The space of probabilistic 1-lipschitz maps

TL;DR

This paper introduces probabilistic 1-Lipschitz maps on probabilistic metric spaces, showing their space forms a probabilistic metric space and, for groups, a monoid, with applications to probabilistic invariant groups.

Contribution

It defines probabilistic 1-Lipschitz maps, studies their structure, and characterizes probabilistic invariant complete Menger groups using these maps, extending classical theorems.

Findings

The space of probabilistic 1-Lipschitz maps is a probabilistic metric space.

When defined on a group, this space has a monoid structure.

Characterization of probabilistic invariant complete Menger groups via these maps.

Abstract

We introduce and study a natural notion of probabilistic 1-Lipschitz maps. We prove that the space of all probabilistic 1-Lipschitz maps defined on a probabilistic metric space G is also a probabilistic metric space. Moreover, when G is a group, then the space of all prob-abilistic 1-Lipschitz maps defined on G can be endowed with a monoid structure. Then, we caracterize the probabilistic invariant complete Menger groups by the space of all probabilistic 1-Lipschitz maps in the sprit of the classical Banach-Stone theorem.