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Cutsets and anti-chains in linear lattices

Authors
Journal
Journal of Combinatorial Theory Series A
0097-3165
Publisher
Elsevier
Publication Date
Volume
113
Issue
8
Identifiers
DOI: 10.1016/j.jcta.2006.03.022
Keywords
  • Cutset
  • Anti-Chain
  • Fibre
  • Maximal Chain
  • Linear Lattice
  • Projective Geometry
  • [Formula Omitted]
Disciplines
  • Mathematics

Abstract

Abstract Consider the poset, ordered by inclusion, of subspaces of a four-dimensional vector space over a field with 2 elements. We prove that, for this poset, any cutset (i.e., a collection of elements that intersects every maximal chain) contains a maximal anti-chain of the poset. In analogy with the same result by Duffus, Sands, and Winkler [D. Duffus, B. Sands, P. Winkler, Maximal chains and anti-chains in Boolean lattices, SIAM J. Discrete Math. 3 (2) (1990) 197–205] for the subset lattice, we conjecture that the above statement holds in any dimension and for any finite base field, and we prove some special cases to support the conjecture.

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