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 Carathéodory's theorem characterizesn-distributivity in such lattices. Several consequences of this result are studied. First, it is shown how infiniten-distributivity and Carathéodory'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.
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Libkin, L. n-distributivity, dimension and Carathéodory's theorem. Algebra Universalis 34, 72–95 (1995). https://doi.org/10.1007/BF01200491
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DOI: https://doi.org/10.1007/BF01200491