Semigroup extensions of Abelian topological groups

Abstract

We show that an Abelian topological group G is absolutely closed in the class of topological semigroups if and only if G is complete and there is $$n\in \mathbb {N}$$ such that the subgroup $$nG=\{nx:x\in G\}$$ is totally bounded. If for every $$n\in \mathbb {N}$$ , the subgroup nG is not totally bounded, then the topology of G can be extended to a semigroup topology $$\mathcal {T}$$ on $$G\times \mathbb {Z}^+$$ in which G is open and dense, and if G is locally compact, so can be chosen $$\mathcal {T}$$ . In particular, the topology of $$\mathbb {R}$$ can be extended to a locally compact semigroup topology on $$\mathbb {R}\times \mathbb {Z}^+$$ in which $$\mathbb {R}$$ is dense. We also show that the topology of G can be extended to a regular (equivalently, Tychonoff) semigroup topology on $$G\times \mathbb {Z}^+$$ in which G is open and dense if and only if there is a neighborhood U of $$0\in G$$ such that for every $$n\in \mathbb {N}$$ , the subgroup nG is U-unbounded.