Ordinal numbers are scattered linear orderings. The class of scattered linear orderings is closed under two natural operations: order reversal and summation along an index set that is also a scattered linear ordering. Hausdorff proved that by closing the class of ordinals under these two operations one obtains all scattered linear orderings.

Uri Abraham and Robert Bonnet extended Hausdorff's theorem to scattered partial orderings with the finite antichain property.

In a slightly different direction, we consider certain uncountable analogs of the rational numbers with a property that Hausdorff introduced as $\eta_\alpha$-orderings. Our notation will be $\mathbb{Q}(\kappa)$ for cardinals $\kappa \geq \omega$. It will be clear from the definition that $\mathbb{Q}(\omega) = \mathbb{Q}$. We say that a partial ordering is $\kappa$-scattered if it does not contain a copy of $\mathbb{Q}(\kappa)$. We will discuss our results on $\kappa$-scattered partial orderings, which generalize those of Abraham and Bonnet.

This work is joint with Mirna Dzamonja.