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<head><title>dualFaceLattice(ZZ,Polyhedron) -- computes the dual face lattice of a polyhedron</title>
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<div><h1>dualFaceLattice(ZZ,Polyhedron) -- computes the dual face lattice of a polyhedron</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 P </tt><br/><tt>L = dualFaceLattice(k,P)</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>X</tt></span></li>
<li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
</ul>
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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 polyhedron <tt>P</tt> displays for each<tt>k</tt> the faces of
dimension <tt>k</tt> as a list of integers, indicating the halfspaceces of <tt>P</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 polyhedron itself.<table class="examples"><tr><td><pre>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</pre>
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<tr><td><pre>i2 : dualFaceLattice(2,P)
o2 = {{0}, {1}, {2}, {3}, {4}}
o2 : List</pre>
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Returns the faces of dimension two where each list of integers gives the rows in the halfspaces
matrix of the polyhedron:<table class="examples"><tr><td><pre>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</pre>
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The complete face lattice is returned if no integer is given:<table class="examples"><tr><td><pre>i4 : faceLattice P
o4 = {{({0}, {}), ({1}, {}), ({2}, {}), ({3}, {})}, {({0}, {0}), ({2}, {0}),
------------------------------------------------------------------------
({0, 2}, {}), ({1}, {0}), ({3}, {0}), ({1, 3}, {}), ({0, 1}, {}), ({2,
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3}, {})}, {({0, 2}, {0}), ({1, 3}, {0}), ({0, 1}, {0}), ({2, 3}, {0}),
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({0, 1, 2, 3}, {})}, {({0, 1, 2, 3}, {0})}}
o4 : List</pre>
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