On the categories of probabilistic approach groups: actions

T M G Ahsanullah ; Fawzi Al-Thukair ; Bhamini Nayar

Filomat, Tome 37 (2023) no. 16, p. 5413

Résumé

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 isomorphic to the category of probabilistic approach groups under so-called triangle function τ : ∆ + × ∆ + −→ ∆ + , where ∆ + is the set of all distance distribution functions that plays a central role for probabilistic metric spaces. Moreover, if we allow this triangle function τ 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 T 1 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.

Détail

DOI : 10.2298/FIL2316413A

Classification : 54A20, 54E70, 54E90, 54H11
Keywords: Topology, Probabilistic metric space, Approach space, Probabilistic approach space, Probabilistic approach group, Probabilistic topological convergence space, Probabilistic metric group, Action, Probabilistic approach transformation group, Probabilistic metric transformation group, Category