Difference between revisions of "ApCoCoA-1:NCo.LWIdeal"
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| − | <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"] | ||
Revision as of 17:40, 9 May 2013
NC.LTIdeal
Leading word ideal of a finitely generated two-sided ideal in a free monoid ring.
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"]
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See also
