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In the paper we describe injective endomorphisms of the inverse semigroup$\boldsymbol{B}_{\omega}^{\mathscr{F}}$, 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. \textbf{90} (2020), 5--19 (in Ukrainian)],in the case when the family $\mathscr{F}_n$ is generated by the set$\{0,1,\ldots,n\}$. In particular we show that the semigroup of injectiveendomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ isisomorphic to $(\omega,+)$. Also we describe the structure of the semigroup$\mathfrak{End}(\mathscr{B}_{\lambda})$ of all endomorphisms of the semigroupof $\lambda{\times}\lambda$-matrix units $\mathscr{B}_{\lambda}$.

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