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?
Tropical-like redefinitions of addition and multiplication?
9
$\begingroup$
algebraic-geometry
algebraic-curves
-
1Every 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