2021/11/28 by Banakh, Taras, Bardyla, Serhii
#20M18 #22A15 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2111.14154
In this paper we establish a connection between categorical closedness and topologizability of semigroups. In particular, for a class \mathsf T 1\mathsf S of T1 topological semigroups we prove that a countable semigroup X with finite-to-one shifts is injectively \mathsf T 1\mathsf S-closed if and only if X is \mathsfT 1S-nontopologizable in the sense that every T1 semigroup topology on X is discrete. Moreover, a countable cancellative semigroup X is absolutely \mathsf T 1\mathsf S-closed if and only if every homomorphic image of X is \mathsf T 1\mathsf S-nontopologizable. Also, we introduce and investigate a notion of a polybounded semigroup. It is proved that a countable semigroup X with finite-to-one shifts is polybounded if and only if X is \mathsf T 1\mathsf S-closed if and only if X is \mathsf T z\mathsf S-closed, where \mathsf T z\mathsf S is a class of zero-dimensional Tychonoff topological semigroups. We show that polyboundedness provides an automatic continuity of the inversion in T1 paratopological groups and prove that every cancellative polybounded semigroup is a group.