I was recently trying to understand how coproducts of Boolean algebras work, since I need them for my research. I came across a StackExchange question with a great constructive answer from "Math Student 020".
However, even after a lot of searching, I can't seem to find a reference to use for such a construction. I don't have access to the one "Math Student 020" mentioned (from P.T. Johnstone). Does anybody know a reference I could use for a constructive characterization of the coproduct of Boolean algebras? Thanks!
The category of Boolean algebras is isomorphic to the category of Boolean rings, so let's work with these.
Every Boolean is a commutative $\mathbb{F}_2$-algebra.
When $R$ is a commutative ring, the category of commutative $R$-algebras has a coproduct. It is the tensor product $A \otimes_R B$ equipped with the multiplication
$$(A \otimes_R B) \otimes_R (A \otimes_R B) \xrightarrow{\cong} (A \otimes_R A) \otimes_R (B \otimes_R B) \xrightarrow{\mu_A \otimes \mu_B} A \otimes_R B$$ and the unit $1_A \otimes 1_B$.
If $A,B$ are Boolean rings, then the tensor product $A \otimes_{\mathbb{F}_2} B$ is again a Boolean ring: In fact, since it is a commutative $\mathbb{F}_2$-algebra, the Frobenius $r \mapsto r^2$ is additive, so it suffices to check $r^2=r$ on elementary tensors $r = a \otimes b$, where it is clear.
It follows formally that $A \otimes_{\mathbb{F}_2} B$ is also the coproduct of $A$ and $B$ in the category of Boolean rings.
Notice that there cannot be a very easy description. In fact, the coproduct of Boolean rings corresponds to the product of Stone spaces (via Stone duality), which is of course easy to describe, but the Boolean algebra of compact-open subsets in a product of Stone spaces is not easy to describe (but of course it exists).