Partitioning products of \({\mathcal P} (\omega)/\text{fin}\)
We generalize the cardinal invariant \({\mathfrak a}\) to products of \({\mathcal P} (\omega)/\text{fin}\) and then sharpen the well-known inequality \({\mathfrak b} \leq {\mathfrak a}\) by proving \({\mathfrak b} \leq {\mathfrak a} (\lambda)\) for every \(\lambda \leq \omega\). Here \({\mathfrak a} (n)\), for \(n< \omega\), is the least size of an infinite partition of \(({\mathcal P} (\omega)/\text{fin})^n\), \({\mathfrak a} (\omega)\) is the least size of an uncountable partition of \(({\mathcal P} (\omega)/ \text{fin})^\omega\), and \({\mathfrak b}\) is the least size of an unbounded family of functions from \(\omega\) to \(\omega\) ordered by eventual dominance. We also prove the consistency of \({\mathfrak b}< {\mathfrak a} (n)\) for every \(n< \omega\).
scientific article; zbMATH DE number 950228