On the probabilistic convergence spaces: Monad and its Eilenberg-Moore category

Journal Article
Ahsanullah, T. M. G. . 2022
Publication Work Type: 
Original Research Article
Magazine \ Newspaper: 
New Math. & Nat. Comput. World Scientific
Issue Number: 
Volume Number: 
Publication Abstract: 

Motivated by the category of probabilistic convergence spaces - a supercategory of the category of topological spaces, recently, we brought to light the categories of probabilistic convergence groups, probabilistic metric probabilistic convergence groups, probabilistic convergence transformation groups along with their underpinning natural examples. The purpose of this paper is, first, to establish a result on the isomorphism between the categories of probabilistic metric groups, and probabilistic metric probabilistic convergence groups. Secondly, among others, we explore various monads in relation with probabilistic convergence groups, and probabilistic convergence spaces, and their related algebras. In so doing, we consider a product of the categories of probabilistic convergence groups and probabilistic convergence spaces in an attempt to construct a monad on it such that the corresponding category of algebras, so-called Eilenberg-Moore category, is isomorphic to the category of probabilistic convergence transformation groups. Finally, invoking so-called Beck's theorem on characterization of algebras, and starting with a particular adjunction we achieved a monad; and, conversely, given a monad, we obtain an adjunction.