On probabilistic convergence rings and their natural uniform convergence structures

Abstract

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 theorems on probabilistic convergence rings, (b) probabilistic uniformizability of probabilistic limit rings, and discuss some results on probabilistic Cauchy rings, and their relationship with probabilistic convergence rings. In doing so, we produce various examples, particularly, from the function space structure of continuous probabilistic convergence. Moreover, we observe that the category of probabilistic convergence rings in the sense of Richardson-Kent is a reflective subcategory of the category of probabilistic convergence rings in our sense.