On variants of the extended bicyclic semigroup

Анотація

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