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

Abstract

In the paper we describe injective endomorphisms of the inverse semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$, 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 injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is isomorphic to $(ω,+)$. Also we describe the structure of the semigroup $\mathfrak{End}(\mathscr{B}_λ)$ of all endomorphisms of the semigroup of $λ{\times}λ$-matrix units $\mathscr{B}_λ$.