On the categories of quantale-valued convergence tower rings
Generalizing the notion of probabilistic convergence ring, where the underlying lattice is a specific quantale, we consider more general quantale to introduce the notions of quantale valued convergence tower ring, quantale-valued limit tower ring, and quantale-valued Cauchy tower ring. Our motivations among others, are (a) to provide characterization theorems on quantale-valued convergence tower rings; this means, on one hand, we provide a necessary and sufficient conditions for a quantale-valued convergence structure on a ring to be quantale-valued convergence tower ring; on the other hand, given a ring structure and a family of mappings satisfying a set of conditions give rise to unique quantale-valued convergence tower structure on a ring leading to a homogeneous quantale-valued convergence tower ring, (b) to show that a quantale-valued limit tower ring admits a quantale-valued uniform convergence structure, meaning, given a quantale-valued limit tower ring, we can obtain a quantale-valued uniform convergence space, and (c) to introduce a category of quantale-valued Cauchy tower rings and discuss some related results and some of their relationships. In doing so, we produce various examples, particularly, from function space structure of continuous quantale-valued convergence tower spaces, and quantale-valued Cauchy tower spaces. Furthermore, we show that the category of quantale-valued Cauchy tower rings is a topological category over the category of rings with respect the forget functor.
Generalizing the notion of probabilistic convergence ring, where the underlying lattice is a specific quantale, we consider more general quantale to introduce the notions of quantale…
Considering a value quantale, we introduce a category of quantale-valued convergence transformation groups on quantale-valued convergence spaces.
In this article, we focus on discussing some subcategories of the category of probabilistic convergence groups, PConvGrp. In so doing, we introduce a category of probabilistic…