Connected extensions of simple semigroups
Karl Heinrich Hofmann, Paul S. Mostert · Czechoslovak Mathematical Journal · 1965
In the study of compact topological semigroups, one learns early that the minimal ideal is a Rees product of a compact group G and a pair of spaces X and Y [2].Under many conditions, this reduces to a direct product, and indeed no examples seem to be in the literature of compact connected semigroups with identity in which the minimal ideal failed to be a direct product.CLIFFORD [1] gave general construction techniques to illustrate all algebraic extensions of a given Rees product (or completely simple semigroup).We have followed his technique to develop methods of finding many examples of compact connected semigroups with non-trivial Rees products (i.e.not direct products) as minimal ideal.In fact we have the following result:Every compact connected semigroup S with identity can be embedded in a compact connected semigroup T with identity whose minimal ideal N is a non-trivial Rees product in such a way that T\S c: N, Insofar as examples are concerned, it is possible in many ways to construct from any given semigroup, another with a non-trivial Rees product for its minimal ideal and such that the Rees quotients of the two semigroups are isomorphic.One will also observe that our techniques are applicable to far wider classes of topological semigroups than compact connected ones.