Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces

TL;DR

This paper generalizes the theory of quantitative algebras to include fuzzy relations and generalized metric spaces, removing restrictions on carriers and operation interpretations, and establishes foundational results like soundness, completeness, and monadicity.

Contribution

It introduces a unified framework for quantitative algebras with fuzzy and generalized metric spaces, along with a complete proof system and existence of free algebras.

Findings

Established a sound and complete proof system.

Proved the existence of free quantitative algebras.

Demonstrated monadicity of the induced adjunction.

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.