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B.2.5 Module orderings

SINGULAR offers also orderings on the set of “monomials” {xaeia Nn,1 i r} in Loc K[x]r = Loc K[x]e1 + +Loc K[x]er, where e1,,er denote the canonical generators of Loc K[x]r, the r-fold direct sum of Loc K[x]. (The function gen(i) yields ei).

We have two possibilities: either to give priority to the component of a vector in Loc K[x]r or (which is the default in SINGULAR) to give priority to the coefficients. The orderings (<,c) and (<,C) give priority to the coefficients; whereas (c,<) and (C,<) give priority to the components.
Let < be any of the monomial orderings of Loc K[x] as above.

(<,C):
< m = (<,C) denotes the module ordering (giving priority to the coefficients):
xαei < mxβej xα < xβ or (xα = xβ and i < j).

Example:

  ring r = 0, (x,y,z), ds;
  // the same as ring r = 0, (x,y,z), (ds, C);
  [x+y2,z3+xy];
→ x*gen(1)+xy*gen(2)+y2*gen(1)+z3*gen(2)
  [x,x,x];
→ x*gen(3)+x*gen(2)+x*gen(1)
(C,<):
< m = (C,<) denotes the module ordering (giving priority to the component):
xαei < mxβej i < j or (i = j and xα < xβ).

Example:

  ring r = 0, (x,y,z), (C,lp);
  [x+y2,z3+xy];
→ xy*gen(2)+z3*gen(2)+x*gen(1)+y2*gen(1)
  [x,x,x];
→ x*gen(3)+x*gen(2)+x*gen(1)
(<,c):
< m = (<,c) denotes the module ordering (giving priority to the coefficients):
xαei < mxβej xα < xβ or (xα = xβ and i > j).

Example:

  ring r = 0, (x,y,z), (lp,c);
  [x+y2,z3+xy];
→ xy*gen(2)+x*gen(1)+y2*gen(1)+z3*gen(2)
  [x,x,x];
→ x*gen(1)+x*gen(2)+x*gen(3)
(c,<):
< m = (c,<) denotes the module ordering (giving priority to the component):
xαei < mxβej i > j or (i = j and xα < xβ).

Example:

  ring r = 0, (x,y,z), (c,lp);
  [x+y2,z3+xy];
→ [x+y2,xy+z3]
  [x,x,x];
→ [x,x,x]
The output of a vector v in K[x]r with components v1,,vr has the format v1 gen(1) + + vr gen(r)

(up to permutation) unless the ordering starts with c. In this case a vector is written as [v1,,vr]. In all cases SINGULAR can read input in both formats.


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