On some subcategories of probabilistic convergence groups

Journal Article

2025

Published in: https://doi.org/10.29020/nybg.ejpam.v18i3.6535

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 neighborhood spaces, PNeigh, and category of probabilistic topological spaces, PTopNeigh. We identify probabilistic metric spaces as probabilistic neighborhood spaces. Introducing a category of probabilistic neighborhood groups, PNeighGrp, and category of probabilistic topological neighborhood groups, PTopNeighGrp, we show that the category of probabilistic neighbor groups, PNeighGrp - a topological category, and the category of probabilistic pretopological groups, PPreTopGrp are isomorphic, and find a categorical relationship with categorical probabilistic neighborhood groups with probabilistic convergence groups. Furthermore, we introduce probabilistic pre Cauchy groups, PPreChyGrp, showing that the category of probabilistic Cauchy groups, a category previously introduced, is a full subcategory of PChyGrp. In this respect, we prove that in presence of probabilistic convergence groups, the category of probabilistic Cauchy groups, PChyGrp, the category of probabilistic strongly normal probabilistic limit groups, SNPLimitGrp are isomorphic. Moreover, following a notion of probabilistic normed group - a notion introduced earlier, we show that the category of probabilistic normed groups, PNormedGrp is isomorphic with the  category of probabilistic metric groups, PMetGrp. Finally, we present probabilistic version of so-called invariance of norm theorem.

 

• Publication Work Type: Original Research
• Publisher Name: European Journal of Pure and Applied Mathematics
• Publishing City: New York, USA
• Volume Number: 18
• Issue Number: 3
• Pages: 1-28