Elements of finite order in Stone-Čech compactifications

Abstract

LetSbe a free semigroup (on any set of generators). WhenSis given the discrete topology, its Stone-Čech compactification has a natural semigroup structure. We give two results about elementspof finite order inβS. The first is that any continuous homomorphism ofβSinto any compact group must sendpto the identity. The second shows that natural extensions, to elements of finite order, of relationships between idempotents and sequences with distinct finite sums, do not hold.