Generators, extremals and bases of max cones

Details Generators, extremals and bases of max cones Generators, extremals and bases of max cones

2006-04-20

arxiv.org/abs/math/0604454v2

Mathematics

Rings and Algebras

15 pages, 1 figure; v2: new layout, several new references, renumbering of results

Scientific paper

Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in ${{\mathbb R}}_+^n$. This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dependent sets and on computational bounds for the problem of finding the (essentially unique) basis of a finitely generated cone.

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