On the Hausdorff variety generated by all locally kω groups and on Schwartz groups

In Außenhofer (2007) [4] it was shown that every kω group is a Schwartz group, a generalization for abelian Hausdorff groups of the setting of a Schwartz topological vector space. Here we strengthen the definition of a Schwartz group slightly – but strictly – and introduce special Schwartz groups. We prove that they form, as Schwartz groups, a Hausdorff variety and that this variety coincides with the Hausdorff variety generated by all abelian locally kω groups.