On the semigroup of injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}_n}$ which is generated by the family ${\mathcal{F}}_n$ of initial finite intervals of $\omega$

In the paper we describe injective endomorphisms of the inverse semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$, which is introduced in the paper [O. Gutik and M. Mykhalenych, \emph{On some generalization of the bicyclic monoid}, Visnyk Lviv. Univ. Ser. Mech.-Mat. $\mathbf{\text{90}}$ (2020), 5–19 (in Ukrainian)], in the case when the family ${\mathcal{F}}_n$ is generated by the set $\{0,1,\dots ,n\}$. In particular we show that the semigroup of injective endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$ is isomorphic to $(\omega ,+)$. Also we describe the structure of the semigroup $\mathfrak{E}\mathfrak{n}\mathfrak{d}({\mathcal{B}}_{\lambda })$ of all endomorphisms of the semigroup of $\lambda \times \lambda$-matrix units ${\mathcal{B}}_{\lambda }$.