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.
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.