On antichain numbers and the splitting ideal

Abstract

In this article, we study combinatorial properties of a certain ideal on $\ddot{I}‰$, called the \emph{Splitting ideal}. We calculate its cardinal invariants and its position in the Katětov order among other definable ideals. We also study the antichain numbers $\mathfrak{a}(\mathcal{J})$ of algebras $\mathcal{P}(\ddot{I}‰)/\mathcal{J}$ for various Borel ideals. We show that $\text{min}\{\mathfrak{b},{\text{cov}}_h^{+}(\mathcal{J})\}\le \mathfrak{a}(\mathcal{J})$ holds for a wide class of ideals, including all $F_{\ddot{I}}ƒ$-ideals, all analytic $P$-ideals and many other examples. We also show that $\mathfrak{b}\le \mathfrak{a}(\mathcal{J})$ 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