Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces
Matteo Mio, Ralph Sarkis, Valeria Vignudelli
TL;DR
This work generalises quantitative algebra beyond metric-space carriers by allowing arbitrary fuzzy relation spaces and arbitrary operation interpretations, removing the nonexpansiveness constraint. It develops a sound and complete deductive system for equations and quantitative equations, proves the existence of free quantitative algebras, and establishes strict monadicity of the free-forgetful adjunction, yielding a precise monad-theoretic presentation. A key contribution is the lifting correspondence: every lifting of a finitary Set monad corresponds to a quantitative equational presentation, and conversely, every quantitative extension presents a lifted monad on fuzzy-relational carriers. The framework naturally extends to generalised metric spaces (GMet) and connects to prior frameworks (MPP16, FordMS21, MioSV22), offering a unified theory with potential applications to program semantics and fuzzy logic. Overall, the paper provides a robust, extensible foundation for equational reasoning about quantitative behavior in very general relational settings, with clear paths to further theoretical and practical developments.
Abstract
We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised metric spaces, and (ii) the interpretations of the algebraic operations are not required to be nonexpansive. Our main results include: a novel sound and complete proof system, the proof that free quantitative algebras always exist, the proof of strict monadicity of the induced Free-Forgetful adjunction, the result that all monads (on fuzzy relations) that lift finitary monads (on sets) admit a quantitative equational presentation.