ApCoCoA-1:NCo.LWIdeal: Difference between revisions
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<command> | <command> | ||
<title> | <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> | </short_description> | ||
<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<X></tt>, and let <tt>Ordering</tt> be a word ordering. 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<X></tt>, and let <tt>Ordering</tt> be a word ordering on <tt><X></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> | ||
NCo.SetX("xyzt"); | NCo.SetX("xyzt"); | ||
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G := [F1,F2,F3,F4]; | G := [F1,F2,F3,F4]; | ||
GB:=NCo.GB(G); | GB:=NCo.GB(G); | ||
[NCo.LW(E) | E In GB]; -- the leading word ideal of <G> | [NCo.LW(E) | E In GB]; -- the leading word ideal of <G> w.r.t. the length-lexicographic word ordering | ||
["yt", "xt", "xy", "xx", "tyy", "yyx"] | ["yt", "xt", "xy", "xx", "tyy", "yyx"] | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
<see>NCo.GB</see> | <see>ApCoCoA-1:NCo.GB|NCo.GB</see> | ||
<see>NCo.LW</see> | <see>ApCoCoA-1:NCo.LW|NCo.LW</see> | ||
<see>NCo.SetOrdering</see> | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> | ||
<see>NCo.SetX</see> | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</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>NCo.LWIdeal</key> | <key>NCo.LWIdeal</key> | ||
<key>LWIdeal</key> | <key>LWIdeal</key> | ||
<wiki-category>Package_gbmr</wiki-category> | <wiki-category>ApCoCoA-1:Package_gbmr</wiki-category> | ||
</command> | </command> | ||
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 <G> w.r.t. the length-lexicographic word ordering
["yt", "xt", "xy", "xx", "tyy", "yyx"]
-------------------------------
See also
