# Singular

### B.2.5 Module orderings

SINGULAR offers also orderings on the set of "monomials" in Loc = Loc Loc , where denote the canonical generators of Loc , the r-fold direct sum of Loc . (The function gen(i) yields ).

We have two possibilities: either to give priority to the component of a vector in Loc 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 as above.

(<,C): denotes the module ordering (giving priority to the coefficients): or ( and ).

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,<): denotes the module ordering (giving priority to the component): or ( and ).

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): denotes the module ordering (giving priority to the coefficients): or ( and ).

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,<): denotes the module ordering (giving priority to the component): or ( and ).

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

### Misc 