Free complete Wasserstein algebras.

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Radu Mardare, Prakash Panangaden, Gordon D. Plotkin

We present an algebraic account of the Wasserstein distances $W_p$ oncomplete metric spaces. This is part of a program of a quantitative algebraictheory of effects in programming languages. In particular, we give axioms,parametric in $p$, for algebras over metric spaces equipped with probabilisticchoice operations. The axioms say that the operations form a barycentricalgebra and that the metric satisfies a property typical of the Wassersteindistance $W_p$. We show that the free complete such algebra over a completemetric space is that of the Radon probability measures on the space with theWasserstein distance as metric, equipped with the usual binary convex sumoperations.

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