Categorically closed countable semigroups

Abstract

AbstractIn this paper, we establish a connection between categorical closedness and nontopologizability of semigroups. In particular, for the class𝖳𝟣⁢𝖲{\mathsf{T_{\!1}S}}ofT1{T_{1}}topological semigroups we prove that a countable semigroupXwith finite-to-one shifts is injectively𝖳𝟣⁢𝖲{\mathsf{T_{\!1}S}}-closed if and only ifXis𝖳𝟣⁢𝖲{\mathsf{T_{\!1}S}}-discrete in the sense that everyT1{T_{1}}semigroup topology onXis discrete. Moreover, a countable cancellative semigroupXis absolutely𝖳𝟣⁢𝖲{\mathsf{T_{\!1}S}}-closed if and only if every homomorphic image ofXis𝖳𝟣⁢𝖲{\mathsf{T_{\!1}S}}-discrete. Also, we introduce and investigate a new notion of a polybounded semigroup. It is proved that a countable semigroupXwith finite-to-one shifts is polybounded if and only ifXis𝖳𝟣⁢𝖲{\mathsf{T_{\!1}S}}-closed if and only ifXis𝖳𝗓⁢𝖲{\mathsf{T_{\!z}S}}-closed, where𝖳𝗓⁢𝖲{\mathsf{T_{\!z}S}}is the class of Tychonoff zero-dimensional topological semigroups. We show that polybounded cancellative semigroups are groups, and polyboundedT1{T_{1}}paratopological groups are topological groups.