Previous: Chains and antichains.
The construction showed in this subsection has inportant applications. In particular, the generalization of Euclides's Algorithm to mulivariables polynomials
demands the definition of an ordering on the variables and their power products. This is a cornerstone of the theory of Gröbner bases ().
Suppose and are posets. Then
has various orders;
two of them being
Exercise: check that these are orders.
Note that there are no infinite descending chains in
under lexicographic order. Such posets are said to be well ordered. The
principle of induction follows from well-ordering as discussed earlier.
- product order:
- lexicographic order:
or if then