- The
**lexicographical ordering***lp*, given by the matrix:

resp.

*ls*:

**Remark 1..1**The positive lexicraphic ordering*lp*on is an elimanation ordering forThe definition of rings with these orderings in SINGULAR:

(Each line starting with`//`is a comment in SINGULAR.)ring R1=0,(x(1..5)),lp; ring R2=0,(x(1..5)),ls;

- The
**weighted degree reverse lexicographical ordering**, given by the matrix

, (resp. ).If

*w*_{i}= 1 (respectively*w*_{i}= -1) for all*i*we obtain the**degree reverse lexicographical ordering, dp**(respectively**ds**).The definition of rings with these orderings in SINGULAR:

ring R3=0,(x(1..5)),wp(2,3,4,5,6); // correspond to w_i:2,3,4,5,6 ring R4=0,(x(1..5)),ws(2,3,4,5,6); // correspond to w_i:-2,-3,-4,-5,-6 ring R5=0,(x(1..4)),dp; ring R6=0,(x(1..4)),ds;

- An example for an
**elimination ordering**for in is given by the matrix

with . In it is given by the same matrix with and .The definition of a polynomial ring with an elimination ordering for

*x*_{3}and*x*_{4}in SINGULAR:ring E=0,(x(1..4)),(a(0,0,1,1),a(1,1),dp); // correspond to w_i=1 for all i, r=2 // or simpler: ring EE=0,(x(1..4)),(a(0,0,1,1),dp);

- The
**product ordering**, given by the matrix

if the*A*_{i}define orderings on monomials given by the corresponding subsets of . Such an ordering can be used to compute in- ( )
- ( )
- ( )

The definition of a ring with this ordering in SINGULAR:

ring P=0,(x(1..6)),(dp(4),ds(2)); // correspond to // a first block of 4 variables with ordering dp // and a second block of 2 variables with ordering ds