The geometry of relations
| dc.creator | Minian, Elias Gabriel | |
| dc.date | 2007-02-07 | |
| dc.date.accessioned | 2026-07-07T07:45:22Z | |
| dc.date.available | 2026-07-07T07:45:22Z | |
| dc.description | There is a canonical way to associate two simplicial complexes K, L to any relation $R\subset X\times Y$. Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker. In this article we prove a Galois-type correspondence for relations $R\subset X\times Y$ when X is fixed and use these constructions to investigate finite posets (or equivalently, finite topological spaces) from a geometrical point of view. Given any poset $(X, \leq)$, we define the simplicial complexes K, L associated to the relation $\leq$. In many cases these polyhedra have the same homotopy type as the standard simplicial complex C of nonempty finite chains in 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 and L are geometrically equivalent to the smaller complexes K',L' induced by the relation <. More precisely, we prove that K (resp. L) simplicially collapses to K' (resp. L'). | |
| dc.description | 10 pages | |
| dc.identifier | https://arxiv.org/abs/math/0702197 | |
| dc.identifier | http://arxiv.org/abs/math/0702197 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123506 | |
| dc.subject | Combinatorics | |
| dc.subject | Geometric Topology | |
| dc.subject | 06A06; 06A11; 06A15; 55U05; 55U10; 57Q10 | |
| dc.title | The geometry of relations | |
| dc.type | text |