Increasing sequences of principal left ideals of $\beta \mathbb{Z}$ are finite

Abstract

We show that increasing sequences of principal left ideals of βZ are finite. As a consequence, βZ\Z is a disjoint union of maximal principal left ideals of βZ. Another consequence is that increasing chains of idempotents (p ≤ q ⇔ p + q = q + p = p) in βZ are finite. All these are answers to long-standing open questions.Addition of integers extends to the Stone-Čech compactification βZ of the discrete space Z so that for each a ∈ Z, the left translation βZ ∋ x → a + x ∈ βZ is continuous, and for each q ∈ βZ, the right translation βZ ∋ x → x + q ∈ βZ is continuous.We take the points of βZ to be the ultrafilters on Z, identifying the principal ultrafilters with the points of Z. The topology of βZ has a base consisting of subsets of the form A = {p ∈ βZ : A ∈ p}, where A ⊆ Z, and A is the closure of A. For p, q ∈ βZ, the ultrafilter p + q has a base consisting of subsets of the form x∈A (x + B x ), where A ∈ p and B x ∈ q.The semigroup βZ is interesting both for its own sake and for its applications to Ramsey theory and to topological dynamics. The first application to Ramsey theory was the proof of Hindman's theorem: whenever the set N of positive integers is finitely colored, there is an infinite sequence all of whose sums are monochrome. Such a sequence was constructed using an idempotent of the semigroup Z * = βZ \ Z. An elementary introduction to βZ can be found in [4].Sometime in the 1970's or 1980's M. Rudin was asked by some, now anonymous, analysts whether every point of Z * is a member of a maximal orbit closure under the continuous extension of the shift function n → 1 + n,