On Some Subcategories of Probabilistic Convergence Groups

Abstract

In this article, we focus on discussing some subcategories of the category of probabilistic convergence groups. In so doing, we introduce a category of probabilistic neighborhood spaces, \textbf{PNeigh}, and a category of probabilistic topological neighborhood spaces, \textbf{PTopNeigh}. We identify probabilistic  metric spaces as probabilistic neighborhood spaces. Introducing a category of probabilistic neighborhood groups, \textbf{PNeighGrp}, we show that the category of probabilistic neighborhood groups, \textbf{PNeighGrp} - a topological category, and the category of probabilistic pretopological groups, \textbf{PPreTopGrp} are isomorphic, and find categorical relationship of probabilistic neighborhood groups with  probabilistic convergence groups. Furthermore, we introduce a category of probabilistic pre Cauchy  groups \textbf{PPreChyGrp} showing  that the category of probabilistic Cauchy groups, \textbf{PChyGrp} - a category introduced earlier,  is a full subcategory  of  \textbf{PPreChyGrp}.  In this respect, we show that in presence of  probabilistic convergence group,  the category of  probabilistic Cauchy groups and the category of strongly normal probabilistic limit groups are isomorphic. Moreover, following the notion of probabilistic normed group, we show that the category of probabilistic normed groups is isomorphic to the category of  probabilistic metric groups; finally, we present probabilistic version of so-called invariance of norm theorem.