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?
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1Not exactly on topic, but there's a question about Strassen on the frontpage right now; It's worth-noting that [Strassen's fast matrix multiplication](http://en.wikipedia.org/wiki/Strassen_algorithm) does not work on tropical *semirings* (and hence [Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm) cannot make use of it). – 2012-08-03
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1The question is vague. To go from the ordinary real numbers to the tropical semiring involves discarding the axiom that additive inverses exist. So what counts as "addition" and "multiplication"? Are you looking for interesting examples of rigs? – 2012-08-03
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1I've wondered whether tropical mathematics might be applicable to origami. I know it's been applied to phylogenetic trees in biology. – 2012-08-03
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0@Zhen: The term "rigs" in this context is new to me---Could you explain? – 2012-08-03
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0@J.D.: Perhaps it is also worth noting that Dijkstra's shortest path algorithm is just raising a matrix to a power in tropical math! – 2012-08-03
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1Indeed, but the point being that fast matrix multiplication algorithms do not work on "semi"rings, since most of the speeding up tricks in, say Strassen, require existence of additive inverse. – 2012-08-03
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0@Michael: An astute question, which is in fact part of my motivation. There are metric trees underlying origami bases, as per the work of Robert Lang, especially [TreeMaker](http://www.langorigami.com/science/computational/treemaker/treemaker.php). – 2012-08-04
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0To give an example of what seems completely sterile: Map $+$ to the arithmetic mean, and $\times$ to geometric mean. Now even the distributive law fails. – 2012-08-04
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1@JosephO'Rourke A rig is also known as a semiring. It is what is obtained from the ring axioms when you drop the existence of additive inverses (and add the axiom that $0$ is absorbent for multiplication). – 2012-08-04
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0@Zhen: Thanks for educating me! Yes, the question is vague. But still I like to think it has substance. Tropical mathematics has a rich algebra, a rich geometry, and many applications (e.g., to phylogenetic trees, as per Michael Hardy). Are there semirings with analogous rich and applicable structures? – 2012-08-04
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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