Well-quasi-ordering

A well-quasi-ordering on a set is a quasi-ordering (i.e., a reflexive, transitive binary relation) such that any infinite sequence of elements, … from contains an increasing pair ≤ with <. The set is said to be well-quasi-ordered, or shortly wqo.

A well partial order, or a wpo, is a wqo that is a proper ordering relation, i.e., it is antisymmetric.

Among other ways of defining wqo's, one is to say that they do not contain infinite strictly decreasing sequences (of the form >>>…) nor infinite sequences of pairwise incomparable elements. Hence a quasi-order (,≤) is wqo if and only if it is well-founded and has no infinite antichains.

Read more about Well-quasi-ordering:  Examples, Wqo's Versus Well Partial Orders, Infinite Increasing Subsequences, Properties of Wqos