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与Plotkin定量代数理论的推广形式，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) 代数运算的语义解释不要求满足非扩张性。主要研究成果包括：构建了新颖的可靠且完备的证明系统，证明了自由定量代数的普遍存在性，论证了所诱导的自由-遗忘伴随具有严格幺半性，并得出所有（在模糊关系上）能提升（集合上）有限幺半的幺半均允许定量等式表示的结论。