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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_monomial_sporderings.html" title="">monomial orderings</a> > <a href="_examples_spof_spspecifying_spalternate_spmonomial_sporders.html" title="">examples of specifying alternate monomial orders</a></div>
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<div><h1>examples of specifying alternate monomial orders</h1>
<div>For definitions of these monomial orders, see <a href="___G__Rev__Lex.html" title="graded reverse lexicographical monomial order.">GRevLex</a>, <a href="___Lex.html" title="lexicographical monomial order.">Lex</a>, <a href="___Weights.html" title="assigning weights to the variables">Weights</a>, <a href="___Eliminate.html" title="elimination order">Eliminate</a>, <a href="___Group__Lex.html" title="defines a ring where some variables are inverted">GroupLex</a>, <a href="___Group__Rev__Lex.html" title="">GroupRevLex</a>, <a href="___Rev__Lex.html" title="reverse lexicographic ordering">RevLex</a>, and <a href="___N__C__Lex.html" title="Non-commutative lexicographical order.">NCLex</a>.<h2>Graded reverse lexicographic order</h2>
<table class="examples"><tr><td><pre>i1 : R = ZZ[a..d];</pre>
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<tr><td><pre>i2 : a+b^100+c*d
100
o2 = b + c*d + a
o2 : R</pre>
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<h2>Lexicographic order</h2>
<table class="examples"><tr><td><pre>i3 : R = ZZ[a..d, MonomialOrder=>Lex];</pre>
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<tr><td><pre>i4 : a+b^100+c*d
100
o4 = a + b + c*d
o4 : R</pre>
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<h2>Weight order</h2>
<table class="examples"><tr><td><pre>i5 : R = ZZ[a..d, MonomialOrder => Weights => {201,2}];</pre>
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<tr><td><pre>i6 : a+b^100+c*d
100
o6 = a + b + c*d
o6 : R</pre>
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<h2>Graded lexicographic order</h2>
<table class="examples"><tr><td><pre>i7 : R = ZZ[a..d, MonomialOrder=>{Weights=>4:1,Lex}];</pre>
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<tr><td><pre>i8 : a+b^100+c*d
100
o8 = b + c*d + a
o8 : R</pre>
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<h2>Elimination order</h2>
To use an elimination order, which eliminates the first 2 variables, use<table class="examples"><tr><td><pre>i9 : R = ZZ[a..f, MonomialOrder=>Eliminate 2];</pre>
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<tr><td><pre>i10 : a+b^100+c*d
100
o10 = b + a + c*d
o10 : R</pre>
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Alternatively, use a weight vector<table class="examples"><tr><td><pre>i11 : R = ZZ[a..f, MonomialOrder=>Weights=>2:1];</pre>
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<tr><td><pre>i12 : a+b^100+c*d
100
o12 = b + a + c*d
o12 : R</pre>
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<h2>Product (block) order</h2>
To make a product order where each block has the GRevLex order:<table class="examples"><tr><td><pre>i13 : R = ZZ[a..f, MonomialOrder=>{2,4}];</pre>
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<tr><td><pre>i14 : a^2*(c+d) + b*(c^100+d^100)*(c + e + f)
2 2 101 100 100 100 100 100
o14 = a c + a d + b*c + b*c*d + b*c e + b*d e + b*c f + b*d f
o14 : R</pre>
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The orders in each block can be other orders as well.<table class="examples"><tr><td><pre>i15 : R = ZZ[a..f, MonomialOrder=>{Weights=>2:1,Lex}]
o15 = R
o15 : PolynomialRing</pre>
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<tr><td><pre>i16 : a^2*(c+d) + b*(c^100+d^100)*(c + e + f)
2 2 101 100 100 100 100 100
o16 = a c + a d + b*c + b*c e + b*c f + b*c*d + b*d e + b*d f
o16 : R</pre>
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<h2>GroupLex</h2>
This order is useful for making degree rings, and allows some variables to appear with negative exponent.<table class="examples"><tr><td><pre>i17 : R = ZZ[a..f, MonomialOrder => GroupLex => 3];</pre>
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<tr><td><pre>i18 : a^-2*(c+d) + b*(c^100+d^100)*(c + e + f)
101 100 100 100 100 100 -2 -2
o18 = b*c + b*c e + b*c f + b*c*d + b*d e + b*d f + a c + a d
o18 : R</pre>
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<h2>GroupRevLex</h2>
This order is useful for making degree rings, and allows some variables to appear with negative exponent. Not implemented yet.<h2>RevLex</h2>
Warning: this is a local ordering, not a global ordering.<table class="examples"><tr><td><pre>i19 : R = ZZ[a..f, MonomialOrder=>RevLex, Global=>false];</pre>
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<tr><td><pre>i20 : a^2*(c+d) + b*(c^100+d^100)*(c + e + f)
100 100 100 100 100 101 2 2
o20 = b*d f + b*d e + b*c*d + b*c f + b*c e + b*c + a d + a c
o20 : R</pre>
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<h2>NCLex</h2>
For non-commutative Gröbner bases. Not implemented yet.</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Singular_sp__Book_sp1.2.13.html" title="monomial orderings">Singular Book 1.2.13</a> -- monomial orderings</span></li>
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