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

Abstract

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

17 pages. arXiv admin note: text overlap with arXiv:2206.12819