On endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup

Abstract

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

arXiv comment10 pages, in Ukrainian