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的量化代数理论的推广，其中（i）量化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）代数运算的解释不再要求是非扩张的。我们的主要成果包括：一套新颖的可靠且完备的证明系统，证明了自由量化代数总是存在，证明了所诱导的自由-遗忘伴随具有严格单子性，并证明了所有（在模糊关系上）提升有限集单子的单子都允许量化等式表示。