On the semigroup of injective monoid endomor\-phisms of the monoid $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with the two-elements family $\mathscr{F}$ of inductive nonempty subsets of $\omega$

TL;DR

This paper characterizes the structure of injective monoid endomorphisms of a specific semigroup formed from inductive subsets of natural numbers, showing that Green's relations are trivial on this endomorphism semigroup.

Contribution

It provides a detailed description of all injective monoid endomorphisms of the semigroup $oldsymbol{B}_{ omannumeral 0}^{ ext{F}}$ and proves that Green's relations are trivial on this set.

Findings

All injective monoid endomorphisms are explicitly described.

Green's relations on the endomorphism semigroup are trivial, coinciding with equality.

The structure of the endomorphism semigroup is fully characterized.

Abstract

We study injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with the two-elements family $\mathscr{F}$ of inductive nonempty subsets of $\omega$. We describe the elements of the semigroup $\boldsymbol{End}^1_*(\boldsymbol{B}_{\omega}^{\mathscr{F}})$ of all injective monoid endomorphisms of the monoid $\boldsymbol{B}_{\omega}^{\mathscr{F}}$, and show that Green's relations $\mathscr{R}$, $\mathscr{L}$, $\mathscr{H}$, $\mathscr{D}$, and $\mathscr{J}$ on $\boldsymbol{End}^1_*(\boldsymbol{B}_{\omega}^{\mathscr{F}})$ coincide with the relation of equality.