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Is there any kind of distributive law for $c\cdot\max(a,b)$, allowing both signs of $c$?

Notation: For any two numbers $a$ and $b$, let the maximum be $a\sqcap b$, and let the minimum be $a\sqcup b$. (No, my symbols aren't upside-down. Compare this with floor notation; $\lfloor a\rfloor\...
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