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I've been impressed at the rich structure of tropical mathematics, all consequences of the seemingly mundane starting point of replacing classical $a+b$ by $\min(a,b)$ (or max), and replacing classical $a \times b$ with $a + b$. My question is: Are there other natural redefinitions of addition and multiplication that lead to rich and useful algebraic and geometric structures and applications?

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    Every distributive lattice is automatically a rig. I dare say the theory of distributive lattices is very rich, as it encompasses, say, boolean algebras, Heyting algebras, pointless topology...2012-08-04

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