Products of posets

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 ([3]). Suppose $A$ and $B$ are posets. Then $A \times B$ has various orders; two of them being

• product order: $(a_1,b_1) \le (a_2,b_2)$ iff $a_1 \le a_2$ and $b_1 \le b_2$,
• lexicographic order: $(a_1,b_1) \le (a_2,b_2)$ if either $a_1 \le a_2$ or if $a_1 = a_2$ then $b_1 \le b_2$.

Exercise: check that these are orders. Note that there are no infinite descending chains in $\mathbb{N}\times \mathbb{N}$ under lexicographic order. Such posets are said to be well ordered. The principle of induction follows from well-ordering as discussed earlier.