On antichain numbers and the splitting ideal

TL;DR

This paper investigates the combinatorial properties and cardinal invariants of the splitting ideal and related ideals, analyzing their positions in the Kattov order and antichain numbers in various algebraic contexts.

Contribution

It computes the cardinal invariants of the splitting ideal, explores their relationships with other ideals, and establishes inequalities and consistency results for antichain numbers.

Findings

Calculated cardinal invariants of the splitting ideal.

Established inequalities relating antichain numbers and bounding numbers.

Proved the consistency of antichain numbers being less than the bounding number for certain ideals.

Abstract

In this article, we study combinatorial properties of a certain ideal on $\omega$, called the \emph{Splitting ideal}. We calculate its cardinal invariants and its position in the Kat\v{e}tov order among other definable ideals. We also study the antichain numbers $\mathfrak{a}(\mathcal{J})$ of algebras $\mathcal{P}(\omega)/\mathcal{J}$ for various Borel ideals. We show that $\textrm{min}\{\mathfrak{b},\textrm{cov}^{+}_{h}(\mathcal{J})\}\leq\mathfrak{a}(\mathcal{J})$ holds for a wide class of ideals, including all $F_{\sigma}$-ideals, all analytic $P$-ideals and many other examples. We also show that $\mathfrak{b}\leq\mathfrak{a}(\mathcal{J})$ holds for \emph{convergent ideal} and for \emph{Boring ideal}. Finally, we will show the consistency of $\mathfrak{a}(\mathcal{J})<\mathfrak{b}$ for the \emph{Van der Waerden's ideal} and the linear growth ideal