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：我们提出了Mardare、Panangaden和Plotkin的定量代数理论的一种推广，其中：(i) 定量代数的载体不限于度量空间，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。我们的主要结果包括：一种新颖的完备且可靠证明系统；自由定量代数始终存在的证明；诱导的自由-遗忘伴随的严格单子性证明；以及所有提升集合上的有限单子的模糊关系上的单子都允许定量等式呈现的结果。