<?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>isWellDefined -- whether a map is well defined</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Iterate.html">next</a> | <a href="_is__Unit.html">previous</a> | <a href="___Iterate.html">forward</a> | <a href="_is__Unit.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>isWellDefined -- whether a map is well defined</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>isWellDefined f</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span> or <span>a <a href="___Ring__Map.html">ring map</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="___Boolean.html">Boolean value</a></span>, whether <tt>f</tt> is a well-defined map</span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>In order to check whether a matrix, whose source module is not free, is well-defined, then a Gröbner basis computation will probably be required.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre> </td></tr> <tr><td><pre>i2 : f = map(R^1,coker vars R,{{1_R}}) o2 = | 1 | o2 : Matrix</pre> </td></tr> <tr><td><pre>i3 : isWellDefined f o3 = false</pre> </td></tr> <tr><td><pre>i4 : isWellDefined map(coker vars R, R^1, {{1_R}}) o4 = true</pre> </td></tr> </table> In order to check whether a ring map is well-defined, it is often necessary to check that the image of an ideal under a related ring map is zero. This often requires a Gröbner basis as well.<table class="examples"><tr><td><pre>i5 : A = ZZ/5[a] o5 = A o5 : PolynomialRing</pre> </td></tr> <tr><td><pre>i6 : factor(a^3-a-2) 3 o6 = (a - a - 2) o6 : Expression of class Product</pre> </td></tr> <tr><td><pre>i7 : B = A/(a^3-a-2);</pre> </td></tr> <tr><td><pre>i8 : isWellDefined map(A,B) o8 = false</pre> </td></tr> <tr><td><pre>i9 : isWellDefined map(B,A) o9 = true</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_map.html" title="make a map">map</a> -- make a map</span></li> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>isWellDefined</tt> :</h2> <ul><li>isWellDefined(Matrix)</li> <li>isWellDefined(RingMap)</li> </ul> </div> </div> </body> </html>