Sublattices of Lattices of Convex Subsets of Vector Spaces

Abstract

Let Co(V) be a lattice of convex subsets of a vector space V over a totally ordered division ring $$\mathbb{F}$$ . We state that every lattice L can be embedded into Co(V), for some space V over $$\mathbb{F}$$ . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional; in this case L also embeds into a finite lower bounded lattice of the form Co(V,Ω)={X⋂Ω | X ∈ Co(V)}, for some finite subset Ω of V. This result yields, in particular, a new universal class of finite lower bounded lattices.