Relative cofinality of ideals

TL;DR

This paper introduces a two-parameter modification of the cofinality invariant for ideals, analyzing its values for critical ideals on ω and classifying pairs of ideals on the real line based on their relative cofinality.

Contribution

It develops a new two-parameter invariant for ideals, providing explicit calculations and a dichotomy classification for pairs of ideals on the real line.

Findings

Calculated the invariant for critical ideals on ω.

Established a dichotomy for pairs of ideals on the real line.

Studied the relative cofinality of maximal ideals.

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 $\omega$. 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.