A splitting property of maximal antichains

Finite distributive lattices and the splitting property

“…In [1], Ahlswede, Erdős and Graham define a dense ordered set P to be one with the property: for all elements a < b in P , if the open interval (a, b) = {x ∈ P : a < x < b} is nonempty then it must contain at least two elements. They then prove that every dense finite ordered set has the splitting property.…”

“…For 3 × 2 4 , we take 2 4 to be the set of all subsets of the set {1, 2, 3, 4}, and look at the maximal antichain It remains to show that 3 × 2 3 has the splitting property. This could of course be done by thorough inspection, but we instead do it by repeating the argument in [1], slightly modified.…”

“…Ahlswede, Erdős and Graham introduced these notions in [1], and proved that every finite Boolean lattice has the splitting property. In the first part of this paper (Sections 1 and 2), we characterize all finite distributive lattices with the splitting property, and find that there are exactly three others.…”

Minimum sized fibres in distributive lattices

“…The important results for us from [1] are that Boolean lattices are dense, and every dense ordered set has the splitting property. In fact it is easy to prove that the only dense distributive lattices are Boolean.…”

“…To prove Theorem 1, we play off the dimension of a distributive lattice against its length and invoke a description of partitions induced by maximal antichains in a certain class of distributive lattices [1]. In the process we prove that for at least some distributive lattices, minimal fibre size is a hereditary property: see Theorem 2.…”