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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 of the weakly almost periodic compactification w[open face N] of the semigroup ([open face N], +) of positive integers has prompted us to see how successful they would be in another context. We consider wG, where G=[oplus B: plus sign in circle]i[set membership]ωGi is the direct sum of a sequence of finite groups with its discrete topology. 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 subsemigroup of wG contains only one idempotent. We also prove that the set of idempotents in wG is not closed. The minimal idempotent of wG can be written as the sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of β[open face N], we find subsemigroups SF of wG which are ‘carried’ by closed subsets F of βω. wG contains the free abelian product of the semigroups SF corresponding to families of disjoint subsets F. Usually SF is a very large semigroup, but for some points p[set membership]βω, Sp can be quite small.