On variants of the extended bicyclic semigroup

Abstract

In the paper we describe the group $\mathbf{Aut}\left({\mathcal{C}}_{\mathbb{Z}}\right)$ of automorphisms of the extended bicyclic semigroup ${\mathcal{C}}_{\mathbb{Z}}$ and study the variants ${\mathcal{C}}_{\mathbb{Z}}^{m,n}$ of the extended bicycle semigroup ${\mathcal{C}}_{\mathbb{Z}}$, where $m,n\in \mathbb{Z}$. In particular, we prove that $\mathbf{Aut}\left({\mathcal{C}}_{\mathbb{Z}}\right)$ is isomorphic to the additive group of integers, the extended bicyclic semigroup ${\mathcal{C}}_{\mathbb{Z}}$ and every its variant are not finitely generated, and describe the subset of idempotents $E({\mathcal{C}}_{\mathbb{Z}}^{m,n})$ and Green's relations on the semigroup ${\mathcal{C}}_{\mathbb{Z}}^{m,n}$. Also we show that $E({\mathcal{C}}_{\mathbb{Z}}^{m,n})$ is an $\omega$-chain and any two variants of the extended bicyclic semigroup ${\mathcal{C}}_{\mathbb{Z}}$ are isomorphic. At the end we discuss shift-continuous Hausdorff topologies on the variant ${\mathcal{C}}_{\mathbb{Z}}^{0,0}$. In particular, we prove that if $\tau$ is a Hausdorff shift-continuous topology on ${\mathcal{C}}_{\mathbb{Z}}^{0,0}$ then each of inequalities $a>0$ or $b>0$ implies that $(a,b)$ is an isolated point of $({\mathcal{C}}_{\mathbb{Z}}^{0,0},\tau )$ and construct an example of a Hausdorff semigroup topology ${\tau }^{\ast }$ on the semigroup ${\mathcal{C}}_{\mathbb{Z}}^{0,0}$ such that all its points with $ab⩽0$ and $a+b⩽0$ are not isolated in $({\mathcal{C}}_{\mathbb{Z}}^{0,0},{\tau }^{\ast })$.