The comparability numbers and the incomparability numbers

TL;DR

This paper introduces new cardinal invariants called the comparability and incomparability numbers for posets, computes their values for various well-known structures, and explores their behavior in different orderings and ideals.

Contribution

It defines and analyzes the comparability and incomparability numbers, providing their values for key posets and orderings, advancing the understanding of their structural properties.

Findings

Determined the invariants for $ ext{omega}^ ext{omega}$ and $ ext{P}( ext{omega})/ ext{fin}$

Computed values for Turing degrees and Borel null algebras

Analyzed invariants for ultrafilters and ideals on $ ext{omega}$ and $ ext{omega}_1$

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 $\omega_1$.