<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>faceLattice(ZZ,Cone) -- computes the face lattice of a cone</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_face__Lattice_lp__Z__Z_cm__Polyhedron_rp.html">next</a> | <a href="_face__Lattice.html">previous</a> | <a href="_face__Lattice_lp__Z__Z_cm__Polyhedron_rp.html">forward</a> | <a href="_face__Lattice.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>faceLattice(ZZ,Cone) -- computes the 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 = faceLattice C </tt><br/><tt>L = faceLattice(k,C)</tt></div> </dd></dl> </div> </li> <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>C</tt></span></li> <li><span><tt>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><p/> The face lattice of a cone <tt>C</tt> displays for each<tt>k</tt> the faces of codimension <tt>k</tt> as a list of integers, indicating the rays of <tt>C</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 face.<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> </td></tr> <tr><td><pre>i2 : faceLattice(1,C) o2 = {{1, 2, 3}, {0, 2, 3}, {0, 1, 3}, {0, 1, 2}} o2 : List</pre> </td></tr> </table> <p/> Returns the faces of codimension one where the integers give the columns in the rays matrix of the cone:<table class="examples"><tr><td><pre>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 QQ <--- QQ</pre> </td></tr> </table> <p/> The complete face lattice is returned if no integer is given:<table class="examples"><tr><td><pre>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</pre> </td></tr> </table> </div> </div> </div> </body> </html>