Group topologies and semigroup topologies on the integers determined by convergent sequences

We address the strongest group topologies and semigroup topologies on the integers with acertain sequence converging to 0as investigated by Protasov and Zelenyuk. These are useful for constructing topological groups with peculiar duality properies and related to exponential Diophatine equations and additive bases. 1Introduction As in [10], for agroup $G $ and asequence $\langle a_{n} : n\in \mathrm{N}\rangle $ of its elements, we denote by $(G|\langle a_{n} : n\in \mathrm{N})) $ the topological group $G $ with the strongest group topology in which $\langle a_{n} : n\in \mathrm{N}\rangle $ converges to the neutral element; it is $G\{a_{n}\} $ in the notation of [14]. We admit non-Hausdorff topologies as well. Similarly for amonoid $G $ , we denote by $(G|\langle a_{n} : n\in \mathrm{N}\rangle)_{\mathrm{s}} $ the topological semigroup with the strongest semigroup topology sat-isfying the convergence condition. This article contains two themes concerning such topological (semi)groups as above; they are relatively independent each other. First we exhibit apair of topological groups which witnesses that certain duality properties are not preserved under direct product