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 $ω^ω$, $\mathcal{P}(ω)/\mathrm{fin}$, the Turing degrees $\mathcal{D}$, the quotient algebra $\mathsf{Borel}(2^ω)/\mathsf{null}$, the ideals $\mathsf{meager}$ and $\mathsf{null}$. Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on $ω$. We also consider these invariants for ideals on $ω$ and on $ω_1$.