Relative cofinality of ideals
Adam Marton, Miroslav Repický
TL;DR
This work introduces the relative cofinality invariants $\mathrm{cof}^{\mathcal{J}}(\mathcal{I})$ and $\mathrm{cof}^{*}(\mathcal{I})$, framing them as two-parameter tools that encode interaction between pairs of ideals and relate to ideal convergence via the $\mathrm{P}(\mathcal{J})$-property. It develops a relational-system framework with Tukey connections to derive bounds, proves a tight inequality diagram among classical and relative invariants, and analyzes the invariants for key ideal pairs on $\omega$ and on $[0,1]$, including a complete calculation for critical ideals on $\omega$ and a dichotomy for the uncountable real line. The paper also studies maximal ideals, establishing a clear criterion when $\mathrm{cof}^{\mathcal{J}}(\mathcal{I})=1$ and describing boundary behavior across cardinal thresholds such as $\mathrm{non}(\mathcal{I})$ and $|X|$, with consistency results like the existence of maximal ideals attaining $\mathrm{cof}(\mathcal{I})=\mathrm{cof}^{*}(\mathcal{I})=2^{|X|}$. Open problems are proposed to classify possible relative cofinalities in various pairings and to characterize the spectrum $\mathrm{Cof}(\mathcal{I})$ for maximal ideals. The work thus deepens understanding of how two interacting ideals shape cofinality and related convergence phenomena in both countable and uncountable contexts.
Abstract
We introduce a two-parameter modification of the cofinality invariant of ideals. This allows us to include the interaction of a pair of ideals in the study of base-like structures. We find the values (cardinal numbers or well-known cardinal invariants) of the invariant for pairs of some critical ideals on $ω$. We also dichotomously divide pairs of known ideals on the real line based on whether their relative cofinality is trivial or uncountable. Finally, we also study the relative cofinality of maximal ideals.