On injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^3}$ with a three-element family ${\mathcal{F}}^3$ of inductive non-empty subsets of $\omega$

We describe injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^3}$ with a three-element family ${\mathcal{F}}^3$ of inductive non-empty subsets of $\omega$. In particular we find endomorphisms ${\varpi }_3$ and $\lambda$ of ${\mathit{B}}_{\omega }^{{\mathcal{F}}^3}$ such that for every injective endomorphism $\epsilon$ of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^3}$ there exists an injective endomorphism $\iota \in ⟨\lambda ,{\varpi }_3⟩$ such that $\epsilon ={\alpha }_{[k]}\circ \iota$ for some positive integer $k$, where ${\alpha }_{[k]}$ is an injective monoid endomorphism of ${\mathit{B}}_{\omega }^{{\mathcal{F}}^3}$.