In the paper "On Families of Mutually Exclusive Sets" by Erdős and Tarski, it is claimed that for every Boolean algebra $B$ and a nondecreasing cardinal function $f\colon B \to \mathrm{Card}$, $B$ can be expressed as a direct product of $f$-homogeneous Boolean subalgebras. A subalgebra $A$ of $B$ is called $f$-homogeneous if $f(A \setminus \{0\})$ is a singleton. The proof would goes like this.
Let $H \subset B$ be a maximal family of pairwise disjoint $f$-homogeneous elements of $B$. $H$ exists by Teichmüller–Tukey lemma. For every nonzero $b \in B$, there exists $a \le b$ such that $B\restriction a$ is $f$-homogeneous, for you can pick any $a \le b$ such that $f(a) = \min f(B \restriction b \setminus \{0\})$. Hence there exists $h \in H$ which is not disjoint with $a$ and, a fortiori, $b$. This implies that the Boolean algebra homomorphism $\phi\colon B \to \prod_{h \in H} B \restriction h$ given by $b \mapsto (b \land h)_{h \in H}$ is injective.
If $B$ is a complete Boolean algebra, $\phi$ is indeed surjective. But I don't see why in noncomplete cases. Is the claim correct?