On automorphisms of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$ in the case when the family $\mathcal{F}$ consists of nonempty inductive subsets of $\omega$

Let $\mathcal{F}$ be a family of nonempty inductive subsets of $\omega$. It is proved that an injective endomorphism $\epsilon$ of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$ is the transformation if and only if $\epsilon$ has three distinct fixed points, which is equivalent to existence non-idempotent element $(i,j,[p))\in {\mathit{B}}_{\omega }^{\mathcal{F}}$ such that $(i,j,[p))\epsilon =(i,j,[p))$.