faceLattice(ZZ,Cone) -- computes the face lattice of a cone

Synopsis

• Function: faceLattice

• Usage:

  L = faceLattice C

  L = faceLattice(k,C)

• Inputs:
  • k, an integer, between 0 and the dimension of C
  • C, a convex rational cone
• Outputs:
  • L, a list

Description

The face lattice of a cone C displays for eachk the faces of codimension k as a list of integers, indicating the rays of C that generate this face together with the lineality space. If no integer is given the function returns the faces of all codimensions in a list, starting with the 0 dimensional face.

i1 : C = posOrthant 4

o1 = {ambient dimension => 4           }
      dimension of lineality space => 0
      dimension of the cone => 4
      number of facets => 4
      number of rays => 4

o1 : Cone

i2 : faceLattice(1,C)

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

o2 : List

Returns the faces of codimension one where the integers give the columns in the rays matrix of the cone:

i3 : R = rays C

o3 = | 1 0 0 0 |
     | 0 1 0 0 |
     | 0 0 1 0 |
     | 0 0 0 1 |

              4        4
o3 : Matrix ZZ  <--- ZZ

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

i4 : faceLattice C

o4 = {{{}}, {{0}, {1}, {2}, {3}}, {{2, 3}, {1, 3}, {1, 2}, {0, 3}, {0, 2}, {0, 1}}, {{1, 2, 3}, {0, 2, 3}, {0, 1, 3}, {0, 1, 2}},
     ----------------------------------------------------------------------------------------------------------------------------
     {{0, 1, 2, 3}}}

o4 : List