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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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_01678094_v27_n2_p213_Minian |
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| Sumario: | 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. |
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