On the closure of the extended bicyclic semigroup

Abstract

In the paper we study the semigroup $mathscr{C}_{mathbb{Z}}$which is a generalization of the bicyclic semigroup. We describemain algebraic properties of the semigroup$mathscr{C}_{mathbb{Z}}$ and prove that every non-trivialcongruence $mathfrak{C}$ on the semigroup$mathscr{C}_{mathbb{Z}}$ is a group congruence, and moreover thequotient semigroup $mathscr{C}_{mathbb{Z}}/mathfrak{C}$ isisomorphic to a cyclic group. Also we show that the semigroup$mathscr{C}_{mathbb{Z}}$ as a Hausdorff semitopological semigroupadmits only the discrete topology. Next we study the closure$operatorname{cl}_Tleft(mathscr{C}_{mathbb{Z}}ight)$ of thesemigroup $mathscr{C}_{mathbb{Z}}$ 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$H(1_T)eqvarnothing$ and $Ieqvarnothing$. In the case when $T$is a locally compact topological inverse semigroup and$Ieqvarnothing$ we prove that an ideal $I$ is topologicallyisomorphic to the discrete additive group of integers and describethe 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 $H(1_T)$ istopologically isomorphic to the discrete additive group ofintegers.