Venn diagrams and symmetric chain decompositions in the Boolean lattice
type: Item
instance of: scholarly article
title: Venn diagrams and symmetric chain decompositions in the Boolean lattice
zbMATH Open document ID: 1034.06001
author: Charles E. Killian
author: Carla D. Savage
author: J. Griggs
published in: The Electronic Journal of Combinatorics
publication date: 2004-02-05T00:00:00Z
full work available at URL: https://eudml.org/doc/123373
full work available at URL: v11i1r2.html
review text: Summary: We show that symmetric Venn diagrams for \(n\) sets exist for every prime \(n\), settling an open question. Until this time, \(n=11\) was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice \(\mathcal B_n\), and prove that such decompositions exist for all prime \(n\). A consequence of the approach is a constructive proof that the quotient poset of \(\mathcal B_n\), under the relation ``equivalence under rotation'', has a symmetric chain decomposition whenever \(n\) is prime. We also show how symmetric chain decompositions can be used to construct, for all \(n\), monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 2038348
zbMATH Keywords: Venn diagrams
zbMATH Keywords: symmetric chain decomposition
zbMATH Keywords: Boolean lattice
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2038348