The comparability numbers and the incomparability numbers

Abstract

We introduce new cardinal invariants of a poset, called the comparability number and the incomparability number. We determine their value for well-known posets, such as ${\omega }^{\omega }$, $\mathcal{P}(\omega )/\mathrm{fin}$, the Turing degrees $\mathcal{D}$, the quotient algebra $\mathsf{Borel}(2^{\omega })/\mathsf{null}$, the ideals $\mathsf{meager}$ and $\mathsf{null}$. Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on $\omega$. We also consider these invariants for ideals on $\omega$ and on ${\omega }_1$.

14 pages