Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Paul Poncet

CMAP – École Polytechnique, Route de Saclay, 91128 Palaiseau, France and: INRIA, Saclay–Île-de-France

poncet@cmap.polytechnique.fr

Keywords

  • Abstract convexity
  • max-plus convexity
  • tropical convexity
  • Krein-Milman theorem
  • convex geometries
  • antimatroids
  • partially ordered sets
  • semilattices
  • Lawson semilattices
  • lattices

Mathematics Subject Classification

22A26; 52A01, 06A06, 06A12, 06B30, 14T05

Suggested citation

P. Poncet. “Convexities on Ordered Structures Have Their Krein-Milman Theorem.” Journal of Convex Analysis 21 (2014), No. 1, 89–120. https://doi.org/10.68381/jca21005

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press.