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<head><title>dual(MonomialIdeal) -- the Alexander dual of a monomial ideal</title>
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<div><h1>dual(MonomialIdeal) -- the Alexander dual of a monomial 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>dual I</tt></div>
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<li><span>Function: <a href="_dual.html" title="dual module or map">dual</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span>, a monomial ideal</span></li>
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<li><div class="single">Outputs:<ul><li><span>the Alexander dual of <tt>I</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_dual_lp__Monomial__Ideal_cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>If <tt>I</tt>is a square free monomial ideal then <tt>I</tt> is the Stanley-Reisner ideal of a simplicial complex. In this case, <tt>dual I</tt> is the Stanley-Reisner ideal associated to the dual complex. In particular, <tt>dual I</tt> is obtained by switching the roles of minimal generators and prime components.<table class="examples"><tr><td><pre>i1 : QQ[a,b,c,d];</pre>
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<tr><td><pre>i2 : I = monomialIdeal(a*b, b*c, c*d)
o2 = monomialIdeal (a*b, b*c, c*d)
o2 : MonomialIdeal of QQ[a, b, c, d]</pre>
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<tr><td><pre>i3 : dual I
o3 = monomialIdeal (a*c, b*c, b*d)
o3 : MonomialIdeal of QQ[a, b, c, d]</pre>
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<tr><td><pre>i4 : intersect(monomialIdeal(a,b),
monomialIdeal(b,c),
monomialIdeal(c,d))
o4 = monomialIdeal (a*c, b*c, b*d)
o4 : MonomialIdeal of QQ[a, b, c, d]</pre>
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<tr><td><pre>i5 : dual dual I
o5 = monomialIdeal (a*b, b*c, c*d)
o5 : MonomialIdeal of QQ[a, b, c, d]</pre>
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<p/>
For a general monomial ideal, the Alexander dual defined as follows: Given two list of nonnegative integers <tt>a</tt> and <tt>b</tt>for which <tt>a_i >= b_i</tt> for all <tt>i</tt> let <tt>a\b</tt> denote the list whose <tt>i</tt>-th entry is <tt>a_i+1-b_i</tt>if <tt>b_i >= 1</tt>and <tt>0</tt>otherwise. The Alexander dual with respect to <tt>a</tt> is the ideal generated by a monomial <tt>x^a\b</tt> for each irreducible component <tt>(x_i^b_i)</tt> of <tt>I</tt>. If <tt>a</tt> is not provided, it is assumed to be the least common multiple of the minimal generators of <tt>I</tt>.<table class="examples"><tr><td><pre>i6 : QQ[x,y,z];</pre>
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<tr><td><pre>i7 : I = monomialIdeal(x^3, x*y, y*z^2)
3 2
o7 = monomialIdeal (x , x*y, y*z )
o7 : MonomialIdeal of QQ[x, y, z]</pre>
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<tr><td><pre>i8 : dual(I, {4,4,4})
2 4 4 3
o8 = monomialIdeal (x y , x z )
o8 : MonomialIdeal of QQ[x, y, z]</pre>
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<tr><td><pre>i9 : intersect( monomialIdeal(x^2),
monomialIdeal(x^4, y^4),
monomialIdeal(y^4, z^3))
2 4 4 3
o9 = monomialIdeal (x y , x z )
o9 : MonomialIdeal of QQ[x, y, z]</pre>
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One always has <tt>dual( dual(I, a), a) == I</tt> however <tt>dual dual I</tt>may not equal <tt>I</tt>.<table class="examples"><tr><td><pre>i10 : QQ[x,y,z];</pre>
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<tr><td><pre>i11 : J = monomialIdeal( x^3*y^2, x*y^4, x*z, y^2*z)
3 2 4 2
o11 = monomialIdeal (x y , x*y , x*z, y z)
o11 : MonomialIdeal of QQ[x, y, z]</pre>
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<tr><td><pre>i12 : dual dual J
3 3
o12 = monomialIdeal (x y, x*y , x*z, y*z)
o12 : MonomialIdeal of QQ[x, y, z]</pre>
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<tr><td><pre>i13 : dual( dual(J, {3,4,1}), {3,4,1})
3 2 4 2
o13 = monomialIdeal (x y , x*y , x*z, y z)
o13 : MonomialIdeal of QQ[x, y, z]</pre>
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<p/>
See Ezra Miller's Ph.D. thesis 'Resolutions and Duality for Monomial Ideals'.<p>Implemented by Greg Smith.</p>
<p>The computation is done by calling the <a href="_frobby.html" title="">frobby</a> library, written by B. H. Roune; setting <a href="_gb__Trace.html" title="provide tracing output during various computations in the engine.">gbTrace</a> to a positive value will cause a message to be printed when it is called.</p>
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