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

Abstract

In the paper we describe injective endomorphisms of the inverse semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$, 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. \textbf{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 ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ is isomorphic to $(\omega ,+)$. Also we describe the structure of the semigroup $\mathfrak{End}({\mathcal{B}}_{\lambda })$ of all endomorphisms of the semigroup of $\lambda \times \lambda$-matrix units ${\mathcal{B}}_{\lambda }$.

11 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2208.09155; text overlap with arXiv:2206.12819