On endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup

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

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