# ApCoCoA-1:NCo.LWIdeal

From ApCoCoAWiki

## NC.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