orderComplex -- produces the order complex of a poset

Synopsis

Usage:

O = orderComplex P
Inputs:
- P, an instance of the type Poset,
Optional inputs:
- VariableName => a symbol, default value v
- CoefficientRing => a ring, default value QQ
Outputs:
- O, an abstract simplicial complex, the order complex of $P$

Description

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 = simplicialComplex | 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 = simplicialComplex | v_0v_1v_2v_3v_4 |

o2 : SimplicialComplex

Caveat

This method renames the vertices with integers $0, 1, \ldots$ corresponding to the index of the vertices in the GroundSet.

Ways to use orderComplex :

orderComplex(Poset)

For the programmer

The object orderComplex is a method function with options.