Abstract:
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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Citas:
---------- APA ----------
(2010)
. The Geometry of Relations. Order, 27(2), 213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4---------- CHICAGO ----------
Minian, E.G.
"The Geometry of Relations"
. Order 27, no. 2
(2010) : 213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4---------- MLA ----------
Minian, E.G.
"The Geometry of Relations"
. Order, vol. 27, no. 2, 2010, pp. 213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4---------- VANCOUVER ----------
Minian, E.G. The Geometry of Relations. Order. 2010;27(2):213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4