The almost disjointness invariant for products of ideals

TL;DR

This paper investigates the almost disjointness numbers related to transfinite products of finite set ideals, establishing bounds and equalities with classical cardinal characteristics, and explores their consistency and open problems.

Contribution

It provides a ZFC lower bound for these characteristics, shows the splitting numbers are equal to the classical splitting number, and demonstrates consistency results for the almost disjointness numbers.

Findings

Established a ZFC lower bound involving classical almost disjointness and splitting numbers.

Proved that splitting numbers for these quotients equal the classical splitting number.

Showed it is consistent that these almost disjointness numbers equal the second uncountable cardinal.

Abstract

The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A $\mathrm{ZFC}$ lower bound involving the minimum of the classical almost disjointness and splitting numbers is proved for these characteristics. En route, it is shown that the splitting numbers associated to these quotients are all equal to the classical splitting number. Finally, it is proved to be consistent that the almost disjointness numbers associated to these quotients are all equal to the second uncountable cardinal while the bounding number is the first uncountable cardinal. Several open problems are considered.