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<head><title>lcmLattice -- returns the LCM lattice of an ideal</title>
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<div><h1>lcmLattice -- returns the LCM lattice of an 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>L = lcmLattice (I)</tt><br/><tt>L = lcmLattice (M)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
<li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>an object of class <a href="___Poset.html" title="a class for partially ordered sets (posets)">Poset</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div><div>This command allows for fast computation of LCM lattices, which are particularly useful in the study of resolutions of monomial ideals. Specifically the LCM lattice is the set of all lcms of subsets of the generators of the ideal, partially ordered by divisability.</div>
<table class="examples"><tr><td><pre>i1 : S = QQ[a,b,c,d];</pre>
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<tr><td><pre>i2 : I = ideal (b^2-a*d, a*d-b*c, c^2-b*d);
o2 : Ideal of S</pre>
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<tr><td><pre>i3 : M = monomialIdeal (b^2, b*c, c^2);
o3 : MonomialIdeal of S</pre>
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<tr><td><pre>i4 : L = lcmLattice (I);</pre>
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<tr><td><pre>i5 : L.GroundSet
2 2 3 2 2 2 2 2
o5 = {b - a*d, - b*c + a*d, c - b*d, - b c + a*b d + a*b*c*d - a d , b c -
------------------------------------------------------------------------
3 2 2 3 2 2 2 3 3 4
b d - a*c d + a*b*d , - b*c + b c*d + a*c d - a*b*d , - b c + b c*d +
------------------------------------------------------------------------
2 2 3 3 2 2 2 2 2 2 2 3
a*b c d + a*b*c d - a*b d - a*b c*d - a c d + a b*d }
o5 : List</pre>
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<tr><td><pre>i6 : L.RelationMatrix
o6 = | 1 0 0 1 1 0 1 |
| 0 1 0 1 0 1 1 |
| 0 0 1 0 1 1 1 |
| 0 0 0 1 0 0 1 |
| 0 0 0 0 1 0 1 |
| 0 0 0 0 0 1 1 |
| 0 0 0 0 0 0 1 |
7 7
o6 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i7 : LM = lcmLattice (M);</pre>
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<tr><td><pre>i8 : LM.GroundSet
2 2 2 2 2 2
o8 = {b , b*c, c , b c, b c , b*c }
o8 : List</pre>
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<tr><td><pre>i9 : LM.RelationMatrix
o9 = | 1 0 0 1 1 0 |
| 0 1 0 1 1 1 |
| 0 0 1 0 1 1 |
| 0 0 0 1 1 0 |
| 0 0 0 0 1 0 |
| 0 0 0 0 1 1 |
6 6
o9 : Matrix ZZ <--- ZZ</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>Note, that at present this command does not efficiently handle ideals with large numbers of generators. This is a problem that should be fixed by the next release of this package.</div>
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<div class="waystouse"><h2>Ways to use <tt>lcmLattice</tt> :</h2>
<ul><li>lcmLattice(Ideal)</li>
<li>lcmLattice(MonomialIdeal)</li>
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