On injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $\omega$

TL;DR

This paper characterizes all injective endomorphisms of a specific semigroup constructed from a three-element family of inductive subsets of natural numbers, identifying key endomorphisms and their compositions.

Contribution

It provides a complete description of injective endomorphisms of the semigroup oldsymbol{B}_{\u2208}^{\u211d^3} with a three-element family alF^3, including explicit endomorphisms and their structure.

Findings

Identified specific endomorphisms ta_{[k]}, \u001lambda, and ar{\u00f6} of the semigroup.

Proved that every injective endomorphism can be expressed as a composition involving these endomorphisms.

Established the algebraic structure of the endomorphism semigroup for oldsymbol{B}_{\u2208}^{\u211d^3}.

Abstract

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $\omega$. In particular we find endomorphisms $\varpi_3$ and $\lambda$ of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ such that for every injective endomorphism $\varepsilon$ of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ there exists an injective endomorphism $\iota\in\left\langle\lambda,\varpi_3\right\rangle$ such that $\varepsilon=\alpha_{[k]}\circ\iota$ for some positive integer $k$, where $\alpha_{[k]}$ is an injective monoid endomorphism of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$.