On the closure of the extended bicyclic semigroup

Abstract

In the paper we study the semigroup CZ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup CZ and prove that every non-trivial congruence C on the semigroup CZ is a group congruence, and moreover the quotient semigroup CZ=C is isomorphic to a cyclic group. Also we show that the semigroup CZ as a Hausdor semitopological semigroup admits only the discrete topology. Next we study the closure clT (CZ) of the semigroup CZ in a topological semigroup T. We show that the non-empty remainder of CZ in a topological inverse semigroup T consists of a group of units H(1T) of T and a two-sided ideal I of T in the case when H(1T) 6 ? and I 6 ?. In the case when T is a locally compact topological inverse semigroup and I 6 ? we prove that an ideal I is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup CZ [ I. Also we show that if the group of units H(1T) of the semigroup T is non-empty, then H(1T) is either singleton or H(1T) is topologically isomorphic to the discrete additive group of integers.