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.

翻译：本文推广了Mardare、Panangaden和Plotkin的定量代数理论，其中：(i) 定量代数的载体不再局限于度量空间，可以是任意模糊关系或广义度量空间；(ii) 代数运算的解释不要求是非扩张的。我们的主要成果包括：一种新颖的可靠且完备的证明系统；证明了自由定量代数总是存在；证明了由此诱导的自由-遗忘伴随对的严格单子性；以及所有（在模糊关系上）提升（在集合上）有限单子的单子都具有定量等式表示这一结论。