partitions of Boolean algebras A partition of a Boolean algebra is a collection of pairwise disjoint nonzero elements with supremum 1. For any infinite Boolean algebra $A$ let $a(A)$ be the least size of an infinite partition of $A$. Is $a(A+B)=\min\{a(A),a(B)\}$, where $A+B$ is the free product of $A$ and $B$?
 A: It seems this is an open question. First note that (since the answer to the corresponding question for lottery sums is yes) the following are equivalent:


*

*$a(A+B)=\min\{a(A),a(B)\}$ for every $A,B$

*$a(A+A)=a(A)$ for every $A$.


Regarding 2, one should read the article The minimal size of infinite maximal antichains in direct products of partial orders by Miloš Kurilić (Order, 2017). The article starts (and ends) with the open question:

Is there a partial order $\mathbb{P}$ such that $a(\mathbb{P})<a(\mathbb{P}\times \mathbb{P})$?

There are several partial results which imply that the answer to the OP´s question is yes in at least two cases: When the boolean algebras are atomic (see Theorem 6.1) and when the boolean algebras have size at most $\aleph_1$ (see Corollary 4.2).
Kurilić also mentions that it is not even clear what happens with the algebra $A:=\mathcal{P}(\omega)/\mathrm{Fin}$. It is known by results of Spinas that $a(A+A)=a(A)$ in models of $\mathfrak{a}=\mathfrak{b}$ (so it is true under $CH$ or $MA$). However, it is not known whether $a(A+A)<a(A)$ is consistent.
