Countably Complementable Linear Orderings

Abstract

We say that a countable linear ordering $\mathcal L$ is countably complementable if there exists a linear ordering $\overline{\mathcal L}$, possibly uncountable, such that for any countable linear ordering $\mathcal B$,$\mathcal L$ does not embed into $\mathcal B$ if and only if $\mathcal B$ embeds into $\overline{\mathcal L}$. We characterize the linear orderings which are countably complementable. We also show that this property is equivalent to the countable version of the finitely faithful extension property introduced by Hagendorf. Using similar methods and introducing the notion of weakly countably complementable linear orderings, we answer a question posed by Rosenstein and prove the countable case of a conjecture of Hagendorf, namely, that every countable linear ordering satisfies the countable version of the totally faithful extension property.