dualFaceLattice(ZZ,Polyhedron) -- computes the dual face lattice of a polyhedron

Synopsis

Function: dualFaceLattice
Usage:

L = dualFaceLattice P

L = dualFaceLattice(k,P)
Inputs:
- k, an integer, between 0 and the dimension of X
- P, a convex polyhedron
Outputs:
- L, a list

Description

The dual face lattice of a polyhedron P displays for eachk the faces of dimension k as a list of integers, indicating the halfspaceces of P that generate this face together with the hyperplanes. If no integer is given the function returns the faces of all dimensions in a list, starting with the polyhedron itself.

i1 : P = convexHull(matrix{{1,1,-1,-1},{1,-1,1,-1},{1,1,1,1}},matrix {{0},{0},{-1}})

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 5
      number of rays => 1
      number of vertices => 4

o1 : Polyhedron

i2 : dualFaceLattice(2,P)

o2 = {{0}, {1}, {2}, {3}, {4}}

o2 : List

Returns the faces of dimension two where each list of integers gives the rows in the halfspaces matrix of the polyhedron:

i3 : V = halfspaces P

o3 = (| -1 0  0 |, | 1 |)
      | 1  0  0 |  | 1 |
      | 0  -1 0 |  | 1 |
      | 0  1  0 |  | 1 |
      | 0  0  1 |  | 1 |

o3 : Sequence

The complete face lattice is returned if no integer is given:

i4 : faceLattice P

o4 = {{({0}, {}), ({1}, {}), ({2}, {}), ({3}, {})}, {({0}, {0}), ({2}, {0}), ({0, 2}, {}), ({1}, {0}), ({3}, {0}), ({1, 3}, {}),
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     ({0, 1}, {}), ({2, 3}, {})}, {({0, 2}, {0}), ({1, 3}, {0}), ({0, 1}, {0}), ({2, 3}, {0}), ({0, 1, 2, 3}, {})}, {({0, 1, 2,
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     3}, {0})}}

o4 : List