For two measurable space $(X,\mathcal X)$ and $(Y,\mathcal Y)$, their product $\sigma$-algebra is $(X\times Y,\mathcal X\otimes \mathcal Y)$. It is the smallest sub $\sigma$-algebra of $\mathcal P(X\times Y)$ such that for any $A\in\mathcal X$ and $B\in\mathcal Y$, $A\times B$ is measurable.
I am wondering if there is a way to generalize this type of product to any Boolean algebra. We may assume that they are complete. The problem here seems to be that we do not have access to underlying sets $X$ and $Y$ to be able to make up a super algebra that we will take sub-algebras of. The product of algebras defined for instance in this book won't work I believe since for two Boolean algebras $A$ and $B$, their product is defined as the Cartesian product and the operations are \begin{align*} \lnot(p,q)&=(\lnot p, \lnot q)\\ (p_1,q_1)\land (p_2,q_2) &= (p_1\land p_2, q_1\land q_2)\\ (p_1,q_1)\lor (p_2,q_2) &= (p_1\lor p_2, q_1\lor q_2)\\ \end{align*}
But this doesn't work for $\sigma$-algebras and the product defined above.
Any answer or references would be much appreciated.