# dualFaceLattice(ZZ,Cone) -- computes the dual face lattice of a cone

## Synopsis

• Function: dualFaceLattice
• Usage:
L = dualFaceLattice C
L = dualFaceLattice(k,C)
• Inputs:
• k, an integer, between 0 and the dimension of C
• Outputs:

## Description

The dual face lattice of a cone C displays for eachk the faces of dimension k as a list of integers, indicating the bounding halfspaces of C 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 Cone itself.

 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 : dualFaceLattice(2,C) o2 = {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}} o2 : List

Returns the faces of dimension two, where the integers give the rows in the halfspaces matrix of the cone:

 i3 : R = halfspaces 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 dual face lattice is returned if no integer is given:

 i4 : dualFaceLattice C o4 = {{{}}, {{0}, {1}, {2}, {3}}, {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, ------------------------------------------------------------------------ {2, 3}}, {{1, 2, 3}, {0, 2, 3}, {0, 1, 3}, {0, 1, 2}}, {{0, 1, 2, 3}}} o4 : List