On the Semigroup of Injective Endomorphisms of the Semigroup \({B}_{\upomega }^{{\mathcal{F}}_{n}}\) Generated by the Family \({\mathcal{F}}_{n}\) of Initial Finite Intervals in ω

We describe injective endomorphisms of the inverse semigroup \({B}_{\upomega }^{\mathcal{F}}\) $B_{\mathrm{\omega }}^{\mathcal{F}}$ introduced in [Visn. Lviv. Univ., Ser. Mech.-Mat., Issue 90, 5 (2020)] for the case where the family \({\mathcal{F}}_{n}\) ${\mathcal{F}}_n$ is generated by the set {0,1,…, n}. In particular, it is shown that the semigroup of injective endomorphisms of the semigroup \({B}_{\upomega }^{\mathcal{F}}\) $B_{\mathrm{\omega }}^{\mathcal{F}}$ is isomorphic to (ω, +). Moreover, we describe the structure of the semigroup \(\mathfrak{E}\mathfrak{n}\mathfrak{d}\left({\mathfrak{B}}_{\uplambda }\right)\) $\mathfrak{E}\mathfrak{n}\mathfrak{d}\left({\mathfrak{B}}_{\mathrm{\lambda }}\right)$ of all endomorphisms of the semigroup of λ × λ -matrix units \({\mathfrak{B}}_{\uplambda }\) ${\mathfrak{B}}_{\mathrm{\lambda }}$ .