On the topology of a compact inverse Clifford semigroup

Abstract

A description of the topology of a compact inverse Clifford semigroup S is given in terms of the topologies of its subgroups and that of the semilattice X of idempotents. It is further shown that the category of compact inverse Clifford semigroups is equivalent to a full subcategory of the category whose objects are inverse limit preserving functors F : X → G F:X \to G , where X is a compact semilattice and G is the category of compact groups and continuous homomorphisms, and where a morphism from F : X → G F:X \to G to G : Y → G G:Y \to G is a pair ( ε , w ) (\varepsilon ,w) such that ε \varepsilon is a continuous homomorphism of X into Y and w is a natural transformation from F to G ε G\varepsilon . Simpler descriptions of the topology of S are given in case the topology of X is first countable and in case the bonding maps between the maximal subgroups of S are open mappings.