On the semigroup of endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $\omega$

TL;DR

This paper investigates the structure of the semigroup of all endomorphisms of a specific bicyclic extension related to inductive nonempty subsets of omega, revealing a unique factorization property within its submonoid.

Contribution

It introduces a detailed analysis of the endomorphism semigroup of the bicyclic extension with a two-element family, including the construction of a submonoid with unique factorization properties.

Findings

Identification of the semigroup structure of endomorphisms.

Construction of a submonoid with unique representation.

Proof of the unique factorization property within the submonoid.

Abstract

We study the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^2})$ of all endomorphisms of the bicyclic extension $\boldsymbol{B}_{\omega}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $\omega$. The submonoid $\left\langle\varpi\right\rangle^1$ of $\overline{\boldsymbol{End}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^2})$ with the property that every element of the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^2})$ has the unique representation as the product of the monoid endomorphism of $\boldsymbol{B}_{\omega}^{\mathscr{F}^2}$ and the element of $\left\langle\varpi\right\rangle^1$ is constructed.