A scattering of orders

A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class $ \mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $ \mathcal B$.

More generally, we say that a partial ordering is $ \kappa $-scattered if it does not contain a copy of any $ \kappa $-dense linear ordering. We prove analogues of Hausdorff's result for $ \kappa $-scattered linear orderings, and for $ \kappa $-scattered partial orderings satisfying the finite antichain condition.

We also study the $ \mathbb{Q}_\kappa $-scattered partial orderings, where $ \mathbb{Q}_\kappa $ is the saturated linear ordering of cardinality $ \kappa $, and a partial ordering is $ \mathbb{Q}_\kappa $-scattered when it embeds no copy of $ \mathbb{Q}_\kappa $. We classify the $ \mathbb{Q}_\kappa $-scattered partial orderings with the finite antichain condition relative to the $ \mathbb{Q}_\kappa $-scattered linear orderings. We show that in general the property of being a $ \mathbb{Q}_\kappa $-scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions.