On the structure of Borel ideals in-between the ideals $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$ in the Kat\v{e}tov order

TL;DR

This paper investigates the complex structure of Borel ideals between two well-known ideals, revealing intricate orderings and the presence of large chains and antichains within the Katětov order.

Contribution

It introduces a new family of ideals $\u03a3(\u03f4)$ to analyze the structure of Borel ideals between $\u210d$ and $ ext{Fin} imes ext{Fin}$, showing the existence of a rich and complicated order structure.

Findings

A copy of $\u2113(\u03f4)$ exists between $\u210d$ and $ ext{Fin} imes ext{Fin}$.

There are chains of length $\u211e$ and antichains of size $\u2102$ within this structure.

The structure contains a complex hierarchy with large chains and antichains.

Abstract

For a family $\mathcal{F}\subseteq \omega^\omega$ we define the ideal $\mathcal{I}(\mathcal{F})$ on $\omega\times\omega$ to be the ideal generated by the family $\{A\subseteq \omega\times\omega:\exists f\in \mathcal{F}\,\forall^\infty n\, (|\{k:(n,k)\in A\}|\leq f(n))\}.$ Using ideals of the form $\mathcal{I}(\mathcal{F})$, we show that the structure of Borel ideals in-between two well known Borel ideals $\mathcal{ED} = \{A\subseteq\omega\times\omega:\exists m \, \forall^\infty n\, (|\{k:(n,k)\in A\}|<m))\}$ and $\mathrm{Fin}\otimes\mathrm{Fin} = \{A\subseteq\omega\times\omega:\forall^\infty n \, (|\{k:(n,k)\in A\}|<\aleph_0))\}$ in the Kat\v{e}tov order is fairly complicated. Namely, there is a copy of $\mathcal{P}(\omega)/\mathrm{Fin}$ in-between $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$, and consequently there are increasing and decreasing chains of length $\mathfrak{b}$ and antichains of size $\mathfrak{c}$.