NCRingElement % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis

Operator: %
Usage:

fred = f % Igb
Inputs:
- f, an instance of the type NCRingElement, or an integer, or a rational number
- Igb, an instance of the type NCGroebnerBasis,
Outputs:
- fred, an instance of the type NCRingElement,

Description

This command reduces the input modulo a noncommutative Groebner basis. It will either reduce it using top-level Macaulay code, or via a call to Bergman, depending on the size and degree of the input element.

i1 : A = QQ{x,y,z}

o1 = A

o1 : NCPolynomialRing

i2 : p = y*z + z*y - x^2

            2
o2 = zy+yz-x

o2 : A

i3 : q = x*z + z*x - y^2

         2
o3 = zx-y +xz

o3 : A

i4 : r = z^2 - x*y - y*x

      2
o4 = z -yx-xy

o4 : A

i5 : I = ncIdeal {p,q,r}

                             2      2      2
o5 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}

o5 : NCIdeal

i6 : Igb = ncGroebnerBasis I
--Calling Bergman for NCGB calculation.
Complete!

      2    2                2
o6 = y x-xy ; Lead Term = (y x, 1)
       2  2                  2
     yx -x y; Lead Term = (yx , 1)
         2
     zx-y +xz; Lead Term = (zx, 1)
            2
     zy+yz-x ; Lead Term = (zy, 1)
      2                      2
     z -yx-xy; Lead Term = (z , 1)

o6 : NCGroebnerBasis

i7 : z^6 % Igb

                     2   2   3 3
o7 = yxyxyx+xyxyxy+3x yxy +3x y

o7 : A

Ways to use this method:

NCRingElement % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
QQ % NCGroebnerBasis
ZZ % NCGroebnerBasis

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NCAlgebra/NCAlgebraDoc.m2:1670:0.