On Countably Compact 0-simple Topological Inverse Semigroups

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups. We follow the terminology of [3, 4, 8]. In this paper all topological spaces are Hausdorff. If S is a semigroup then we denote the subset of idempotents of S by E(S). A topological space S that is algebraically a semigroup with a continuous semigroup operation is called a topological semigroup. A topological inverse semigroup is a topological semigroup S that is algebraically an inverse semigroup with continuous inversion. If Y is a subspace of a topological space X and A⊆Y , then we denote by clY (A) the topological closure of A in Y . The bicyclic semigroup C (p, q) is the semigroup with the identity 1 generated by two elements p and q, subject only to the condition pq=1. The bicyclic semigroup plays an important role in the algebraic theory of semigroups and in the theory of topological semigroups. For example, the well-known Andersen’s result [1] states that a (0–) simple semigroup is completely (0–) simple if and only if it does not contain the bicyclic semigroup. The bicyclic semigroup admits only the discrete topology and a topological semigroup S can contain C (p, q) only as an open subset [7]. Neither stable nor Γ-compact topological semigroups can contain a copy of the bicyclic semigroup [2, 12]. Let S be a semigroup and Iλ a non-empty set of cardinality λ. We define the semigroup operation ′ · ′ on the set Bλ(S)= Iλ×S ×Iλ∪{0} as follows (α, a, β)·(γ, b, δ) = { (α, ab, δ), if β = γ,