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<head><title>restriction -- restriction of arrangement to flat/hyperplane</title>
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<div><h1>restriction -- restriction of arrangement to flat/hyperplane</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>restriction(A,F) or restriction(A,x) or restriction(A,I)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="___Arrangement.html">hyperplane arrangement</a></span>, a hyperplane arrangement (optional)</span></li>
<li><span><tt>F</tt>, <span>an <a href="___Flat.html">intersection of hyperplane(s)</a></span>, flat to which you restrict</span></li>
<li><span><tt>x</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, equation of hyperplane to which you restrict</span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, an ideal defining a subspace to which you restrict</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Arrangement.html">hyperplane arrangement</a></span>, the restriction of <tt>A</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The restriction of an arrangement to the subspace <tt>X</tt> indexed by a flat is the (multi)set of hyperplanes <tt>H intersect X</tt> for all <tt>H</tt> in the arrangement <tt>A</tt>. In the first case, one can also write <a href="___Arrangement_sp^_sp__Flat.html">A^F</a>.<table class="examples"><tr><td><pre>i1 : A := typeA(3)
o1 = {x - x , x - x , x - x , x - x , x - x , x - x }
1 2 1 3 1 4 2 3 2 4 3 4
o1 : Hyperplane Arrangement </pre>
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<tr><td><pre>i2 : L := flats(2,A)
o2 = {{0, 1, 3}, {0, 2, 4}, {0, 5}, {1, 4}, {1, 2, 5}, {2, 3}, {3, 4, 5}}
o2 : List</pre>
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<tr><td><pre>i3 : A' := restriction first L
o3 = {x - x , x - x , x - x }
3 4 3 4 3 4
o3 : Hyperplane Arrangement </pre>
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<tr><td><pre>i4 : x := (ring A)_0 -- the subspace need not be in the arrangement
o4 = x
1
o4 : QQ[x , x , x , x ]
1 2 3 4</pre>
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<tr><td><pre>i5 : restriction(A,x)
o5 = {-x , -x , -x , x - x , x - x , x - x }
2 3 4 2 3 2 4 3 4
o5 : Hyperplane Arrangement </pre>
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The restriction is, in general, a multiarrangement. Use <a href="../../Macaulay2Doc/html/_trim.html" title="minimize generators and relations">trim</a> to eliminate repeated hyperplanes. For example,<table class="examples"><tr><td><pre>i6 : trim A'
o6 = {x - x }
3 4
o6 : Hyperplane Arrangement </pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Arrangement_sp^_sp__Flat.html" title="restriction of arrangement to flat">Arrangement ^ Flat</a> -- restriction of arrangement to flat</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>restriction</tt> :</h2>
<ul><li>restriction(Arrangement,Flat)</li>
<li>restriction(Arrangement,Ideal)</li>
<li>restriction(Arrangement,RingElement)</li>
<li>restriction(Flat)</li>
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