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

Abstract

In answer to a question on Mathoverflow we show that the Boolean algebra $\mathcal{P}(\omega )/fin$ 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 $X⊈Y$ 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.

Version 2: added a description of a 40-years old answer to the original question Version 3: final version after referee's report; submitted to the journal. 2026-04-06: Fixed a few annoying typos