Stone space partitions indexed by a poset

Abstract

Stone space partitions \(\{X_{p}\mid p\in P\}\) satisfying conditions like \(\overline{X_{p}}=\bigcup _{q\leqslant p}X_{q}\) for all \(p\in P\), where P is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of \(\omega \)-categorical structures. A key concept for studying such partitions is that of a p-trim open set which meets precisely those \(X_{q}\) for which \(q\geqslant p\); for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition \(\{X_{r}\mid r\in [0,1]\}\) of the Cantor set such that \(\overline{X_{r}}=\bigcup _{s\leqslant r}X_{s}\text { for all }r\in [0,1]\).