Separating linear expressions in the Stone–Čech compactification of direct sums

Abstract

A finite sequence a→=〈ai〉i=1m in Z∖{0} is compressed provided ai≠ai+1 for i<m. It is known that if a→=〈ai〉i=1m and b→=〈bi〉i=1k are compressed sequences in Z∖{0}, then there exist idempotents p and q in βQd∖{0} such that a1p+a2p+…+amp=b1q+b2q+…+bkq if and only if b→ is a rational multiple of a→. In fact, if b→ is not a rational multiple of a→, then there is a partition of Q∖{0} into two cells, neither of which is a member of a1p+a2p+…+amp and a member of b1q+b2q+…+bkq for any idempotents p and q in βQd∖{0}. (Here βQd is the Stone–Čech compactification of the set of rational numbers with the discrete topology.)In this paper we extend these results to direct sums of Q. As a corollary, we show that if b→ is not a rational multiple of a→ and G is any torsion free commutative group, then there do not exist idempotents p and q in βGd∖{0} such that a1p+a2p+…+amp=b1q+b2q+…+bkq. We also show that for direct sums of finitely many copies of Q we can separate the corresponding Milliken–Taylor systems, with a similar but weaker result for the direct sum of countably many copies of Q.