The order complex of a poset is the SimplicialComplex with vertices corresponding to the ground set of P and faces corresponding to the chains of P.
i1 : orderComplex booleanLattice 3 o1 = | v_0v_4v_6v_7 v_0v_2v_6v_7 v_0v_4v_5v_7 v_0v_1v_5v_7 v_0v_2v_3v_7 v_0v_1v_3v_7 | o1 : SimplicialComplex |
The minimal non-faces are given by the incomparable pairs of vertices in P. Thus the order complex is the independence complex of the incomparabilityGraph of P and the clique complex of the comparabilityGraph of P. Moreover, the facets are given by the maximalChains of P. Thus, the order complex of a chain poset is a simplex.
i2 : orderComplex chain 5 o2 = | v_0v_1v_2v_3v_4 | o2 : SimplicialComplex |
This method renames the vertices with integers 0, 1, ... corresponding to the index of the vertices in the GroundSet.