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

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Leading word ideal of a finitely generated two-sided ideal in a non-commutative polynomial ring.
 
Leading word ideal of a finitely generated two-sided ideal in a non-commutative polynomial ring.
 
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<em>Proposition:</em> Let <tt>I</tt> be a finitely generated two-sided ideal in a non-commutative polynomial ring <tt>K&lt;x[1],...,x[n]&gt;</tt>, and let <tt>Ordering</tt> be a word ordering on <tt>&lt;x[1],...,x[n]&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 non-commutative polynomial ring <tt>K&lt;x[1],...,x[n]&gt;</tt>, and let <tt>Ordering</tt> be a word ordering on <tt>&lt;x[1],...,x[n]&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>
<see>NC.GB</see>
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<see>ApCoCoA-1:NC.GB|NC.GB</see>
<see>NC.LW</see>
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<see>ApCoCoA-1:NC.LW|NC.LW</see>
<see>NC.SetOrdering</see>
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<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
<see>Introduction to CoCoAServer</see>
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<type>non_commutative</type>
 
<type>non_commutative</type>
 
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<key>ncpoly.LWIdeal</key>
 
<key>NC.LWIdeal</key>
 
<key>NC.LWIdeal</key>
 
<key>LWIdeal</key>
 
<key>LWIdeal</key>
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Latest revision as of 13:35, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.LWIdeal

Leading word ideal of a finitely generated two-sided ideal in a non-commutative polynomial ring.

Syntax

Description

Proposition: Let I be a finitely generated two-sided ideal in a non-commutative polynomial ring K<x[1],...,x[n]>, and let Ordering be a word ordering on <x[1],...,x[n]>. 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

Use QQ[x,y,z,t];
NC.SetOrdering("LLEX");
F1 := [[x^2], [-y,x]];    
F2 := [[x,y], [-t,y]];   
F3 := [[x,t], [-t,x]];   
F4 := [[y,t], [-t,y]];   
G := [F1,F2,F3,F4];
GB:=NC.GB(G);
[NC.LW(E) | E In GB]; -- the leading word ideal of &lt;G&gt; w.r.t. the length-lexicographic word ordering

[[y, t], [x, t], [x, y], [x^2], [t, y^2], [y^2, x]]
-------------------------------

See also

NC.GB

NC.LW

NC.SetOrdering

Introduction to CoCoAServer