Minimum sized fibres in distributive lattices

Abstract

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 / (£>), 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 f(D)>c-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.2000 Mathematics subject classification: primary 06A07,06D99.

Finite distributive lattices and the splitting property

“…In an earlier paper [4], we found that the splitting property was precisely the idea we needed in our investigation of sizes of minimal fibres of distributive lattices. For other papers on the splitting property, see [2], [5], [6].…”

Optimal sets of questions for Twenty Questions

“…); we leave the details to Section 3. Dagan et al showed that the bound 1.25 n is tight, by constructing an optimal set of questions of this size for every n. The bound is not tight for fibers of 2 Xn : Luczak improved the lower bound Ω(1.25 n ) to Ω(2 n/3 ) = Ω(1.2599 n ), as described in Duffus and Sands [DS01]. Lonc and Rival conjectured that the optimal size of a fiber of 2 Xn is Θ(2 n/2 ), realized by the collection of all sets comparable with {x 1 , .…”

Packing and covering k-chain free subsets in Boolean lattices

“…[6,5,4,3,9,8,11]. Duffus et al [3] proved that τ 2 (P) ≤ 2 3 |P|, while in [11] it was shown that the coefficient 2 3 cannot be improved below 8 15 .…”