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<head><title>Lex -- lexicographical monomial order.</title>
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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="___Lex.html" title="lexicographical monomial order.">Lex</a></div>
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<div><h1>Lex -- lexicographical monomial order.</h1>
<div class="single"><h2>Description</h2>
<div>The lexicographic order is defined by: x<sup>A</sup> > x<sup>B</sup> if the FIRST non-zero entry of the vector of integers A-B is POSITIVE.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d, MonomialOrder => Lex];</pre>
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<tr><td><pre>i2 : a^3 + a^2*b^2 + b*c
3 2 2
o2 = a + a b + b*c
o2 : R</pre>
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The largest possible exponent of variables in <tt>Lex</tt> order is 2^31-1. For efficiency reasons, the size of the exponents of variables may be restricted. Then instead of <tt>Lex</tt>, one can use <tt> MonomialSize=>16</tt>, which allows maximal exponent 2^15-1, or <tt>MonomialSize=>8</tt>, which allows maximal exponent 2^7-1.<table class="examples"><tr><td><pre>i3 : B = QQ[a..d,MonomialOrder=>Lex,MonomialSize=>16];</pre>
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<tr><td><pre>i4 : a^(2^15-1)
32767
o4 = a
o4 : B</pre>
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<tr><td><pre>i5 : C = QQ[a..d,MonomialOrder=>Lex,MonomialSize=>8];</pre>
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<tr><td><pre>i6 : try a^(2^15-1) else "failed"
o6 = failed</pre>
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<tr><td><pre>i7 : a^(2^7-1)
127
o7 = a
o7 : C</pre>
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Any of these versions of <tt>Lex</tt> order may be combined, for example, with a weight order given by a weight vector: x^A > x^B if weight(x^A) > weight(x^B) or if weight(x^A) = weight(x^B) and if the FIRST non-zero entry of the vector of integers A-B is POSITIVE.<table class="examples"><tr><td><pre>i8 : B = QQ[a..d,MonomialSize=>16,MonomialOrder=>{Weights => {1,2,3,4}, Lex}];</pre>
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<tr><td><pre>i9 : a^2 + b+ c + b*d
2
o9 = b*d + c + a + b
o9 : B</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Weights.html" title="assigning weights to the variables">Weights</a> -- assigning weights to the variables</span></li>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Lex.html" title="lexicographical monomial order.">Lex</a> is <span>a <a href="___Symbol.html">symbol</a></span>.</p>
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