# 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<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>. |

## Revision as of 19:38, 14 May 2013

## NC.LTIdeal

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(<quotes>LLEX</quotes>); 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