Lower bound on the number of meet-irreducible elements in extremal lattices
Abstract
Extremal lattices are lattices maximal in size with respect to the number n of their join-irreducible elements with bounded Vapnik-Chervonekis dimension k. It is natural, however, to estimate the size of a lattice also with respect to the number of its meet-irreducible elements. Although this number may differ for nonequivalent (n, k + 1)-extremal lattices, we show that each (n, k + 1)-extremal lattice has k disjoint chains of meet-irreducible elements, each of length n − k + 2.
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References
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