n-distributivity, dimension and Carathéodory's theorem

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, ndistributivity 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. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-92-71. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/377 n-distributivity, dimension and CarathBodory's theorem MS-CIS-92-71 LOGIC & COMPUTATION 51