Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariants for metrisable locally compact Boolean spaces

Andrew B. Apps

TL;DR

The paper expands Pierce’s invariants to locally compact metrisable Boolean spaces by introducing the compact rank $\rho(W)$ and showing that the tuple $[(K(W),r_W),\nu(W),\rho(W),n(W)]$ uniquely determines an $\omega$-Stone space up to homeomorphism. It then develops a robust framework linking primitive $\omega$-Stone spaces to extended PO systems $(P,L,f)$ via trim partitions, demonstrating that many invariants can be recovered from the PO-system Cantor–Bendixson structure and illustrating how primitivity corresponds to additive-measure self-similarity. The work further characterizes strongly uniform spaces through rank-based measures $\sigma_r$ and establishes criteria for primitivity in terms of $\sigma$-PI/$\sigma$-primitive conditions, while also providing a method to construct non-primitive spaces using incompatible measures. Throughout, the results unify Cantor–Bendixson invariants, PO-system structure, and measure-theoretic notions to give a comprehensive description of metrisable Boolean spaces and their primitivity properties, with explicit decomposition and uniqueness results for both primitive and non-primitive cases.

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.

Invariants for metrisable locally compact Boolean spaces

TL;DR

The paper expands Pierce’s invariants to locally compact metrisable Boolean spaces by introducing the compact rank and showing that the tuple uniquely determines an -Stone space up to homeomorphism. It then develops a robust framework linking primitive -Stone spaces to extended PO systems via trim partitions, demonstrating that many invariants can be recovered from the PO-system Cantor–Bendixson structure and illustrating how primitivity corresponds to additive-measure self-similarity. The work further characterizes strongly uniform spaces through rank-based measures and establishes criteria for primitivity in terms of -PI/-primitive conditions, while also providing a method to construct non-primitive spaces using incompatible measures. Throughout, the results unify Cantor–Bendixson invariants, PO-system structure, and measure-theoretic notions to give a comprehensive description of metrisable Boolean spaces and their primitivity properties, with explicit decomposition and uniqueness results for both primitive and non-primitive cases.

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.

Paper Structure

This paper contains 21 sections, 44 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Invariants for $\omega$-Stone spaces
  3. Scattered $\omega$-Stone spaces
  4. Structure results for general $\omega$-Stone spaces
  5. Primitive spaces, trim partitions and their invariants
  6. Primitive Boolean rings and spaces
  7. Trim partitions
  8. Trim partitions of scattered spaces
  9. Trim partition preliminaries
  10. PO system invariants
  11. Primitive space invariants matched to those of an underlying extended PO system
  12. Uniqueness results for trim $(P,L,f)$-partitions
  13. Uniform $\omega$-Stone spaces and primitivity
  14. Non-primitive $\omega$-Stone spaces: bad points
  15. A construction method for Boolean spaces with bad points
  16. ...and 6 more sections

Key Result

theorem 1

Suppose $\sim$ is a relation between the elements of countable Boolean algebras $R$ and $S$ such that: (i) $1_{R}\sim1_{S}$; (ii) if $C\sim0_{S}$, then $C=0_{R}$; and vice versa; (iii) if $C\sim D_{1}\dotplus D_{2}$ ($D_{1},D_{2}\in S$), then we can write $C=C_{1}\dotplus C_{2}$, with $C_{i}\in R$ a

Figures (1)

  • Figure 1: Structure of a typical Boolean space $W$

Theorems & Definitions (111)

  • theorem 1: Vaught
  • definition 1
  • remark 1
  • proposition 1
  • proof
  • corollary 1
  • proof
  • definition 2
  • remark 2
  • theorem 2: Flum, Martinez
  • ...and 101 more