So we can call addition in R "multiplication" in terms of the preimage and we can also define "addition" in terms of the pre-image. The interesting thing is what becomes of this when we take the "classical limit" of h->0: Then "addition" is nothing but the maximim and this "addition" is idempotent: a "+" a = a.
This is an example of an idempotent semiring and in fact it is the generic one: Besides idempotency, it satisfies many of the usual laws: associativity, distributional law, commutativity. Thus you can carry over much of the usual stuff you can do with fields to this extreme limit. Other examples of this structure are Boolean algebras or compact convex sets where "multiplication" is the usual sum of sets and "addition" is the convex hull (obviously, the above example is a special case). Another example are polynomials with non-negative coefficients and for these the degree turns out to be a homomorphism! The obvious generalization of the integral is the supremum and the Fourier transform becomes the Legendre transform (you have to work out what the characters of the addition are!).
This theory has many applications, it seems especially strong for optimization problems. But you can also apply this limiting procedure to algebraic varieties under which they are turned into Newton polytopes.
I enjoyed this seminar especially because it made clear that many constructions can be thought of extreme limits of some even more common, linear constructions.
But now for something completely different: When I came back to my computer, I had received the following email:
Dear Mr. Helling
I would greatly appreciate your response.
Please what is interrelation mutually
fractal attractor of the black hole condensation,
Bott spectrum of the homotopy groups
and moduli space of the nonassociative geometry?
Thank you very much obliged.
I have no idea what he is talking about but maybe one of my readers has. I googled for a passage from the question and found that exactly the same question has also been posted in the comment sections of the Coffee Table and Lubos's blog.