N-distributivity, dimension and Carath�odory's theorem

Abstract

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\(\mathbb{E}^n \) isn+1-distributive but notn-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 Caratheodory's theorem characterizesn-distributivity in such lattices. Several consequences of this result are studied. First, it is shown how infiniten-distributivity and Caratheodory's theorem are related. Then the main result is applied to prove that for a large class of lattices beingn-distributive means being in the variety generated by the finiten-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.