On the closure of the extended bicyclic semigroup
In the paper, we study the semigroup C_Z which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup C_Z and prove that every non-trivial congruence C on the semigroup C_Z is a group congruence, and moreover the quotient semigroup C_Z /C is isomorphic to a cyclic group. Also we show that the semigroup C_Z as a Hausdorff semitopological semigroup admits only the discrete topology. Next we study the closure cl_T(C_Z ) of the semigroup C_Z in a topological semigroup T. We show that the non-empty remainder of C_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) eq and I eq. In the case when T is a locally compact topological inverse semigroup and I eq we prove that an ideal I is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup C_Z 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. , number=2, journal=Carpathian Mathematical Publications, author=Fihel, I.R. and Gutik, O.V., year=2011, month=Dec., pages=131–157