Let $A$ be a fixed boolean ring. Is there a sort of classification of boolean rings $B$ with $A \subseteq B$? For example, if $A=\mathbb{F}_2$, the answer would be Stone duality: $B=C(Spec(B),\mathbb{F}_2)$. Also in the general case, Stone duality tells us that the inclusion $A \subseteq B$ corresponds to a surjective map of Stone spaces $Spec(B) \to Spec(A)$. Remark that $B=C(T,A)$ for some Stone space $T$ if and only if $Spec(A)$ is a direct factor of $Spec(B)$, which does not hold in general. Anyway, is there another description of $B$ which involves $A$ and perhaps a kind of relative spectrum "$Spec(B/A)$"?
For example if $A$ is finite, then $X=Spec(A)$ is finite discrete and it's not hard to see that $B \to (B/\mathfrak{p}B)_{\mathfrak{p} \in X}$ and $(B^{\mathfrak{p}})_{\mathfrak{p} \in X} \mapsto \prod_{\mathfrak{p} \in X} B^{\mathfrak{p}}$ is an equivalence of categories between those boolean rings $B$ with $A \subseteq B$ and $X$-tuples $(B^{\mathfrak{p}})$ of boolean rings such that $A/\mathfrak{p} \subseteq B^{\mathfrak{p}}$. And this category is dual to the category of $X$-tuples of Stone Spaces.
What happens when $A$ is infinite? I hope it's clear what I'm looking for. Otherwise feel free to comment. Please don't answer "There is no such description".
