Brandt Extensions and Primitive Topological Inverse Semigroups

In the paper we study (countably) compact and (absolutely) H-closed primitive topological inverse semigroups. We describe the structure of compact and countably compact primitive topological inverse semigroups and show that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup. In this paper all spaces are Hausdorff. A semigroup is a non-empty set with a binary associative operation. A semigroup S is called inverse if for any x ∈ S there exists a unique y ∈ S such that x · y ·x = x and y ·x · y = y. Such an element y in S is called inverse to x and denoted by x. The map defined on an inverse semigroup S which maps to any element x of S its inverse x is called the inversion. A topological semigroup is a Hausdorff topological space with a jointly continuous semigroup operation. A topological semigroup which is an inverse semigroup is called an inverse topological semigroup. A topological inverse semigroup is an inverse topological semigroup with continuous inversion. A topological group is a topological space with a continuous group operation and an inversion. We observe that the inversion on a topological inverse semigroup is a homeomorphism (see [6, Proposition II.1]). A Hausdorff topology τ on a (inverse) semigroup S is called (inverse) semigroup if (S, τ) is a topological (inverse) semigroup. Further we shall follow the terminology of [2, 3, 7, 17, 20]. If S is a semigroup, then by E(S) we denote the band (the subset of idempotents) of S, and by S [S] we denote the semigroup S with the adjoined unit [zero] (see [17, p. 2]). Also if a semigroup S has zero 0S, then for any A ⊆ S we denote A = A \ {0S}. If Y is a subspace of a topological space X and A ⊆ Y , then by clY (A) we denote the topological closure of A in Y . The set of positive integers is denoted by N. If E is a semilattice, then the semilattice operation on E determines the partial order 6 on E: e 6 f if and only if ef = fe = e. This order is called natural. An element e of a partially ordered set X is called minimal if f 6 e implies f = e for f ∈ X . An idempotent e of a semigroup S without zero (with zero) is called primitive if e is a minimal element in E(S) (in (E(S))). Let S be a semigroup with zero and Iλ be a set of cardinality λ > 1. On the set Bλ(S) = (Iλ × S × Iλ) ∪ {0} we define the semigroup operation as follows (α, a, β) · (γ, b, δ) = { (α, ab, δ), if β = γ; 0, if β 6= γ, and (α, a, β) · 0 = 0 · (α, a, β) = 0 · 0 = 0, for all α, β, γ, δ ∈ Iλ and a, b ∈ S. If S = S 1 then the semigroup Bλ(S) is called the Brandt λ−extension of the semigroup S [9]. Obviously, Date: May 25, 2010. 2000 Mathematics Subject Classification. 22A15, 54G12, 54H10, 54H12.