# Commutative quantale (Q,+) with a binary operation distributing over +

Context: In Flagg's "Quantales and continuity spaces" the notion of a continuity spaces enriched in a certain quantale is introduced. Abstracting from the set $[0,\infty]$ the essential properties identified are that of a value quantale: a quantale with a completely distributive underlying lattice.

This point of view neglects the operation of multiplication in $[0,\infty]$. Multiplication is used for instance in the definition of Lipschitz maps. So, my question is whether the notion of a commutative quantale $(Q,+)$ together with an operation $\cdot:Q\times Q\to Q$ satisfying certain conditions modelled after $([0,\infty],+,\cdot)$ exists?

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