A note on infinite partitions of free products of Boolean algebras

Mario Jardón Santos

Vol. 337 (2025), No. 2, 245–256

DOI: 10.2140/pjm.2025.337.245

Abstract

The cardinal invariants $\mathfrak{𝔞}(n)$, for $1\le n<\omega$, multidimensional generalizations of the mad family number $\mathfrak{𝔞}$, were proved by Spinas (Pacific J. Math. 176:1 (1996), 249–262) to be greater than or equal to the bounding number $\mathfrak{𝔟}$. The lack of knowledge of other lower bounds for these cardinal invariants was noted in the same article. We present a couple of more general results that give lower bounds to the cardinal $\mathfrak{𝔞}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$. One of them, when restricted to the free products of $\mathcal{𝒫}(\omega )∕\mathrm{fin}<!--FUNCTION\; APPLICATION-->$, gives a new proof of the known result. The other has as a corollary a newer lower bound that adds relevant information to the open question of the consistency of $\mathfrak{𝔞}(n+1)<\mathfrak{𝔞}(n)$, for some $1\le n<\omega$.

Keywords

Boolean algebras, free products, mad families, cardinal invariants

Mathematical Subject Classification

Primary: 03E05, 03E17

Milestones

Received: 6 February 2024

Revised: 1 May 2025

Accepted: 3 May 2025

Published: 3 June 2025