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

We describe injective monoid endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^3}$ with a three element family ${\mathcal{F}}^3$ of inductive nonempty subsets of $\omega$. Also, we show that the monoid ${}_{\ast }^1({\mathit{B}}_{\omega }^{\mathcal{F}})$ of all injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^3}$ is isomorphic to the multiplicative semigroup of positive integers.