I think the following is true but I can't find a reference. If $A$ and $A'$ are Boolean algebras, and $A \subset A'$, then $S_A$ can be embedded in $S_{A'}$, where $S_A$ and $S_{A'}$ are the associated Stone spaces.
If anyone can point me in the right direction I'd be very grateful.
Greg
This is false. For instance, let $A'=P(\mathbb{N})$ and let $A$ be the subalgebra consisting of finite or cofinite sets. Then $S_A\cong\mathbb{N}\cup\{\infty\}$ and $S_{A'}\cong\beta\mathbb{N}$, but $\mathbb{N}\cup\{\infty\}$ does not embed in $\beta\mathbb{N}$. (As a sketch of how to prove this, suppose $i:\mathbb{N}\to\beta\mathbb{N}$ is an embedding. Then we can find disjoint subsets $A_n\subset\mathbb{N}$ such that $A_n$ is in the ultrafilter $i(n)$ for each $n$. This means the set $A=\bigcup_n A_{2n}$ is in $i(n)$ iff $n$ is even, and so the sequence $(i(n))$ cannot converge in $\beta\mathbb{N}$ and $i$ cannot be extended to $\infty$.)