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.
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E.G. Minian’s research is partially supported by Conicet and by grant ANPCyT PICT 17-38280.
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Minian, E.G. The Geometry of Relations. Order 27, 213–224 (2010). https://doi.org/10.1007/s11083-010-9146-4
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DOI: https://doi.org/10.1007/s11083-010-9146-4