Well-founded order

Remarks

Example: the set $\N^2$ with the product order, $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$, is well-founded. Check: in a nonempty $A \subseteq \N^2$, take the least first coordinate $a_0$ occurring in $A$, and then the least $b_0$ with $(a_0, b_0) \in A$; no element of $A$ is strictly below $(a_0, b_0)$.

Non-example: the set $\set{1/n : n \ge 1}$ of rational numbers with the usual order is not well-founded, since the whole set has no minimal element.

The order is well-founded exactly when the strict order $<$ is a well-founded relation. With the axiom of dependent choice, the condition is equivalent to the absence of an infinite sequence $x_0 > x_1 > x_2 > \cdots$. The same definition applies to a preorder, with $x < y$ meaning $x \le y$ and not $y \le x$.