The Geometry of Relations

The classical way to study a finite poset (X, ≤) using topology is by means of the simplicial complex Δ X of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by...

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Detalles Bibliográficos
Autor principal: Minian, Elias Gabriel
Publicado: 2010
Materias:
Collapses
Finite spaces
Nerves
Posets
Relations
Simplicial complexes
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01678094_v27_n2_p213_Minian
http://hdl.handle.net/20.500.12110/paper_01678094_v27_n2_p213_Minian
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_01678094_v27_n2_p213_Minian
record_format dspace
spelling paper:paper_01678094_v27_n2_p213_Minian2023-06-08T15:16:58Z The Geometry of Relations Minian, Elias Gabriel Collapses Finite spaces Nerves Posets Relations Simplicial complexes The classical way to study a finite poset (X, ≤) using topology is by means of the simplicial complex Δ X of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84-95, 1952), we define the simplicial complexes K X and L X associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex Δ X . We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K X and L X are topologically equivalent to the smaller complexes K′ X , L′ X induced by the relation ≤. More precisely, we prove that K X (resp. L X ) simplicially collapses to K′ X (resp. L′ X ). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y. © 2010 Springer Science+Business Media B.V. Fil:Minian, E.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2010 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01678094_v27_n2_p213_Minian http://hdl.handle.net/20.500.12110/paper_01678094_v27_n2_p213_Minian
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Collapses
Finite spaces
Nerves
Posets
Relations
Simplicial complexes
spellingShingle Collapses
Finite spaces
Nerves
Posets
Relations
Simplicial complexes
Minian, Elias Gabriel
The Geometry of Relations
topic_facet Collapses
Finite spaces
Nerves
Posets
Relations
Simplicial complexes
description The classical way to study a finite poset (X, ≤) using topology is by means of the simplicial complex Δ X of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84-95, 1952), we define the simplicial complexes K X and L X associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex Δ X . We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K X and L X are topologically equivalent to the smaller complexes K′ X , L′ X induced by the relation ≤. More precisely, we prove that K X (resp. L X ) simplicially collapses to K′ X (resp. L′ X ). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y. © 2010 Springer Science+Business Media B.V.
author Minian, Elias Gabriel
author_facet Minian, Elias Gabriel
author_sort Minian, Elias Gabriel
title The Geometry of Relations
title_short The Geometry of Relations
title_full The Geometry of Relations
title_fullStr The Geometry of Relations
title_full_unstemmed The Geometry of Relations
title_sort geometry of relations
publishDate 2010
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01678094_v27_n2_p213_Minian
http://hdl.handle.net/20.500.12110/paper_01678094_v27_n2_p213_Minian
work_keys_str_mv AT minianeliasgabriel thegeometryofrelations
AT minianeliasgabriel geometryofrelations
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