Pairings from down-sets and up-sets in distributive lattices

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Abstract

If D is a set of subsets of a finite set such that a ϵ D,bab ϵ D, then D is called a down-set. Berge proved that any down-set D is the disjoint union of pairs {a, b} such that ab = /b/ together with the singleton {⊘} if |D| is odd. A proof is given of the corresponding result for finite distributive lattices, together with a number of generalizations and analogues of it; when specialized to the lattice of all subsets of a finite set, this proof is rather simpler than was Berge's.

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