ApCoCoA-1:NC.LWIdeal: Difference between revisions
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<command> | <command> | ||
<title>NC. | <title>NC.LWIdeal</title> | ||
<short_description> | <short_description> | ||
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. | ||
</short_description> | </short_description> | ||
<syntax></syntax> | |||
<description> | <description> | ||
<em>Proposition:</em> Let <tt>I</tt> be a finitely generated two-sided ideal in a non-commutative polynomial ring <tt>K<x[1],...,x[n]></tt>, and let <tt>Ordering</tt> be a word ordering on <tt><x[1],...,x[n]></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<x[1],...,x[n]></tt>, and let <tt>Ordering</tt> be a word ordering on <tt><x[1],...,x[n]></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>. | ||
<example> | <example> | ||
Use QQ[x,y,z,t]; | Use QQ[x,y,z,t]; | ||
NC.SetOrdering( | NC.SetOrdering("LLEX"); | ||
F1 := [[x^2], [-y,x]]; | F1 := [[x^2], [-y,x]]; | ||
F2 := [[x,y], [-t,y]]; | F2 := [[x,y], [-t,y]]; | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
<see>NC.GB</see> | <see>ApCoCoA-1:NC.GB|NC.GB</see> | ||
<see>NC.LW</see> | <see>ApCoCoA-1:NC.LW|NC.LW</see> | ||
<see>NC.SetOrdering</see> | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> | ||
<see>Introduction to CoCoAServer</see> | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> | ||
</seealso> | </seealso> | ||
<types> | <types> | ||
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<key>NC.LWIdeal</key> | <key>NC.LWIdeal</key> | ||
<key>LWIdeal</key> | <key>LWIdeal</key> | ||
<wiki-category>Package_ncpoly</wiki-category> | <wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category> | ||
</command> | </command> | ||
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 <G> w.r.t. the length-lexicographic word ordering
[[y, t], [x, t], [x, y], [x^2], [t, y^2], [y^2, x]]
-------------------------------
See also
