On the semigroup of monoid endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$ with a two-element family $\mathcal{F}$ of inductive nonempty subsets of $\omega$

We study the semigroup of non-injective monoid endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$ with a two-elements family $\mathcal{F}$ of inductive nonempty subsets of $\omega$. We describe the structure of elements of the semigroup ${}_0^{\ast }({\mathit{B}}_{\omega }^{\mathcal{F}})$ of non-injective monoid endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$. In particular we show that its subsemigroup ${}^{\ast }({\mathit{B}}_{\omega }^{\mathcal{F}})$ of non-injective non-annihilating monoid endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{\mathcal{F}}$ is isomorphic to the direct product of the two-element left-zero semigroup and the multiplicative semigroup of positive integers and describe Green’s relations on ${}^{\ast }({\mathit{B}}_{\omega }^{\mathcal{F}})$.