FUNCTORS AND SPACES IN IDEMPOTENT MATHEMATICS

Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction. The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.