The Comparability Numbers and the Incomparability Numbers

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 \(\omega ^\omega \) ${\omega }^{\omega }$ , \(\mathcal {P}(\omega )/\textrm{fin}\) $\mathcal{P}(\omega )/\text{fin}$ , the Turing degrees \(\mathcal {D}\) $\mathcal{D}$ , the quotient algebra \(\textsf {Borel}(2^\omega )/\textsf {null}\) $\mathsf{Borel}(2^{\omega })/\mathsf{null}$ , the ideals \(\textsf {meager}\) $\mathsf{meager}$ and \(\textsf {null}\) $\mathsf{null}$ . Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on \(\omega \) $\omega$ . We also consider these invariants for ideals on \(\omega \) $\omega$ and on \(\omega _1\) ${\omega }_1$ .