  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.
