# 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"] -------------------------------

### See also