Academic Journal
Conjugating trivial automorphisms of \mathcal P(\mathbb N) / \mathrm{fin}.
| Τίτλος: | Conjugating trivial automorphisms of \mathcal P(\mathbb N) / \mathrm{fin}. |
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| Συγγραφείς: | Brian, Will1 (AUTHOR), Farah, Ilijas2 (AUTHOR) |
| Πηγή: | Transactions of the American Mathematical Society. Apr2026, Vol. 379 Issue 4, p2919-2954. 36p. |
| Θεματικοί όροι: | *Automorphisms, *Continuum hypothesis, *First-order logic, *Forcing (Model theory), *Boolean algebra |
| Περίληψη: | A trivial automorphism of the Boolean algebra \mathcal P(\mathbb N) / \mathrm {Fin} is an automorphism induced by the action of some function \mathbb N \rightarrow \mathbb N. The forcing axiom \mathsf {OCA}_{\mathrm {T}} implies all automorphisms are trivial, and therefore two trivial automorphisms are conjugate if and only if they have the same (modulo finite) orbit structure. We show that the Continuum Hypothesis implies that two trivial automorphisms are conjugate if and only if there are no first-order obstructions to their conjugacy and their indices have the same parity, if and only if the given trivial automorphisms are conjugate in some forcing extension of the universe. To each automorphism \alpha of \raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}} we associate the first-order structure \mathfrak {A}_\alpha =(\raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}},\alpha) and compute the existential theories of these structures. These results are applied to resolve a question of Braga, Farah, and Vignati and prove that there are coarse, uniformly locally finite, metric spaces X and Y such that the isomorphism between their uniform Roe coronas is independent of \operatorname {\mathsf {ZFC}}. [ABSTRACT FROM AUTHOR] |
| Βάση Δεδομένων: | Academic Search Index |
| FullText | Text: Availability: 0 CustomLinks: – Url: http://www.jstor.org/openurl?issn=00029947&date=2026&volume=379&issue=4&spage=2919 Name: JSTOR (all content) (s7799221) Category: fullText Text: Icon: https://imageserver.ebscohost.com/branding/jstor/iconJSTOR.gif MouseOverText: Full Text from JSTOR |
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| Header | DbId: asx DbLabel: Academic Search Index An: 192044503 RelevancyScore: 1411 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 1411.07458496094 |
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| Items | – Name: Title Label: Title Group: Ti Data: Conjugating trivial automorphisms of \mathcal P(\mathbb N) / \mathrm{fin}. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Brian%2C+Will%22">Brian, Will</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Farah%2C+Ilijas%22">Farah, Ilijas</searchLink><relatesTo>2</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Transactions+of+the+American+Mathematical+Society%22">Transactions of the American Mathematical Society</searchLink>. Apr2026, Vol. 379 Issue 4, p2919-2954. 36p. – Name: Subject Label: Subject Terms Group: Su Data: *<searchLink fieldCode="DE" term="%22Automorphisms%22">Automorphisms</searchLink><br />*<searchLink fieldCode="DE" term="%22Continuum+hypothesis%22">Continuum hypothesis</searchLink><br />*<searchLink fieldCode="DE" term="%22First-order+logic%22">First-order logic</searchLink><br />*<searchLink fieldCode="DE" term="%22Forcing+%28Model+theory%29%22">Forcing (Model theory)</searchLink><br />*<searchLink fieldCode="DE" term="%22Boolean+algebra%22">Boolean algebra</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: A trivial automorphism of the Boolean algebra \mathcal P(\mathbb N) / \mathrm {Fin} is an automorphism induced by the action of some function \mathbb N \rightarrow \mathbb N. The forcing axiom \mathsf {OCA}_{\mathrm {T}} implies all automorphisms are trivial, and therefore two trivial automorphisms are conjugate if and only if they have the same (modulo finite) orbit structure. We show that the Continuum Hypothesis implies that two trivial automorphisms are conjugate if and only if there are no first-order obstructions to their conjugacy and their indices have the same parity, if and only if the given trivial automorphisms are conjugate in some forcing extension of the universe. To each automorphism \alpha of \raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}} we associate the first-order structure \mathfrak {A}_\alpha =(\raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}},\alpha) and compute the existential theories of these structures. These results are applied to resolve a question of Braga, Farah, and Vignati and prove that there are coarse, uniformly locally finite, metric spaces X and Y such that the isomorphism between their uniform Roe coronas is independent of \operatorname {\mathsf {ZFC}}. [ABSTRACT FROM AUTHOR] |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=asx&AN=192044503 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1090/tran/9573 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 36 StartPage: 2919 Subjects: – SubjectFull: Automorphisms Type: general – SubjectFull: Continuum hypothesis Type: general – SubjectFull: First-order logic Type: general – SubjectFull: Forcing (Model theory) Type: general – SubjectFull: Boolean algebra Type: general Titles: – TitleFull: Conjugating trivial automorphisms of \mathcal P(\mathbb N) / \mathrm{fin}. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Brian, Will – PersonEntity: Name: NameFull: Farah, Ilijas IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Text: Apr2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00029947 Numbering: – Type: volume Value: 379 – Type: issue Value: 4 Titles: – TitleFull: Transactions of the American Mathematical Society Type: main |
| ResultId | 1 |