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 $\mathfrak{ c}$.
Categories: math.GN math.LO