Pairings from down-sets and up-sets in distributive lattices |
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| Authors: | DE Daykin AJW Hilton D Miklós |
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| Institution: | 1. Department of Mathematics, University of Reading, Whiteknights, Reading RG6 2AX, England;2. Faculty of Mathematics, The Open University, Walton Hall, Milton Keynes, MK England;3. Mathematical Institute, Hungarian Academy of Science, Reáltanoda utca 13-15, Budapest H-1053, Hungary |
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| Abstract: | If D is a set of subsets of a finite set such that a?D,b ? a ? b?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 a ∩ b = /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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