Title: On the semigroup injective monoid endomorphisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}}$ with a three element family $\mathscr{F}$ of inductive nonempty subsets of $ω$

Abstract: We study monoid non-injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with an infinite family $\mathscr{F}$ of inductive nonempty subsets of $\omega$.