# orderComplex -- produces the order complex of a poset

## Synopsis

• Usage:
O = orderComplex P
• Inputs:
• P, an instance of the type Poset,
• Optional inputs:
• VariableName => , default value v
• CoefficientRing => a ring, default value QQ
• Outputs:
• O, , 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 = | 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

## Caveat

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