Singular Manual: Module orderings

  | ### B.2.5 Module orderings SINGULAR offers also orderings on the set of "monomials" 572#572 in Loc 573#573 = Loc 574#574Loc 575#575, where 576#576 denote the canonical generators of Loc 573#573, the r-fold direct sum of Loc 551#551. (The function gen(i) yields 415#415). We have two possibilities: either to give priority to the component of a vector in Loc 573#573or (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 551#551as above. (<,C): 577#577 denotes the module ordering (giving priority to the coefficients): 578#578 or ( 579#579 and 340#340). 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,<): 580#580 denotes the module ordering (giving priority to the component): 581#581 or (582#582 and 544#544). 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): 583#583 denotes the module ordering (giving priority to the coefficients): 578#578 or ( 579#579 and 584#584). 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,<): 585#585 denotes the module ordering (giving priority to the component): 586#586 or (582#582 and 544#544). 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 333#333 in 573#573 with components 587#587 has the format 588#588(up to permutation) unless the ordering starts with `c`. In this case a vector is written as 589#589.In all cases SINGULAR can read input in both formats.

User manual for Singular version 4.4.1, 2025, generated by texi2html.