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# 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
• Outputs:

## 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