On the semigroup of injective monoid endomor\-phisms of the monoid ${\mathit{B}}_{\omega }^{\mathcal{F}}$ with the two-elements family $\mathcal{F}$ of inductive nonempty subsets of $\omega$

We study injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$ with the two-elements family $\mathcal{F}$ of inductive nonempty subsets of $\omega$. We describe the elements of the semigroup ${}_{\ast }^1({\mathit{B}}_{\omega }^{\mathcal{F}})$ of all injective monoid endomorphisms of the monoid ${\mathit{B}}_{\omega }^{\mathcal{F}}$, and show that Green’s relations $\mathcal{R}$, $\mathcal{L}$, $\mathcal{H}$, $\mathcal{D}$, and $\mathcal{J}$ on ${}_{\ast }^1({\mathit{B}}_{\omega }^{\mathcal{F}})$ coincide with the relation of equality.