Hate to disillusion you, Art, but I read about the duality and separation thing in metaphysical books at least as far back as the early '90s. It was worded in different terminology, of course, but was the same basic idea.Well, at least since 2002.http://arxiv.org/abs/math/0212294We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.Yup. Slightly difference terminology. ;-)
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