On injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^4}$ with a four-element family ${\mathcal{F}}^4$ of inductive non-empty subsets of $\omega$

We describe injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^4}$ with a four-element family ${\mathcal{F}}^4$ of inductive non-empty subsets of $\omega$. In particular we proved that every injective monoid endomorphism of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^4}$ is the identity transformation. Also we describe all (not necessary monoid) injective endomorphism of ${\mathit{B}}_{\omega }^{{\mathcal{F}}^4}$.