Journal of Convex Analysis 21 (2014), No. 1, 089--120
Copyright Heldermann Verlag 2014
Convexities on Ordered Structures Have Their Krein-Milman Theorem
Paul Poncet
CMAP -- École Polytechnique, Route de Saclay, 91128 Palaiseau, France
poncet@cmap.polytechnique.fr
We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex subsets, and for our proofs we use arguments from continuous lattice theory and abstract convexity theory.
Keywords: Abstract convexity, max-plus convexity, tropical convexity, Krein-Milman theorem, convex geometries, antimatroids, partially ordered sets, semilattices, Lawson semilattices, lattices.
MSC: 22A26; 52A01, 06A06, 06A12, 06B30, 14T05
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