Partitions of primitive Boolean spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915140367024128 |
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| author | Apps, Andrew B. |
| author_facet | Apps, Andrew B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01702 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Partitions of primitive Boolean spaces Apps, Andrew B. Logic General Topology Rings and Algebras 06E15 (primary), 06A06 (secondary) 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. |
| title | Partitions of primitive Boolean spaces |
| topic | Logic General Topology Rings and Algebras 06E15 (primary), 06A06 (secondary) |
| url | https://arxiv.org/abs/2306.01702 |