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

Abstract

We study the semigroup of non-injective monoid endomorphisms of the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ with a two-elements family $\mathcal{F}$ of inductive nonempty subsets of $\omega$. We describe the structure of elements of the semigroup ${\mathbf{End}}_0^{\ast }({\mathbf{B}}_{\omega }^{\mathcal{F}})$ of non-injective monoid endomorphisms of the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$. In particular we show that its subsemigroup ${\mathbf{End}}^{\ast }({\mathbf{B}}_{\omega }^{\mathcal{F}})$ of non-injective non-annihilating monoid endomorphisms of the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ 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 ${\mathbf{End}}^{\ast }({\mathbf{B}}_{\omega }^{\mathcal{F}})$.

11 pages

Cited by in corpus (1)

• On injective endomorphisms of the semigroup ${\mathbf{B}}_{\mathbb{Z}}^{{\mathcal{F}}^2}$ with the two-element family ${\mathcal{F}}^2$ of inductive nonempty subsets of $\omega$