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<head><title>faceLattice(ZZ,Polyhedron) -- computes the face lattice of a polyhedron</title>
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<div><h1>faceLattice(ZZ,Polyhedron) -- computes the 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 = faceLattice P </tt><br/><tt>L = faceLattice(k,P)</tt></div>
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<li><span>Function: <a href="_face__Lattice.html" title="computes the face lattice of a cone or polyhedron">faceLattice</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>P</tt></span></li>
<li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</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 face lattice of a polyhedron <tt>P</tt> displays for each<tt>k</tt> the faces of
codimension <tt>k</tt> as two lists of integers, the first indicating the vertices of <tt>P</tt> and
the second indicating the rays of <tt>P</tt> 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 faces<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 : faceLattice(1,P)
o2 = {({0, 2}, {0}), ({1, 3}, {0}), ({0, 1}, {0}), ({2, 3}, {0}), ({0, 1, 2,
------------------------------------------------------------------------
3}, {})}
o2 : List</pre>
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Returns the faces of codimension one where the first list of integers give the columns in the vertices
matrix of the polyhedron and the second list the columns in the rays matrix of the polyhedron:<table class="examples"><tr><td><pre>i3 : V = vertices P
o3 = | -1 1 -1 1 |
| -1 -1 1 1 |
| 1 1 1 1 |
3 4
o3 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i4 : R = rays P
o4 = | 0 |
| 0 |
| -1 |
3 1
o4 : Matrix QQ <--- QQ</pre>
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The complete face lattice is returned if no integer is given:<table class="examples"><tr><td><pre>i5 : faceLattice P
o5 = {{({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})}}
o5 : List</pre>
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