On the categories of probabilistic approach groups: Actions by T. M. G. Ahsanullah , Fawzi Al-Thukair (KSU), and Bhamini Nayar (Morgan State University, Baltimore, USA)
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 conference groups, we show that the category of probabilistic topological convergence groups is isomorphic to the category of probabilistic approach groups under so-called triangle function $\tau: \Delta^+\times \Delta^+\longrightarrow \Delta^+, where $\Delta^+$ is the set of all distance distributive functions that play central role for probabilistic metric spaces. Moreover, if we allow the triangle function $\tau$ to be 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.
Introducing the notion of probabilistic convergence ring, and probabilistic limit ring, our motivations among others are, to focus at two vital issues, such as, (a) to provide characterization…
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…
Starting with an approximation space as the underlying structure, we
look at the rough uniformity of a topological rough group. Next, taking L as a
complete residuated lattice, we…