Autostable of Boolean Algebra $\mathfrak{B}_{\omega}$, Enriched by Automorphism

This paper is devoted to researching of boolean algebra enriched by automorphism autostable. In this paper there are examples of autostable boolean algebra $(\mathfrak{B}_{\omega}, \phi)$ and some case of nonautosability are derived. We use theorem about unbounded models and branching theorem for taking this results.