A Poset Hierarchy

Abstract

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. Abraham and Bonnet gave a poset hierarchy that characterised the class of scattered posets which do not have infinite antichains (abbreviated FAC for finite antichain condition). An antichain here is taken in the sense of incomparability. We define a larger poset hierarchy than that of Abraham and Bonnet, to include a broader class of ``scattered'' posets that we call $κ$-scattered. These posets cannot embed any order such that for every two subsets of size $ < κ$, one being strictly less than the other, there is an element in between. If a linear order has this property and has size $κ$ we call this set $\qkappa$. Such a set only exists when $κ^{<κ}=κ$. Partial orders with the property that for every $a<b$ the set $\{x: a<x<b\}$ has size $\geq κ$ are called weakly $κ$-dense, and partial orders that do not have a weakly $κ$-dense subset are called strongly $κ$-scattered. We prove that our hierarchy includes all strongly $κ$-scattered FAC posets, and that the hierarchy is included in the class of all FAC $κ$-scattered posets. In addition, we prove that our hierarchy is in fact the closure of the class of all $κ$-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. For $κ=\aleph_0$ our hierarchy agrees with the one from the Abraham-Bonnet theorem.