In classical mathematics, a well-quasi-order is a preorder$(P, \leq)$ such that for any infinite sequence $x_i$ in $P$, there exist $i, j$ with $i \lt j$ and $x_i \leq x_j$. Another way of expressing this condition is:

There are no infinite antichains in $P$: if $x_i$ is a sequence in $P$, then there exist some pair of elements $x_i$, $x_j$ that are comparable, i.e., either $x_i \leq x_j$ or $x_j \leq x_i$, and

There is no strictly decreasing sequence in $P$: if $x_0 \geq x_1 \geq \ldots$ in $P$, then there exists $n$ such that $x_i \leq x_{i+1}$ for all $i \geq n$.

One motivation for this notion is given by the following theorem:

Theorem

Let $(X, \leq)$ be a quasi-order (i.e., a preorder). Define a partial order on the power set$P(X)$ by $A \leq^+ B$ if $\forall_{a\colon A} \exists_{b\colon B} (a \leq b)$. Then $\leq^+$ is a well-founded relation if and only if $\leq$ is a well-quasi-ordering.

Example

The collection of finite simple graphs is well-quasi-ordered by the graph minor relation. This is the celebrated Robertson-Seymour Graph Minor Theorem.