Embedding the Bicyclic Semigroup into Countably Compact Topological Semigroups

We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p, q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p, q). On the other hand, we construct a consistent example of a Tychonov countably compact semigroup containing a copy of C(p, q). In this paper we study the structural properties of topological semigroups that contain a copy of the bicyclic semigroup C(p, q) and present a consistent example of Tychonov countably compact semigroup S that contains C(p, q). This example shows that the theorem of Koch and Wallace [11] saying that compact topological semigroups do not contain the bicyclic semigroup cannot be generalized to the class of countably compact topological semigroups. The presence or absence of bicyclic subsemigroups in a given (topological) semigroup S has important implications for understanding the algebraic (and topological) structure of S. For example, the well-known Andersen Theorem [3, 2.54] and [1] says that a simple semigroup with an idempotent but without a copy of C(p, q) is completely simple and hence by the Rees-Suschkewitsch Theorem [15], has the structure of a sandwich product [X,H, Y ]σ of two sets X,Y and a group H connected by a suitable sandwich function σ : Y ×X → H . Having in mind the mentioned result of Koch and Wallace [11], I.I. Guran asked is the bicyclic semigroup can be embedded into a countably compact topological semigroup. In this paper we shall find many conditions on a topological semigroup S which forbid S to contain a bicyclic subsemigroup. One of the simplest conditions is the countable compactness of the square S × S. On the other hand, assuming the existence of a Tkachenko-Tomita group (which is a countably compact abelian torsion-free topological group without convergent sequences) we shall construct an example of a Tychonov countably compact topological semigroup that contains a copy of the bicyclic semigroup. To construct such an example we shall study the operation of attaching a discrete semigroup D to a topological semigroup X along a homomorphism π : D → X . This construction has two ingredients: topological and algebraic, discussed in the next four sections. In section 5 we establish some structure properties of topological semigroups that contain a copy of the bicyclic subsemigroup and in Section 6 we construct our main counterexample. Our method of constructing this counterexample is rather standard and exploits the ideas of D.Robbie, S.Svetlichny [17] (who constructed a countably compact cancellative semigroup under CH) and A.Tomita 2000 Mathematics Subject Classification. 22A15, 54C25, 54D35, 54H15.