Invariants for metrisable locally compact Boolean spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910838901702656 |
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| author | Apps, Andrew B. |
| author_facet | Apps, Andrew B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15492 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariants for metrisable locally compact Boolean spaces Apps, Andrew B. Logic General Topology Rings and Algebras 06E15, 06E05, 06A06 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. |
| title | Invariants for metrisable locally compact Boolean spaces |
| topic | Logic General Topology Rings and Algebras 06E15, 06E05, 06A06 |
| url | https://arxiv.org/abs/2502.15492 |