Universal Algebra for Generalised Metric Spaces

04/27/2023

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by   Matteo Mio, et al.

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We study in this work a generalisation of the framework of quantitative algebras, where we allow for generalised metric spaces and where operations need not to be nonexpansive. We introduce a sound and complete Birkhoff-style deductive system where judgements are between equations or quantitative equations. We show that (quantitative) equationally defined classes of quantitative algebras have free objects and we prove monadicity theorems.