On variants of the extended bicyclic semigroup

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