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 B of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in B. More generally, we say that a partial ordering is κ-scattered if it does not contain a copy of any κ-dense linear ordering. We prove analogues of Hausdorff's result for κ-scattered linear orderings, and for κ-scattered partial orderings satisfying the finite antichain condition. We also study the ℚκ-scattered partial orderings, where ℚκ is the saturated linear ordering of cardinality κ, and a partial ordering is ℚκ-scattered when it embeds no copy of ℚκ. We classify the ℚκ-scattered partial orderings with the finite antichain condition relative to the ℚκ-scattered linear orderings. We show that in general the property of being a ℚκ-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.