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<head><title>degreesMonoid -- get the monoid of degrees</title>
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<div><h1>degreesMonoid -- get the monoid of degrees</h1>
<div class="single"><h2>Description</h2>
<div><div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>degreesMonoid x</tt></div>
</dd></dl>
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<li>Inputs:<ul><li><span><tt>x</tt>, <span>a <a href="___List.html">list</a></span> or <span>an <a href="___Z__Z.html">integer</a></span>a list of integers, or a single integer</span></li>
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</li>
<li>Outputs:<ul><li><span>the monoid with inverses whose variables have degrees given by the elements of <tt>x</tt>, and whose weights in the first component of the monomial ordering are minus the degrees. If <tt>x</tt> is an integer, then the number of variables is <tt>x</tt>, the degrees are all <tt>{}</tt>, and the weights are all <tt>-1</tt>.</span></li>
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<p>This is the monoid whose elements correspond to degrees of rings with heft vector <tt>x</tt>, or, in case <tt>x</tt> is an integer, of rings with degree rank <tt>x</tt> and no heft vector; see <a href="_heft_spvectors.html" title="">heft vectors</a>. Hilbert series and polynomials of modules over such rings are elements of its monoid ring over <a href="___Z__Z.html" title="the class of all integers">ZZ</a>; see <a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> and <a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a> The monomial ordering is chosen so that the Hilbert series, which has an infinite number of terms, is bounded above by the weight.</p>
<table class="examples"><tr><td><pre>i1 : degreesMonoid {1,2,5}
o1 = [T , T , T , Degrees => {1..2, 5}, MonomialOrder => {MonomialSize => 32 }, DegreeRank => 1, Inverses => true, Global => false]
0 1 2 {Weights => {-1, -2, -5}}
{GroupLex => 3 }
{Position => Up }
o1 : GeneralOrderedMonoid</pre>
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<tr><td><pre>i2 : degreesMonoid 3
o2 = [T , T , T , Degrees => {3:{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
0 1 2 {Weights => {3:-1} }
{GroupLex => 3 }
{Position => Up }
o2 : GeneralOrderedMonoid</pre>
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<div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>degreesMonoid M</tt></div>
</dd></dl>
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</li>
<li>Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, <span>a <a href="___Polynomial__Ring.html">polynomial ring</a></span>, or <span>a <a href="___Quotient__Ring.html">quotient ring</a></span></span></li>
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<li>Outputs:<ul><li><span>the degrees monoid for (the ring of) <tt>M</tt></span></li>
</ul>
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<table class="examples"><tr><td><pre>i3 : R = QQ[x,y,Degrees => {{1,-2},{2,-1}}];</pre>
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<tr><td><pre>i4 : heft R
o4 = {1, 0}
o4 : List</pre>
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<tr><td><pre>i5 : degreesMonoid R
o5 = [T , T , Degrees => {1, 0}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, Inverses => true, Global => false]
0 1 {Weights => {-1..0}}
{GroupLex => 2 }
{Position => Up }
o5 : GeneralOrderedMonoid</pre>
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<tr><td><pre>i6 : S = QQ[x,y,Degrees => {-2,1}];</pre>
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<tr><td><pre>i7 : heft S</pre>
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<tr><td><pre>i8 : degreesMonoid S^3
o8 = [T, Degrees => {{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
{Weights => {-1} }
{GroupLex => 1 }
{Position => Up }
o8 : GeneralOrderedMonoid</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_heft.html" title="heft vector of ring, module, graded module, or resolution">heft</a> -- heft vector of ring, module, graded module, or resolution</span></li>
<li><span><a href="_use.html" title="install or activate object">use</a> -- install or activate object</span></li>
<li><span><a href="_degrees__Ring.html" title="the ring of degrees">degreesRing</a> -- the ring of degrees</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>degreesMonoid</tt> :</h2>
<ul><li>degreesMonoid(GeneralOrderedMonoid)</li>
<li>degreesMonoid(List)</li>
<li>degreesMonoid(Module)</li>
<li>degreesMonoid(PolynomialRing)</li>
<li>degreesMonoid(QuotientRing)</li>
<li>degreesMonoid(ZZ)</li>
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