Sublattices of lattices of convex subsets of vector spaces

Details Sublattices of lattices of convex subsets of vector spaces Sublattices of lattices of convex subsets of vector spaces

2005-01-20

arxiv.org/abs/math/0501324v1

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

Mathematics

General Mathematics

Scientific paper

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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