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

Abstract

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