Many subalgebras of $\mathcal{P}(\omega )\mathrm{/}fin$

In answer to a question on Mathoverflow we show that the Boolean algebra $\mathcal{P}(\omega )\mathrm{/}fin$ contains a family $\{{\mathcal{B}}_X:X\subseteq \mathfrak{c}\}$ of subalgebras with the property that $X\subseteq Y$ implies ${\mathcal{B}}_Y$ is a subalgebra of ${\mathcal{B}}_X$ and if $X\subseteq ̸Y$ then ${\mathcal{B}}_Y$ is not embeddable into~${\mathcal{B}}_X$. The proof proceeds by Stone duality and the construction of a suitable family of separable zero-dimensional compact spaces.