### Abstract

We present an algebraic account of the Wasserstein distances W_{p} on complete metric spaces, for p ≥ 1. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in p, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance W_{p}. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order p, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.

Original language | English |
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Article number | 19 |

Number of pages | 16 |

Journal | Logical Methods in Computer Science |

Volume | 14 |

Issue number | 3 |

DOIs | |

Publication status | Published - 14 Sep 2018 |

### Keywords

- Wasserstein distances
- quantitative algebraic theory
- programming languages

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## Cite this

*Logical Methods in Computer Science*,

*14*(3), [19]. https://doi.org/10.23638/LMCS-14(3:19)2018