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7.6.4 Example of use of LETTERPLACE

First, define a commutative ring $K[X]$ in SINGULAR and equip it with a degree well-ordering. Then, decide what should be the degree bound $d$ and run the procedure makeLetterplaceRing(d) from the library freegb_lib. This procedure creates a Letterplace ring with an ordering, induced from the given commutative ring $K[X]$. In this $K$-algebra, define an ideal I as a list of polynomials in the free algebra (x*y and y*x are different) and run std. The answer is a two-sided Groebner bases of the two-sided ideal $I$.

We illustrate the approach with the following example:

 
LIB "freegb.lib";
ring r = 0,(x,y,z),dp;
def R = makeLetterplaceRing(4);  // with degree bound 4
setring R;
ideal I = x*y + y*z, x*x + x*y - y*x - y*y;
ideal J = std(I);
J;
==> J[1]=x*y+y*z
==> J[2]=x*x-y*x-y*y-y*z
==> J[3]=y*y*y-y*y*z+y*z*y-y*z*z
==> J[4]=y*y*x+y*y*z+y*z*x+y*z*z
==> J[5]=y*z*y*y-y*z*y*z+y*z*z*y-y*z*z*z
==> J[6]=y*z*y*x+y*z*y*z+y*z*z*x+y*z*z*z
==> J[7]=y*y*z*y-y*y*z*z+y*z*z*y-y*z*z*z
==> J[8]=2*y*y*z*x+2*y*y*z*z-y*z*y*x-y*z*y*z+y*z*z*x+y*z*z*z

There are various conversion routines in the library freegb_lib (see freegb_lib). Many algebras are predefined in the library fpalgebras_lib (see fpalgebras_lib). We work further on implementing more algorithms for non-commutative ideals and modules over free associative algebra.


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