The Rees-suschkewitsch Theorem for Simple Topological Semigroups

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product [X × H × Y ]σ of a topological group H over topological spaces X, Y with a continuous sandwich function σ : Y × X → H. We prove that a simple topological semigroup S is a topological paragroup if one of the following conditions is satisfied: (1) S is completely simple and the maximal subgroups of S are topological groups, (2) S contains an idempotent and the square S ×S is countably compact or pseudocompact, (3) S is sequentially compact or the power S c is countably compact. The last item generalizes an old Wallace’s result saying that each simple compact topological semigroup is a topological paragroup. This paper was motivated by the classical Rees-Suschkewitsch Theorem that describes the algebraic structure of completely simple semigroups and the topological versions of this theorem proved for compact topological semigroups by Wallace [28], for compact semitopological semigroups by Ruppert [25], and for sequential countably compact topological semigroups by Gutik, Pagon and Repovš [14]. All topological semigroups considered in this paper are Hausdorff. We recall that a semigroup S is simple if S contains no proper two-sided ideal. A simple semigroup S is called completely simple if the set E = {e ∈ S : ee = e} of idempotents of S contains a primitive idempotent, that is, a minimal idempotent with respect to the partial order e ≤ f on E defined by ef = fe = e. In this case all the idempotents are primitive and He = eSe is a group for every e ∈ E, see [7, Section 2.7, Ex. 6(b)] or [29]. Let us observe that each group is a completely simple semigroup. Less trivial examples of such semigroups appear as minimal ideals in compact right-topological semigroups, see [25, Theorem I.3.13], [16, Theorem 2.9] and [24]. A generic example of a completely simple semigroup can be constructed as follows. Take any group H and a function σ : Y ×X → H defined on the product of two sets. This function σ induces the semigroup operation (x, h, y) · (x, h, y) = (x, hσ(y, x)h, y) on the product X ×H × Y turning it into a completely simple semigroup, called the Rees product of H over X and Y relative to the sandwich map σ [17] or a paragroup [25] and denoted by [X,H, Y ]σ. The Rees-Suschkewitsch Structure Theorem [22] says that the converse is also true: each completely simple semigroup S is isomorphic to the paragroup [Xe, He, Ye]σ where e is any idempotent of S, He = eSe is the maximal subgroup of S containing Date: February 24, 2009. 2000 Mathematics Subject Classification. Primary 22A15, 20M20. Secondary 20M18, 54H15.