Polyboundedness of zero-closed semigroups

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.