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Journal Articles Aequationes Mathematicae Year : 2019

## The space of probabilistic $1$-Lipschitz maps

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Mohammed Bachir
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#### Abstract

We introduce and study the natural notion of probabilistic $1$-Lipschitz maps. We use the space of all probabilistic $1$-Lipschitz maps to give a new method for the construction of probabilistic metric completion (respectively of probabilistic invariant metric group completion). Our construction is of independent interest. We prove that the space of all probabilistic $1$-Lipschitz maps defined on a probabilistic invariant metric group can be endowed with a monoid structure. Next, we explecite the set of all invertible element of this monoid and characterize the probabilistic invariant complete Menger groups by the space of all probabilistic $1$-Lipschitz maps in the sprit of the classical Banach-Stone theorem.

### Dates and versions

hal-01973212 , version 1 (08-01-2019)

### Identifiers

• HAL Id : hal-01973212 , version 1

### Cite

Mohammed Bachir. The space of probabilistic $1$-Lipschitz maps. Aequationes Mathematicae, 2019, 93 (5), pp.955-983. ⟨hal-01973212⟩

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