Singular Manual: Module orderings

  | ### 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.

User manual for Singular version 4.4.0, 2024, generated by texi2html.