Some examples of quantifier elimination and o-minimality

Detalhes bibliográficos
Autor(a) principal: Schimura, Ricardo Mateus
Data de Publicação: 2021
Tipo de documento: Dissertação
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10451/51256
Resumo: Tese de mestrado, Matemática, Universidade de Lisboa, Faculdade de ciências, 2021
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spelling Some examples of quantifier elimination and o-minimalityo-minimalidadeeliminação de quantificadoresaxiomatização universalcorpos reais fechados ordenadoscorpos analíticos restritosTeses de mestrado - 2021Departamento de MatemáticaTese de mestrado, Matemática, Universidade de Lisboa, Faculdade de ciências, 2021A structure with a total order that is dense without end-points is o-minimal if every definable set in dimension 1is a finite union of intervals and points. This notion materialized from observations that many of the proprieties of semialgebraic sets were deduced from very simple axioms, the ones that now define o-minimal structures. Indeed, o-minimality establishes strong regularity results of the definable sets. In this way, o-minimality can be viewed as a candidate to “topologie mod´er´ee” mentioned by Grothendieck in his Esquisse d’un programme. In the context of this dissertation, despite of its intrinsic richness, we study the property of quantifier elimination (abbreviated QE) as a way of proving o-minimality of a given structure. The goal of this dissertation was to study proofs of o-minimality and QE by studying a concrete example, the real closed ordered fields (abbreviated rcof). In Chapter 1, we begin by defining basic notions of first-order logic. We present some examples that will be useful later, such as the theory of rcof. We alude to the usefulness of different axiomatizations, such as the universal axiomatization, and simplifications of formulas, such as QE, that make the theories much more easier to understand. We present some criterias for a theory to admit QE. We present a geometrical perspective of the definable sets in general and the special case of o-minimality. In Chapter 2 we prove that the theory of rcof has QE. As a consequence we prove that every rcof is o-minimal. In Chapter 3 we study proprieties of o-minimal structures. In Chapter 4 we study the theory Tan of rcof with restricted analytic functions. We state that Tan has QE in the language Lan(1) and as a consequence we show that Tan admits an universal axiomatization in the language Lan(1; ( np)n=2;3;:::). In Chapter 5 we establish the result that every model of Tan can be seen as a substructure of a power series field R((t)). We use this fact to deduce key results concerning valuations on these structures and use these to prove that Tan(exp) admits QE and universal axiomatization, both in the language Lan(exp; log). In the last section of this chapter we begin by noting that, provided the theory admits QE, o-minimality is equivalent to regularity of the signal (whether it is greater, less or equal to zero) “at infinity” of the definable functions in one variable. This leads us to consider Hardy fields and using properties from these fields we prove that Tan(exp) is o-minimal.Edmundo, Mário JorgeRepositório da Universidade de LisboaSchimura, Ricardo Mateus2022-02-14T09:42:25Z202120212021-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttp://hdl.handle.net/10451/51256TID:202933350enginfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-08T16:55:50Zoai:repositorio.ul.pt:10451/51256Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T22:02:34.639587Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Some examples of quantifier elimination and o-minimality
title Some examples of quantifier elimination and o-minimality
spellingShingle Some examples of quantifier elimination and o-minimality
Schimura, Ricardo Mateus
o-minimalidade
eliminação de quantificadores
axiomatização universal
corpos reais fechados ordenados
corpos analíticos restritos
Teses de mestrado - 2021
Departamento de Matemática
title_short Some examples of quantifier elimination and o-minimality
title_full Some examples of quantifier elimination and o-minimality
title_fullStr Some examples of quantifier elimination and o-minimality
title_full_unstemmed Some examples of quantifier elimination and o-minimality
title_sort Some examples of quantifier elimination and o-minimality
author Schimura, Ricardo Mateus
author_facet Schimura, Ricardo Mateus
author_role author
dc.contributor.none.fl_str_mv Edmundo, Mário Jorge
Repositório da Universidade de Lisboa
dc.contributor.author.fl_str_mv Schimura, Ricardo Mateus
dc.subject.por.fl_str_mv o-minimalidade
eliminação de quantificadores
axiomatização universal
corpos reais fechados ordenados
corpos analíticos restritos
Teses de mestrado - 2021
Departamento de Matemática
topic o-minimalidade
eliminação de quantificadores
axiomatização universal
corpos reais fechados ordenados
corpos analíticos restritos
Teses de mestrado - 2021
Departamento de Matemática
description Tese de mestrado, Matemática, Universidade de Lisboa, Faculdade de ciências, 2021
publishDate 2021
dc.date.none.fl_str_mv 2021
2021
2021-01-01T00:00:00Z
2022-02-14T09:42:25Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10451/51256
TID:202933350
url http://hdl.handle.net/10451/51256
identifier_str_mv TID:202933350
dc.language.iso.fl_str_mv eng
language eng
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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