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A poset hierarchy. (English) Zbl 1105.03038
Summary: This article extends a paper of U. Abraham and R. Bonnet [Fundam. Math. 159, 51–69 (1999; Zbl 0934.06005)] which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy \(^\kappa{\mathcal H}^*\) which characterises the closure of the class of all \(\kappa\)-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call \(\kappa\)-scattered. These posets cannot embed any order such that for every two subsets of size \(<\kappa\), one being strictly less than the other, there is an element in between. If a linear order has this property and has size \(\kappa\) it is unique and called \(\mathbb{Q}(\kappa)\). Partial orders such that for every \(a<b\) the set \(\{x:a<x<b\}\) has size \(\geq\kappa\) are called weakly \(\kappa\)-dense, and posets that do not have a weakly \(\kappa\)-dense subset are called strongly \(\kappa\)-scattered. We prove that \(^\kappa{\mathcal H}^*\) includes all strongly \(\kappa\)-scattered FAC posets and is included in the class of all FAC \(\kappa\)-scattered posets. For \(\kappa=\aleph_0\) the notions of scattered and strongly scattered coincide and our hierarchy is exactly \(\text{aug}({\mathcal H})\) from the Abraham-Bonnet theorem.

MSC:
03E04 Ordered sets and their cofinalities; pcf theory
06A05 Total orders
06A06 Partial orders, general
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