2022/12/03 by Тарас Банах, Banakh, Taras, Andriy Rega +1
Mathematics · Computer Science · Decision Sciences · #Advanced Topology and Set Theory #semigroups and automata theory #Fuzzy and Soft Set Theory
paper · pdf · doi:10.48550/arxiv.2212.01604
The polyboundedness number cov(\mathcal AX) of a semigroup X is the smallest cardinality of a cover of X by sets of the form \x∈ X:a0xa1⋯ xan=b\ for some n≥ 1, b∈ X and a0,…,an∈ X1=X∪\1\. Semigroups with finite polyboundedness number are called polybounded. A semigroup X is called zero-closed if X is closed in its 0-extension X0=\0\∪ X endowed with any Hausdorff semigroup topology. We prove that any zero-closed infinite semigroup X has cov(\mathcal AX)<|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.