On extendability to $F_σ$ ideals
Adam Kwela
TL;DR
The paper addresses M. Hrušák's question on when a Borel ideal can be extended to a $\boldsymbol{\Sigma^0_2}$ (and related) ideal by exploiting the Katětov order and the $\mathrm{conv}$ construction. It introduces $\mathrm{conv}(\mathcal{I},(\alpha_n))$, proves an equivalence: $\mathcal{I}$ extends to a $P^+$-ideal iff $\mathrm{conv}(\mathcal{I},(\alpha_n))$ does, and establishes a criterion for $\mathrm{conv}\le_K\mathcal{J}$. Using this framework, the authors build a concrete counterexample: $\mathcal{I}=\mathrm{conv}(\mathcal{I}_d,(1/2^{n+1}))$ is $\boldsymbol{\Sigma^0_6}$, not extendable to any $P^+$-ideal, and satisfies $\mathrm{conv}\not\le_K\mathcal{I}$. This yields a negative answer to the question and highlights limitations of the method, noting that the constructed counterexample may not be Borel in general and suggesting further avenues to refine the characterization via $\mathrm{conv}$.
Abstract
Answering in negative a question of M. Hrušák, we construct a Borel ideal not extendable to any $F_σ$ ideal and such that it is not Katětov above the ideal $\mathrm{conv}$.