About Stone space of one Boolean algebra

We consider the Boolean algebra of the same type as algebra constructed by Bell, and the Stone space of this Boolean algebra. This space is a compactification of a countable discrete space $N$. We prove that there are isolated points in a remainder of this compactification, which are limits of some convergent sequences. We prove that a clopen subset of our space, which is homeomorphic to $\beta\omega$, is a closure of the union of finitely many antichains from $N$. We construct two examples: a clopen subset of the remainder without isolated points, which is not homeomorphic to $\beta\omega\setminus\omega$; a subset of the remainder which is homeomorphic to $\beta\omega\setminus\omega$, but is not a clopen.