A structure theorem for posets admitting a “strong” chain partition:: A generalization of a conjecture of Daykin and Daykin (with connections to probability correlation inequalities)

Abstract

Suppose a finite poset P is partitioned into three non-empty chains so that, whenever p, q ∈ P lie in distinct chains and p < q , then every other element of P is either above p or below q. In 1985, the following conjecture was made by David Daykin and Jacqueline Daykin: such a poset may be decomposed into an ordinal sum of posets ⊕ i = 1 n R i such that, for 1 ⩽ i ⩽ n , one of the following occurs: (1) R i is disjoint from one of the chains of the partition; or (2) if p, q ∈ R i are in distinct chains, then they are incomparable. The conjecture is related to a question of R. L. Graham's concerning probability correlation inequalities for linear extensions of finite posets. In 1996, a proof of the Daykin–Daykin conjecture was announced (by two other mathematicians), but their proof needs to be rectified. In this note, a generalization of the conjecture is proven that applies to finite or infinite posets partitioned into a (possibly infinite) number of chains with the same property. In particular, it is shown that a poset admits such a partition if and only if it is an ordinal sum of posets, each of which is either a width 2 poset or else a disjoint sum of chains. A forbidden subposet characterization of these partial orders is also obtained.