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A. Huhn proved that the dimension of Euclidean spaces can be characterized through algebraic properties of the lattices of convex sets. In fact, the lattice of convex sets of IEn is n+1-distributive but not n-distributive. In this paper his result is generalized for a class of algebraic lattices generated by their completely join-irreducible elements. The lattice theoretic form of Carathédory's theorem characterizes n-distributivity in such lattices. Several consequences of this result are studies. First, it is shown how infinite n-distributivity and Carathédory's theorem are related. Then the main result is applied to prove that for a large class of lattices being n-distributive means being in a variety generated by the finite n-distributive lattices. Finally, n-distributivity is studied for various classes of lattices, with particular attention being paid to convexity lattices of Birkhoff and Bennett for which a Helly type result is also proved.
Leonid Libkin, "n-Distributivity, Dimension and Carathéodory's Theorem", . September 1992.
Date Posted: 08 August 2007