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On the closure of the extended bicyclic semigroup

In the paper we study the semigroup $\mathscr{C}_{\mathbb{Z}}$ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup $\mathscr{C}_{\mathbb{Z}}$ and prove that every non-trivial congruence $\mathfrak{C}$ on the semigroup $\mathscr{C}_{\mathbb{Z}}$ is a group congruence, and moreover the quotient semigroup $\mathscr{C}_{\mathbb{Z}}/\mathfrak{C}$ is isomorphic to a cyclic group. Also we show that the semigroup $\mathscr{C}_{\mathbb{Z}}$ as a Hausdorff semitopological semigroup admits only the discrete topology. Next we study the closure $\operatorname{cl}_T(\mathscr{C}_{\mathbb{Z}})$ of the semigroup $\mathscr{C}_{\mathbb{Z}}$ in a topological semigroup $T$. We show that the non-empty remainder of $\mathscr{C}_{\mathbb{Z}}$ in a topological inverse semigroup $T$ consists of a group of units $H(1_T)$ of $T$ and a two-sided ideal $I$ of $T$ in the case when $H(1_T)\neq\varnothing$ and $I\neq\varnothing$. In the case when $T$ is a locally compact topological inverse semigroup and $I\neq\varnothing$ we prove that an ideal $I$ is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup $\mathscr{C}_{\mathbb{Z}}\cup I$. Also we show that if the group of units $H(1_T)$ of the semigroup $T$ is non-empty, then $H(1_T)$ is either singleton or $H(1_T)$ is topologically isomorphic to the discrete additive group of integers.