On the categories of probabilistic approach groups: Actions

Abstract

Starting with the category of probabilistic approach groups, we show that the category of approach groups can be embedded into the category of probabilistic approach groups as a bicoreflective subcategory; further, considering a category of probabilistic topological convergence groups, we show that the category of probabilistic topological convergence groups is {\em isomorphic} to the category of probabilistic approach groups under so-called triangle function
$\tau\colon \Delta^+\times\Delta^+\longrightarrow \Delta^+$, where $\Delta^+$ is the set of all {\em distance distribution functions} that plays a central role for probabilistic metric spaces. Moreover, if we allow this triangle function $\tau$ to be {\em sup-continuous}, then we can show that the category of probabilistic metric groups can be embedded into the category of probabilistic approach groups as a coreflective subcategory.
Furthermore, we demonstrate that every $T1$ probabilistic topological convergence group satisfying so-called (PM) axiom is probabilistic metrizable.
Finally, among others, introducing a category of probabilistic approach transformation groups, we show that the category of probabilistic topological convergence transformation groups is isomorphic to the category of probabilistic approach transformation groups; this solves an open problem that proposed in one of our earlier papers. Moreover, we prove that the category of probabilistic metric transformation groups is isomorphic to the category of probabilistic metric probabilistic convergence transformation groups.