On injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^4}$ with a four-element family $\mathscr{F}^4$ of inductive non-empty subsets of $ω$

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^4}$ with a four-element family $\mathscr{F}^4$ of inductive non-empty subsets of $ω$. In particular we proved that every injective monoid endomorphism of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^4}$ is the identity transformation. Also we describe all (not necessary monoid) injective endomorphism of $\boldsymbol{B}_ω^{\mathscr{F}^4}$.