A note on infinite partitions of free products of Boolean algebras

TL;DR

This paper investigates the minimal size of infinite partitions in free products of Boolean algebras, providing new lower bounds and exploring whether this invariant equals the minimum of the individual algebras' invariants.

Contribution

It offers new lower bounds for the cardinal invariant of free products of Boolean algebras and discusses conditions under which equality holds.

Findings

Lower bounds for (A B) are established.

The equality (A B)=min((A),(B)) is not proven for all cases.

The dual topological space of the free product relates to the product of individual spaces.

Abstract

If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$ (whose dual topological space is the product of the dual topological spaces of $A$ and $B$), is below both $\mathfrak{a}(A)$ and $\mathfrak{a}(B)$. The equality $\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$ is not known to hold for all infinite Boolean algebras $A$ and $B$. Here some lower bounds of $\mathfrak{a}(A\oplus B)$ are provided.