic properties of polycyclic monoids to infinite cardinals and characterizes their topological behaviors, including the nature of semitopological and compact topologies.

Findings

$P_{\lambda}$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup.

Every non-zero element in $P_{\lambda}$ is isolated under Hausdorff semitopological topologies.

No non-discrete locally compact Hausdorff semigroup topology exists on $P_{\lambda}$.

Abstract

We study algebraic structure of the $\lambda$-polycyclic monoid $P_{\lambda}$ and its topologizations. We show that the $\lambda$-polycyclic monoid for an infinite cardinal $\lambda\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $\lambda$ the polycyclic monoid $P_{\lambda}$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_{\lambda},\tau)$ for every Hausdorff topology $\tau$ on $P_{\lambda}$, such that $(P_{\lambda},\tau)$ is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on $P_\lambda$ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies $\tau$ on $P_{\lambda}$ such that $\left(P_{\lambda},\tau\right)$ is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal $\lambda\geqslant 2$ any continuous homomorphism from a topological semigroup $P_\lambda$ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains $P_{\lambda}$ as a dense subsemigroup.