Difference between revisions of "ApCoCoA-1:NCo.LWIdeal"

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<title>NC.LTIdeal</title>
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<title>NCo.LWIdeal</title>
 
<short_description>
 
<short_description>
 
Leading word ideal of a finitely generated two-sided ideal in a free monoid ring.
 
Leading word ideal of a finitely generated two-sided ideal in a free monoid ring.
 
</short_description>
 
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<syntax></syntax>
 
<description>
 
<description>
 
<em>Proposition:</em> Let <tt>I</tt> be a finitely generated two-sided ideal in a free monoid ring <tt>K&lt;X&gt;</tt>, and let <tt>Ordering</tt> be a word ordering on <tt>&lt;X&gt;</tt>. If <tt>G</tt> is a Groebner basis of <tt>I</tt> with respect to <tt>Ordering</tt>. Then the leading word set <tt>LW{G}:={LW(g): g in G}</tt> is a generating system of the leading word ideal <tt>LW(I)</tt> with respect to <tt>Ordering</tt>.
 
<em>Proposition:</em> Let <tt>I</tt> be a finitely generated two-sided ideal in a free monoid ring <tt>K&lt;X&gt;</tt>, and let <tt>Ordering</tt> be a word ordering on <tt>&lt;X&gt;</tt>. If <tt>G</tt> is a Groebner basis of <tt>I</tt> with respect to <tt>Ordering</tt>. Then the leading word set <tt>LW{G}:={LW(g): g in G}</tt> is a generating system of the leading word ideal <tt>LW(I)</tt> with respect to <tt>Ordering</tt>.
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<seealso>
 
<seealso>
<see>NCo.GB</see>
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<see>ApCoCoA-1:NCo.GB|NCo.GB</see>
<see>NCo.LW</see>
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<see>ApCoCoA-1:NCo.LW|NCo.LW</see>
<see>NCo.SetOrdering</see>
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<see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see>
<see>NCo.SetX</see>
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<see>ApCoCoA-1:NCo.SetX|NCo.SetX</see>
<see>Introduction to CoCoAServer</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
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<key>NCo.LWIdeal</key>
 
<key>NCo.LWIdeal</key>
 
<key>LWIdeal</key>
 
<key>LWIdeal</key>
<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_gbmr</wiki-category>
 
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Latest revision as of 10:21, 7 October 2020

This article is about a function from ApCoCoA-1.

NCo.LWIdeal

Leading word ideal of a finitely generated two-sided ideal in a free monoid ring.

Syntax

Description

Proposition: Let I be a finitely generated two-sided ideal in a free monoid ring K<X>, and let Ordering be a word ordering on <X>. If G is a Groebner basis of I with respect to Ordering. Then the leading word set LW{G}:={LW(g): g in G} is a generating system of the leading word ideal LW(I) with respect to Ordering.

Example

NCo.SetX("xyzt");
NCo.SetOrdering("LLEX");
F1 := [[1,"xx"], [-1,"yx"]];    
F2 := [[1,"xy"], [-1,"ty"]];   
F3 := [[1,"xt"], [-1,"tx"]];   
F4 := [[1,"yt"], [-1,"ty"]];   
G := [F1,F2,F3,F4];
GB:=NCo.GB(G);
[NCo.LW(E) | E In GB]; -- the leading word ideal of &lt;G&gt; w.r.t. the length-lexicographic word ordering

["yt", "xt", "xy", "xx", "tyy", "yyx"]
-------------------------------

See also

NCo.GB

NCo.LW

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer