Article Dans Une Revue Algebra and Logic Année : 2004

Sublattices of lattices of convex subsets of vector spaces

Résumé

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 $Co(V,Z)=\{X\cap Z | X\in Co(V)\}$, for some finite subset $Z$ of $V$. In particular, we obtain a new universal class for finite lower bounded lattices.

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hal-00003955 , version 1 (20-01-2005)

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Friedrich Wehrung, Marina V. Semenova. Sublattices of lattices of convex subsets of vector spaces. Algebra and Logic, 2004, 43 (no. 3 (May-June 2004)), pp.145--161. ⟨10.1023/B:ALLO.0000028929.28946.d6⟩. ⟨hal-00003955⟩
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