Polyboundedness of zero-closed semigroups

TL;DR

This paper investigates the polyboundedness property of zero-closed semigroups, establishing bounds on their covering numbers and conditions under which they are polybounded, especially under additional topological assumptions.

Contribution

It proves that infinite zero-closed semigroups have a polyboundedness number smaller than their size and identifies conditions under which such semigroups are polybounded.

Findings

Infinite zero-closed semigroups have cov(A_X) < |X|.

Under Martin's Axiom, zero-closed semigroups are polybounded if they admit certain topologies.

Polyboundedness relates to the existence of specific covers in semigroup topology.

Abstract

The polyboundedness number $\mathrm{cov}(\mathcal A_X)$ of a semigroup $X$ is the smallest cardinality of a cover of $X$ by sets of the form $\{x\in X:a_0xa_1\cdots xa_n=b\}$ for some $n\ge 1$, $b\in X$ and $a_0,\dots,a_n\in X^1=X\cup\{1\}$. Semigroups with finite polyboundedness number are called polybounded. A semigroup $X$ is called zero-closed if $X$ is closed in its $0$-extension $X^0=\{0\}\cup X$ endowed with any Hausdorff semigroup topology. We prove that any zero-closed infinite semigroup $X$ has $\mathrm{cov}(\mathcal A_X)<|X|$. Under Martin's Axiom, a zero-closed semigroup is polybounded if $X$ admits a compact Hausdorff semigroup topology or $X$ has a separable complete subinvariant metric.