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 $\big(\mathscr{ C}_{ \mathbb{ Z}}^{ 0,0},\tau^*\big)$.
Categories: math.GR math.GN