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 $\omega$

TL;DR

This paper characterizes the injective endomorphisms of a specific inverse semigroup generated by finite initial intervals, showing they form a structure isomorphic to the additive semigroup of natural numbers.

Contribution

It provides a detailed description of the injective endomorphisms of the semigroup $oldsymbol{B}_{oldsymbol{ ext{omega}}}^{oldsymbol{ ext{F}}_n}$ and establishes their isomorphism to $(oldsymbol{ ext{omega}},+)$, extending understanding of such algebraic structures.

Findings

Injective endomorphisms form a semigroup isomorphic to $(oldsymbol{ ext{omega}},+)$.

Describes the structure of endomorphisms of the matrix units semigroup $oldsymbol{ ext{B}}_{oldsymbol{ ext{lambda}}}$.

Provides structural insights into the semigroup of injective endomorphisms of $oldsymbol{B}_{oldsymbol{ ext{omega}}}^{oldsymbol{ ext{F}}}$.

Abstract

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