Pic.3). More generally, if is well-quasi-order, then is also a well-quasi-order for all . Let be an arbitrary finite set with at least two elements. The set of words over ordered lexicographically (as in a dictionary) is not a well-quasi-order because it contains the infinite decreasing sequence . Similarly, ordered by the prefix relation is not a well-quasi-order, because the previous sequence is an infinite antichain of this partial order. However, ordered by the subsequence relation is a well partial order. (If has only one element, these three partial orders are identical.) More generally, , the set of finite -sequences ordered by embedding is a well-quasi-order if and only if is a well-quasi-order (Higman's lemma). Recall that one embeds a sequence into a sequence by finding a subsequence of that has the same length as and that dominates it term by term. When is an unordered set, if and only if is a subsequence of . , the set of infinite sequences over a well-quasi-order , ordered by embedding, is not a well-quasi-order in general. That is, Higman's lemma does not carry over to infinite sequences. Better-quasi-orderings have been introduced to generalize Higman's lemma to sequences of arbitrary lengths. Embedding between finite trees with nodes labeled by elements of a wqo is a wqo (Kruskal's tree theorem). Embedding between infinite trees with nodes labeled by elements of a wqo is a wqo (Nash-Williams' theorem). Embedding between countable scattered linear order types is a well-quasi-order (Laver's theorem). Embedding between countable boolean algebras is a well-quasi-order. This follows from Laver's theorem and a theorem of Ketonen. Finite graphs ordered by a notion of embedding called "graph minor" is a well-quasi-order (Robertson–Seymour theorem). Graphs of finite tree-depth ordered by the induced subgraph relation form a well-quasi-order, as do the cographs ordered by induced subgraphs. Wqo's versus well partial orders In practice, the wqo's one manipulates are quite often not orderings (see examples above), and the theory is technically smoother if we do not require antisymmetry, so it is built with wqo's as the basic notion. On the other hand, according to Milner 1985, no real gain in generality is obtained by considering quasi-orders rather than partial orders... it is simply more convenient to do so. Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes induced by the kernel of the wqo. For example,