<?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>ideal(SimplicialComplex) -- the ideal of minimal nonfaces (the Stanley-Reisner ideal)</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_is__Pure.html">next</a> | <a href="_f__Vector.html">previous</a> | <a href="_is__Pure.html">forward</a> | <a href="_f__Vector.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>ideal(SimplicialComplex) -- the ideal of minimal nonfaces (the Stanley-Reisner ideal)</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>ideal D</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="../../Macaulay2Doc/html/_ideal.html" title="make an ideal">ideal</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>D</tt>, <span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which is generated by monomials representing the minimal nonfaces of <tt>D</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>In Macaulay2, every <a href="___Simplicial__Complex.html">simplicial complex</a> is equipped with a polynomial ring, and the Stanley-Reisner ideal is contained in this ring.<table class="examples"><tr><td><pre>i1 : loadPackage "SimplicialComplexes";</pre>
</td></tr>
</table>
The 3-dimensional sphere has a unique minimal nonface which corresponds to the interior.<table class="examples"><tr><td><pre>i2 : R = ZZ[a..e];</pre>
</td></tr>
<tr><td><pre>i3 : sphere = simplicialComplex {b*c*d*e,a*c*d*e,a*b*d*e,a*b*c*e,a*b*c*d}
o3 = | bcde acde abde abce abcd |
o3 : SimplicialComplex</pre>
</td></tr>
<tr><td><pre>i4 : ideal sphere
o4 = ideal(a*b*c*d*e)
o4 : Ideal of R</pre>
</td></tr>
</table>
The simplicial complex from example 1.8 in Miller-Sturmfels, Combinatorial Commutative Algebra, consists of a triangle (on vertices <tt>a,b,c</tt>), two edges connecting <tt>c</tt> to <tt>d</tt> and <tt>b</tt> to <tt>d</tt>, and an isolated vertex <tt>e</tt>.<table class="examples"><tr><td><pre>i5 : D = simplicialComplex {e, c*d, b*d, a*b*c}
o5 = | e cd bd abc |
o5 : SimplicialComplex</pre>
</td></tr>
<tr><td><pre>i6 : ideal D
o6 = ideal (a*d, b*c*d, a*e, b*e, c*e, d*e)
o6 : Ideal of R</pre>
</td></tr>
</table>
There are six minimal nonfaces of <tt>D</tt>.<p/>
This routine is identical to <a href="_monomial__Ideal_lp__Simplicial__Complex_rp.html" title="the monomial ideal of minimal nonfaces (the Stanley-Reisner ideal)">monomialIdeal(SimplicialComplex)</a>, except for the <a href="../../Macaulay2Doc/html/___Type.html">type</a> of the output.<p/>
Note that no computatation is performed by this routine; all the computation was done while constructing the simplicial complex.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="simplicial complexes">SimplicialComplexes</a> -- simplicial complexes</span></li>
<li><span><a href="_simplicial__Complex.html" title="create a simplicial complex">simplicialComplex</a> -- create a simplicial complex</span></li>
<li><span><a href="_facets.html" title="the facets of a simplicial complex">facets</a> -- the facets of a simplicial complex</span></li>
<li><span><a href="_monomial__Ideal_lp__Simplicial__Complex_rp.html" title="the monomial ideal of minimal nonfaces (the Stanley-Reisner ideal)">monomialIdeal(SimplicialComplex)</a> -- the monomial ideal of minimal nonfaces (the Stanley-Reisner ideal)</span></li>
</ul>
</div>
</div>
</body>
</html>
