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3.8.7. Predefined Term-Orderings
The predefined term-orderings are:

 * degree reverse lexicographic: DegRevLex  (the default ordering)
 * degree lexicographic: DegLex
 * pure lexicographic: Lex
 * pure xel: Xel
 * elimination term-ordering: Elim(X:INDETS)

The first two term-orderings use the weights of the indeterminates for
computing the degree of a monomial. If the indeterminates are given in
the order x_1, ..., x_n, then x_1 > ... > x_n with respect to Lex, but
x_1 < ... < x_n with respect to Xel.

In the last ordering, X specifies the variables that are to be
eliminated.  It may be a single indeterminate or a range of
indeterminates.  However, X may not be an arbitrary list of
indeterminates; for that, see the command 'Elim' (as opposed to the
modifier 'Elim' being discussed here).  A range of indeterminates can
be specified using the syntax < first-indet >..< last-indet >.
Another shortcut: if there are indexed variables of the form x[i,j],
then 'Elim(x)' specifies a term-ordering for eliminating all of the
x[i,j].

Example

Use R ::= Q[x,y,z], Lex;
x+y+z;
x + y + z
-------------------------------
Use R ::= Q[x,y,z], Xel;
x+y+z;
z + y + x
-------------------------------
Use R ::= Q[t,x,y,z], Elim(t);
I := Ideal(t-x,t-y^2,t^2-xz^3);
GBasis(I);
[t - x, -y^2 + x, xz^3 - x^2]
-------------------------------
Use R ::= Q[x[1..5],y,z], Elim(x); -- term-ordering for eliminating all
                                 -- of the x[i,j]'s
Ord();
Mat[
  [1, 1, 1, 1, 1, 0, 0],
  [0, 0, 0, 0, 0, 1, 1],
  [0, 0, 0, 0, 0, 0, -1],
  [0, 0, 0, 0, -1, 0, 0],
  [0, 0, 0, -1, 0, 0, 0],
  [0, 0, -1, 0, 0, 0, 0],
  [0, -1, 0, 0, 0, 0, 0]
]
-------------------------------
See also: