# variant on ring objects in the category of complete lattices

Let $L$ be a complete lattice and denote its top and bottom elements by $0$ and $\infty$ respectively. Consider two binary operations $+$ and $\times$ defined on $L$ such that $(L^{op},+,0)$ is a commutative unital quantale, $(L,\times)$ is a (not-necessarily unital) quantale, and so that a suitable (i.e., there is some freedom for play) distributivity condition holds.

Is there a name for such a structure?

The motivation for the question is that I am interested in categorifying Lipschitz mappings. Thus, enriching over the structure above I can use the multiplicative structure to talk about a Lipschitz condition.

For clarify, the additive structure $(L^{op},+,0)$ being a commutative unital quantale means that $a+0=a$ and that $a+\bigwedge S=\bigwedge(a+S)$, and so, taking $S=\emptyset$ we obtain $a+\infty =\infty$. In contrast, the multiplicative structure $(L,\times)$ being a quantale means that $a \times \bigvee S = \bigvee (a\times S)$ and so $a\times 0=0$. Hence this structure is a bit like a ring object in the category of complete lattices.

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