The weakly almost periodic compactification of a direct sum of finite groups

The success of the methods of [24] and [4] in investigating the structure o f the weakly almost periodic compactification wN of the semigroup (N, +) of positive integers has prompted us to see how successful they would be in a nother context. We consider wG, where G = circle plus(i is an element of w) G(i) is the direct sum of a sequence of finite groups with its discrete to pology. We discover a large class of weakly almost periodic functions on G, and we use them to prove the existence of a large number of long chains of idempotents in wG. However, the closure of any singly generated subsemigro up of wG contains only one idempotent. We also prove that the set of idempo tents in wG is not closed. The minimal idempotent of wG can be written as t he sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of beta N, we fin d subsemigroups S-F of wG which are 'carried' by closed subsets F of beta w . wG contains the free abelian product of the semigroups SF corresponding t o families of disjoint subsets F. Usually SF is a very large semigroup, but for some points p is an element of beta w, S-p can be quite small.