On endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup

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.