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Minimum sized fibres in distributive lattices

Part of: Ordered sets Distributive lattices Boolean algebras (Boolean rings)

Published online by Cambridge University Press:  09 April 2009

Dwight Duffus  and
Bill Sands
Dwight Duffus
Affiliation:
Mathematics and Computer Science Department Emory UniversityAtlanta GA 30322USA e-mail: dwight@mathcs.emory.edu
Bill Sands
Affiliation:
Mathematics and Statistics Department The University of CalgaryCalgary AB T2N 1N4Canada e-mail: sands@math.ucalgary.ca
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Abstract

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A subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function ƒ (D), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.

Keywords

distributive latticefibremaximal antichainsplitting property

MSC classification

Secondary: 06A07: Combinatorics of partially ordered sets 06D99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 3 , June 2001 , pp. 337 - 350
Copyright
Copyright © Australian Mathematical Society 2001

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