On endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup

TL;DR

This paper characterizes the structure of endomorphism semigroups of the bicyclic and extended bicyclic semigroups, showing they are isomorphic to specific semidirect products, thus advancing understanding of their algebraic properties.

Contribution

It provides a novel algebraic description of endomorphism semigroups of these semigroups as semidirect products, revealing their structural isomorphisms.

Findings

Endomorphism semigroup of the bicyclic semigroup is isomorphic to a specific semidirect product.

Endomorphism semigroup of the extended bicyclic semigroup is isomorphic to a different semidirect product.

The structural isomorphisms clarify the algebraic nature of these semigroups.

Abstract

It is proved that the semigroups $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\omega})$ and $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\mathbb{Z}})$ of the endomorphisms of the bicyclic semigroup $\boldsymbol{B}_{\omega}$ and the endomorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ are isomorphic to the semidirect products $(\omega,+)\rtimes_\varphi(\omega,*)$ and $\mathbb{Z}(+)\rtimes_\varphi(\omega,*)$, respectively.