Invariants for metrisable locally compact Boolean spaces

TL;DR

This paper extends invariants used to classify metrisable Boolean spaces from compact to locally compact cases, introduces the compact rank invariant, and explores the structure of primitive Boolean spaces via PO systems and measures.

Contribution

It introduces the compact rank invariant for locally compact spaces and links primitivity of Boolean spaces to measure self-similarity, advancing classification methods.

Findings

Four invariants determine a locally compact metrisable Boolean space up to homeomorphism.

Most invariant information for primitive spaces can be derived from associated PO systems.

A method for constructing non-primitive spaces based on measure properties is developed.

Abstract

Pierce identified 3 invariants of a compact metrisable Boolean space, derived from its Cantor-Bendixson sequence, that determine the space up to homeomorphism. For locally compact spaces we define an additional invariant, the compact rank, and show that these 4 invariants determine a locally compact metrisable Boolean space up to homeomorphism. We also identify which combinations of the 4 invariants can arise in practice. A Boolean ring and its associated Boolean space are primitive if the ring is disjointly generated by its pseudo-indecomposable (PI) elements. Spaces in this important sub-class of Boolean spaces can be well described (uniquely in the case of compact spaces) by an extended PO system (poset with a distinguished subset). We define the Cantor-Bendixson sequence and associated invariants for a PO system, and show that almost all of the invariant information for a primitive space can be recovered from that of an associated extended PO system. We also show how the primitivity of a Boolean space corresponds to a notion of primitivity of the additive measure associated with the rank function of a space, which in turn depends on the additive measure being sufficiently 'self-similar'. We use these ideas to develop a method for constructing non-primitive spaces.