Sublattices of lattices of convex subsets of vector spaces

2005/01/20 by Friedrich Wehrung, Marina V. Semenova
Mathematics · #math.GM #msc:06B15 #msc:52C10

paper · pdf

published as Algebra and Logic 43, no. 3 (May-June 2004) (2004) 145--161

arxiv created 2005/01/20 · arxiv updated 2009/12/01

Abstract

For a left vector space V over a totally ordered division ring F, let Co(V) denote the lattice of convex subsets of V. We prove that every lattice L can be embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Z)=\X∩ Z | X∈ Co(V)\, for some finite subset Z of V. In particular, we obtain a new universal class for finite lower bounded lattices.

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