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<head><title>basisElements -- extract the matrix of generators from an involutive basis or factor module basis</title>
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<div><h1>basisElements -- extract the matrix of generators from an involutive basis or factor module basis</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>B = basisElements J or B = basisElements F</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>J</tt>, <span>an object of class <a href="___Involutive__Basis.html" title="the class of all involutive bases">InvolutiveBasis</a></span></span></li>
<li><span><tt>F</tt>, <span>an object of class <a href="___Factor__Module__Basis.html" title="the class of all factor module bases">FactorModuleBasis</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>B => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></span></li>
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<div class="single"><h2>Description</h2>
<div><p>If the argument of basisElements is <span>an object of class <a href="___Involutive__Basis.html" title="the class of all involutive bases">InvolutiveBasis</a></span>, then the columns of B are generators for the module spanned by the involutive basis. These columns form a Gr\"obner basis for this module.</p>
<p>If the argument of basisElements is <span>an object of class <a href="___Factor__Module__Basis.html" title="the class of all factor module bases">FactorModuleBasis</a></span>, then the columns of B are generators for the monomial cones in the factor module basis.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
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<tr><td><pre>i2 : I = ideal(x^3,y^2);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : J = janetBasis I;</pre>
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<tr><td><pre>i4 : basisElements J
o4 = | y2 xy2 x3 x2y2 |
1 4
o4 : Matrix R <--- R</pre>
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<table class="examples"><tr><td><pre>i5 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i6 : M = matrix {{x*y,x^3*z}};
1 2
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : J = janetBasis M;</pre>
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<tr><td><pre>i8 : F = factorModuleBasis J
+--+------+
o8 = |1 |{z, y}|
+--+------+
|x |{z} |
+--+------+
| 2| |
|x |{z} |
+--+------+
| 3| |
|x |{x} |
+--+------+
o8 : FactorModuleBasis</pre>
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<tr><td><pre>i9 : basisElements F
o9 = | 1 x x2 x3 |
1 4
o9 : Matrix R <--- R</pre>
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<tr><td><pre>i10 : multVar F
o10 = {set {y, z}, set {z}, set {z}, set {x}}
o10 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_mult__Var.html" title="extract the sets of multiplicative variables for each generator (in several contexts)">multVar</a> -- extract the sets of multiplicative variables for each generator (in several contexts)</span></li>
<li><span><a href="_janet__Basis.html" title="compute Janet basis for an ideal or a submodule of a free module">janetBasis</a> -- compute Janet basis for an ideal or a submodule of a free module</span></li>
<li><span><a href="_factor__Module__Basis.html" title="enumerate standard monomials">factorModuleBasis</a> -- enumerate standard monomials</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>basisElements</tt> :</h2>
<ul><li>basisElements(FactorModuleBasis)</li>
<li>basisElements(InvolutiveBasis)</li>
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