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

Abstract

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.