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 \(\omega ^\omega \), \(\mathcal {P}(\omega )/\textrm{fin}\), the Turing degrees \(\mathcal {D}\), the quotient algebra \(\textsf {Borel}(2^\omega )/\textsf {null}\), the ideals \(\textsf {meager}\) and \(\textsf {null}\). 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 \(\omega _1\).