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<head><title>dualFaceLattice(ZZ,Cone) -- computes the dual face lattice of a cone</title>
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<div><h1>dualFaceLattice(ZZ,Cone) -- computes the dual face lattice of a cone</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt> L = dualFaceLattice C </tt><br/><tt>L = dualFaceLattice(k,C)</tt></div>
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<li><span>Function: <a href="_dual__Face__Lattice.html" title="computes the dual face lattice of a cone or polyhedron">dualFaceLattice</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>k</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, between 0 and the dimension of <tt>C</tt></span></li>
<li><span><tt>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div><p/>
The dual face lattice of a cone <tt>C</tt> displays for each<tt>k</tt> the faces of
dimension <tt>k</tt> as a list of integers, indicating the bounding halfspaces of <tt>C</tt> 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.<table class="examples"><tr><td><pre>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</pre>
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<tr><td><pre>i2 : dualFaceLattice(2,C)
o2 = {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}}
o2 : List</pre>
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<p/>
Returns the faces of dimension two, where the integers give the rows in the halfspaces
matrix of the cone:<table class="examples"><tr><td><pre>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 QQ <--- QQ</pre>
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The complete dual face lattice is returned if no integer is given:<table class="examples"><tr><td><pre>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</pre>
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