On Rayner rngs of formal power series

Detalhes bibliográficos
Autor(a) principal: Machado, Geovani Pereira
Data de Publicação: 2023
Tipo de documento: Tese
Idioma: eng
Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-28022023-170129/
Resumo: Resumo Rayner rngs are rngs (rings without unity) whose elements are formal power series whose coefficients lie in a rng and the exponents lie in an additive ordered group, such that the supports of these series belong to a predetermined ideal constrained by a set of axioms. The work presents an inspection of the interplay between the algebraic, topological and categorical properties of the Rayner rngs, the rngs of coefficients and the ordered groups of exponents, studying the Rayner rngs under varied theoretical perspectives and seeking universal relations between them. Two key topologies on these structures are systematically analysed, the so-called weak and strong topologies, and a version of the Intermediate Value Theorem is obtained for the weak topology. Special attention is given to rngs of Levi-Civita, Puiseux and Hahn series, which are prominent instances of Rayner rngs.
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spelling On Rayner rngs of formal power seriesSobre os rngs de Rayner de séries de potências formaisFormal power seriesHahn seriesIntermediate Value TheoremLevi-Civita seriesPuiseux seriesRayner seriesSéries de HahnSéries de Levi-CivitaSéries de potências formaisSéries de PuiseuxSéries de RaynerTeorema do Valor IntermediárioResumo Rayner rngs are rngs (rings without unity) whose elements are formal power series whose coefficients lie in a rng and the exponents lie in an additive ordered group, such that the supports of these series belong to a predetermined ideal constrained by a set of axioms. The work presents an inspection of the interplay between the algebraic, topological and categorical properties of the Rayner rngs, the rngs of coefficients and the ordered groups of exponents, studying the Rayner rngs under varied theoretical perspectives and seeking universal relations between them. Two key topologies on these structures are systematically analysed, the so-called weak and strong topologies, and a version of the Intermediate Value Theorem is obtained for the weak topology. Special attention is given to rngs of Levi-Civita, Puiseux and Hahn series, which are prominent instances of Rayner rngs.Os rngs de Rayner são rngs (anéis sem unidade) cujos elementos são séries formais de potências cujos coeficientes pertencem a um rng e os expoentes pertencem a um grupo ordenado aditivo tais que os suportes dessas series pertencem a um predeterminado ideal que satisfaz um conjunto de axiomas. O trabalho apresenta uma inspeção das relações diretas entre as propriedades algébricas, topológicas e categóricas dos rngs de Rayner, dos rngs de coeficientes e dos grupos ordenados de expoentes, estudando os rngs de Rayner sob diferentes perspectivas teóricas e buscando relações universais entre eles. Duas topologias essenciais nessas estruturas são sistematicamente analisadas, as topologias forte e fraca, e uma versão do Teorema do Valor Intermediário é obtida para a topologia fraca. Atenção especial é dada aos rngs de séries de Levi-Civita, Puiseux e Hahn, os quais são instâncias proeminentes de rngs de Rayner.Biblioteca Digitais de Teses e Dissertações da USPFajardo, Rogerio Augusto dos SantosMachado, Geovani Pereira2023-01-09info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/45/45131/tde-28022023-170129/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2023-03-02T14:38:47Zoai:teses.usp.br:tde-28022023-170129Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212023-03-02T14:38:47Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false
dc.title.none.fl_str_mv On Rayner rngs of formal power series
Sobre os rngs de Rayner de séries de potências formais
title On Rayner rngs of formal power series
spellingShingle On Rayner rngs of formal power series
Machado, Geovani Pereira
Formal power series
Hahn series
Intermediate Value Theorem
Levi-Civita series
Puiseux series
Rayner series
Séries de Hahn
Séries de Levi-Civita
Séries de potências formais
Séries de Puiseux
Séries de Rayner
Teorema do Valor Intermediário
title_short On Rayner rngs of formal power series
title_full On Rayner rngs of formal power series
title_fullStr On Rayner rngs of formal power series
title_full_unstemmed On Rayner rngs of formal power series
title_sort On Rayner rngs of formal power series
author Machado, Geovani Pereira
author_facet Machado, Geovani Pereira
author_role author
dc.contributor.none.fl_str_mv Fajardo, Rogerio Augusto dos Santos
dc.contributor.author.fl_str_mv Machado, Geovani Pereira
dc.subject.por.fl_str_mv Formal power series
Hahn series
Intermediate Value Theorem
Levi-Civita series
Puiseux series
Rayner series
Séries de Hahn
Séries de Levi-Civita
Séries de potências formais
Séries de Puiseux
Séries de Rayner
Teorema do Valor Intermediário
topic Formal power series
Hahn series
Intermediate Value Theorem
Levi-Civita series
Puiseux series
Rayner series
Séries de Hahn
Séries de Levi-Civita
Séries de potências formais
Séries de Puiseux
Séries de Rayner
Teorema do Valor Intermediário
description Resumo Rayner rngs are rngs (rings without unity) whose elements are formal power series whose coefficients lie in a rng and the exponents lie in an additive ordered group, such that the supports of these series belong to a predetermined ideal constrained by a set of axioms. The work presents an inspection of the interplay between the algebraic, topological and categorical properties of the Rayner rngs, the rngs of coefficients and the ordered groups of exponents, studying the Rayner rngs under varied theoretical perspectives and seeking universal relations between them. Two key topologies on these structures are systematically analysed, the so-called weak and strong topologies, and a version of the Intermediate Value Theorem is obtained for the weak topology. Special attention is given to rngs of Levi-Civita, Puiseux and Hahn series, which are prominent instances of Rayner rngs.
publishDate 2023
dc.date.none.fl_str_mv 2023-01-09
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/doctoralThesis
format doctoralThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv https://www.teses.usp.br/teses/disponiveis/45/45131/tde-28022023-170129/
url https://www.teses.usp.br/teses/disponiveis/45/45131/tde-28022023-170129/
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv
dc.rights.driver.fl_str_mv Liberar o conteúdo para acesso público.
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Liberar o conteúdo para acesso público.
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.coverage.none.fl_str_mv
dc.publisher.none.fl_str_mv Biblioteca Digitais de Teses e Dissertações da USP
publisher.none.fl_str_mv Biblioteca Digitais de Teses e Dissertações da USP
dc.source.none.fl_str_mv
reponame:Biblioteca Digital de Teses e Dissertações da USP
instname:Universidade de São Paulo (USP)
instacron:USP
instname_str Universidade de São Paulo (USP)
instacron_str USP
institution USP
reponame_str Biblioteca Digital de Teses e Dissertações da USP
collection Biblioteca Digital de Teses e Dissertações da USP
repository.name.fl_str_mv Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)
repository.mail.fl_str_mv virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br
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