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Partitions of primitive Boolean spaces

Andrew B. Apps

TL;DR

We address the classification of primitive Boolean spaces and their Stone spaces beyond compact PI cases by developing trim $P$-partitions and tying them to extended PO systems. The authors prove a quasi-order isomorphism between well-behaved trim partitions of a primitive space and a class of extended PO systems, and introduce rank partitions as ideal completions of trim partitions. They extend classical results on primitive Boolean algebras and compact primitive spaces to locally compact spaces, including a generalized interpretation of structure diagrams, orbit diagrams, and the prime topological Boolean algebra. The work unifies algebraic and topological characterizations of primitiveness and provides tools for distinguishing spaces via their partition hierarchies.

Abstract

A Boolean ring and its Stone space (Boolean space) are primitive if the ring is disjointly generated by its pseudo-indecomposable (PI) elements. Hanf showed that a primitive PI Boolean algebra can be uniquely defined by a structure diagram. In a previous paper we defined trim $P$-partitions of a Stone space, where $P$ is a PO system (poset with a distinguished subset), and showed how they provide a physical representation within the Stone space of these structure diagrams. In this paper we study the class of trim partitions of a fixed primitive Boolean space, which may not be compact, and show how they can be structured as a quasi-ordered set via an appropriate refinement relation. This refinement relation corresponds to a surjective morphism of the associated PO systems, and we establish a quasi-order isomorphism between the class of well-behaved partitions of a primitive space and a class of extended PO systems. We also define rank partitions, which generalise the rank diagrams introduced by Myers, and the ideal completion of a trim $P$-partition, whose underlying PO system is the ideal completion of $P$, and show that rank partitions are just the ideal completions of trim partitions. In the process, we extend a number of existing results regarding primitive Boolean algebras or compact primitive Boolean spaces to locally compact Boolean spaces.

Partitions of primitive Boolean spaces

TL;DR

We address the classification of primitive Boolean spaces and their Stone spaces beyond compact PI cases by developing trim -partitions and tying them to extended PO systems. The authors prove a quasi-order isomorphism between well-behaved trim partitions of a primitive space and a class of extended PO systems, and introduce rank partitions as ideal completions of trim partitions. They extend classical results on primitive Boolean algebras and compact primitive spaces to locally compact spaces, including a generalized interpretation of structure diagrams, orbit diagrams, and the prime topological Boolean algebra. The work unifies algebraic and topological characterizations of primitiveness and provides tools for distinguishing spaces via their partition hierarchies.

Abstract

A Boolean ring and its Stone space (Boolean space) are primitive if the ring is disjointly generated by its pseudo-indecomposable (PI) elements. Hanf showed that a primitive PI Boolean algebra can be uniquely defined by a structure diagram. In a previous paper we defined trim -partitions of a Stone space, where is a PO system (poset with a distinguished subset), and showed how they provide a physical representation within the Stone space of these structure diagrams. In this paper we study the class of trim partitions of a fixed primitive Boolean space, which may not be compact, and show how they can be structured as a quasi-ordered set via an appropriate refinement relation. This refinement relation corresponds to a surjective morphism of the associated PO systems, and we establish a quasi-order isomorphism between the class of well-behaved partitions of a primitive space and a class of extended PO systems. We also define rank partitions, which generalise the rank diagrams introduced by Myers, and the ideal completion of a trim -partition, whose underlying PO system is the ideal completion of , and show that rank partitions are just the ideal completions of trim partitions. In the process, we extend a number of existing results regarding primitive Boolean algebras or compact primitive Boolean spaces to locally compact Boolean spaces.
Paper Structure (17 sections, 33 theorems, 2 equations)

This paper contains 17 sections, 33 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Trim partitions: definitions and basic properties
  3. Primitive spaces and trim partitions
  4. Morphisms and simple images of PO systems
  5. Regular refinements and consolidations of trim partitions
  6. Structure diagrams and the canonical trim partition of a primitive space
  7. Extended PO systems and regular refinements of a partition
  8. Some classification results
  9. The class of bounded trim partitions of a fixed Boolean space
  10. Complete partitions of primitive spaces
  11. Rank diagrams and rank partitions
  12. Regular extensions of a semi-trim partition
  13. The ideal completion of a trim partition
  14. Images of $\mathop{\mathrm{id}}\nolimits(P)$ and complete extensions of a trim $P$-partition
  15. The prime topological Boolean algebra of a primitive space
  16. ...and 2 more sections

Key Result

Proposition 2.3

Let $P$ be a PO system and $\mathscr{X}=\{X_{p}\mid p\in P\}^{*}$ a trim $P$-partition of the Stone space $W$ of the countable Boolean ring $R$.

Theorems & Definitions (104)

  • Definition 2.1: PO systems
  • Definition 2.2: Partitions
  • Proposition 2.3: Trim partitions
  • Definition 3.1: Boolean rings
  • Remark 3.2
  • Theorem 3.3: Apps Apps-Stone
  • Definition 3.4: Morphisms and congruence relations
  • Remark 3.5
  • Theorem 3.6: Pierce Pierce, Hansoul Hansoul2
  • Definition 3.7
  • ...and 94 more