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B.2.5 Module orderings

SINGULAR offers also orderings on the set of "monomials" 570#570 in Loc 571#571 = Loc 572#572Loc 573#573, where 574#574 denote the canonical generators of Loc 571#571, the r-fold direct sum of Loc 549#549. (The function gen(i) yields 413#413).

We have two possibilities: either to give priority to the component of a vector in Loc 571#571or (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 549#549as above.

(<,C):
575#575 denotes the module ordering (giving priority to the coefficients):
         576#576 or ( 577#577 and 338#338).

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,<):
578#578 denotes the module ordering (giving priority to the component):
         579#579 or (580#580 and 542#542).

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):
581#581 denotes the module ordering (giving priority to the coefficients):
         576#576 or ( 577#577 and 582#582).

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,<):
583#583 denotes the module ordering (giving priority to the component):
         584#584 or (580#580 and 542#542).

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 331#331 in 571#571 with components 585#585 has the format 586#586(up to permutation) unless the ordering starts with c. In this case a vector is written as 587#587.In all cases SINGULAR can read input in both formats.


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