B.2.6 Matrix orderings
----------------------

Let 
$M$
be an invertible 
$(n \times n)$-matrix
 with integer coefficients and
$M_1, \ldots, M_n$ the rows of $M$.

The M-ordering < is defined as follows:


\quad \quad $x^a < x^b \Leftrightarrow \exists\  1 \leq i \leq n :
M_1 a = \; M_1 b, \ldots, M_{i-1} a = \; M_{i-1} b$ and $M_i a < \; M_i b$.

Thus,
$x^a < x^b$
if and only if $M a$ is smaller than $M b$
with respect to the lexicographical ordering.

The following matrices represent (for 3 variables) the global and
local orderings defined above (note that the matrix is not uniquely determined
by the ordering):


$\quad$ lp:
$\left(\matrix{
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 0 & 0 & 1 \cr
 }\right)$
\quad dp:
$\left(\matrix{
 1 & 1 & 1 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Dp:
$\left(\matrix{
 1 & 1 & 1 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$

$\quad$ wp(1,2,3):
$\left(\matrix{
 1 & 2 & 3 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Wp(1,2,3):
$\left(\matrix{
 1 & 2 & 3 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$

$\quad$ ls:
$\left(\matrix{
-1 & 0 & 0 \cr
 0 &-1 & 0 \cr
 0 & 0 &-1 \cr
 }\right)$
\quad ds:
$\left(\matrix{
-1 &-1 &-1 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Ds:
$\left(\matrix{
-1 &-1 &-1 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$

$\quad$ ws(1,2,3):
$\left(\matrix{
-1 &-2 &-3 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Ws(1,2,3):
$\left(\matrix{
-1 &-2 &-3 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$

Product orderings (see next section) represented by  a matrix:

$\quad$ (dp(3), wp(1,2,3)):
$\left(\matrix{
1&  1&  1&  0&  0&  0 \cr
0&  0&  -1&  0&  0&  0 \cr
0&  -1&  0&  0&  0&  0 \cr
0&  0&  0&  1&  2&  3 \cr
0&  0&  0&  0&  0&  -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$

$\quad$ (Dp(3), ds(3)):
$\left(\matrix{
1&  1&  1&  0&  0&  0 \cr
1&  0&  0&  0&  0&  0 \cr
0&  1&  0&  0&  0&  0 \cr
0&  0&  0&  -1&  -1&  -1 \cr
0&  0&  0&  0&  0&  -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$

Orderings with extra weight vector (see below) represented by  a matrix:

$\quad$ (dp(3), a(1,2,3),dp(3)):
$\left(\matrix{
1&  1&  1&  0&  0&  0 \cr
0&  0&  -1&  0&  0&  0 \cr
0&  -1&  0&  0&  0&  0 \cr
0&  0&  0&  1&  2&  3 \cr
0&  0&  0&  1&  1&  1 \cr
0&  0&  0&  0&  0&  -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$

$\quad$ (a(1,2,3,4,5),Dp(3), ds(3)):
$\left(\matrix{
1&  2&  3&  4&  5&  0 \cr
1&  1&  1&  0&  0&  0 \cr
1&  0&  0&  0&  0&  0 \cr
0&  1&  0&  0&  0&  0 \cr
0&  0&  0&  -1&  -1&  -1 \cr
0&  0&  0&  0&  0 & -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$


Example:
  ring r = 0, (x,y,z), M(1, 0, 0,   0, 1, 0,   0, 0, 1);

which may also be written as:
  intmat m[3][3]=1, 0, 0, 0, 1, 0, 0, 0, 1;
  m;
==> 1,0,0,
==> 0,1,0,
==> 0,0,1 
  ring r = 0, (x,y,z), M(m);
  r;
==> //   characteristic : 0
==> //   number of vars : 3
==> //        block   1 : ordering M
==> //                  : names    x y z 
==> //                  : weights  1 0 0 
==> //                  : weights  0 1 0 
==> //                  : weights  0 0 1 
==> //        block   2 : ordering C

If the ring has 
$n$
variables and the matrix contains less than 
$n \times n$
entries an error message is given, if there are more entries,
the last ones are ignored.

WARNING: SINGULAR
does not check whether the matrix has full rank.   In such a case some
computations might not terminate, others might give a nonsense result.

Having these matrix orderings SINGULAR can compute standard bases for
any monomial ordering which is compatible with the natural semigroup structure.
In practice the global and local orderings together with block orderings should be
sufficient in most cases. These orderings are faster than the corresponding
matrix orderings, since evaluating a matrix product is time consuming.

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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
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