Well-quasi-ordering

In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements $x_{0}$ , $x_{1}$ , $x_{2}$ , … from $X$ contains an increasing pair $x_{i}\leq x_{j}$ with $i<j$ .

Motivation

Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. However the class of well-founded quasiorders is not closed under certain operations - that is, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, this quasiorder is found to be not well-founded. By placing stronger restrictions on the original well-founded quasiordering one can hope to ensure that our derived quasiorderings are still well-founded.

An example of this is the power set operation. Given a quasiordering $\leq$ for a set $X$ one can define a quasiorder $\leq ^{+}$ on $X$ 's power set $P(X)$ by setting $A\leq ^{+}B$ if and only if for each element of $A$ one can find some element of $B$ which is larger than it under $\leq$ . One can show that this quasiordering on $P(X)$ needn't be well-founded, but if one takes the original quasi-ordering to be a well-quasi-ordering, then it is.

Formal definition

A well-quasi-ordering on a set $X$ is a quasi-ordering (i.e., a reflexive, transitive binary relation) such that any infinite sequence of elements $x_{0}$ , $x_{1}$ , $x_{2}$ , … from $X$ contains an increasing pair $x_{i}$ ≤ $x_{j}$ with $i$ < $j$ . The set $X$ 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 are quasi-orderings which do not contain infinite strictly decreasing sequences (of the form $x_{0}$ > $x_{1}$ > $x_{2}$ >…) nor infinite sequences of pairwise incomparable elements. Hence a quasi-order ( $X$ ,≤) is wqo if and only if it is well-founded and has no infinite antichains.

Examples

$(\mathbb {N} ,\leq )$ , the set of natural numbers with standard ordering, is a well partial order (in fact, a well-order). However, $(\mathbb {Z} ,\leq )$ , the set of positive and negative integers, is not a well-quasi-order, because it is not well-founded.
$(\mathbb {N} ,\mid )$ , the set of natural numbers ordered by divisibility, is not a well partial order: the prime numbers are an infinite antichain.
$(\mathbb {N} ^{k},\leq )$ , the set of vectors of $k$ natural numbers (where $k$ is finite) with component-wise ordering, is a well partial order (Dickson's lemma). More generally, if $(X,\leq )$ is well-quasi-order, then $(X^{k},\leq ^{k})$ is also a well-quasi-order for all $k$ .
Let $X$ be an arbitrary finite set with at least two elements. The set $X^{*}$ of words over $X$ ordered lexicographically (as in a dictionary) is not a well-quasi-order because it contains the infinite decreasing sequence $b,ab,aab,aaab,\dots$ . Similarly, $X^{*}$ ordered by the prefix relation is not a well-quasi-order, because the previous sequence is an infinite antichain of this partial order. However, $X^{*}$ ordered by the subsequence relation is a well partial order.^[1] (If $X$ has only one element, these three partial orders are identical.)
More generally, $(X^{*},\leq )$ , the set of finite $X$ -sequences ordered by embedding is a well-quasi-order if and only if $(X,\leq )$ is a well-quasi-order (Higman's lemma). Recall that one embeds a sequence $u$ into a sequence $v$ by finding a subsequence of $v$ that has the same length as $u$ and that dominates it term by term. When $(X,=)$ is a finite unordered set, $u\leq v$ if and only if $u$ is a subsequence of $v$ .
$(X^{\omega },\leq )$ , the set of infinite sequences over a well-quasi-order $(X,\leq )$ , 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 $(X,\leq )$ is a wqo (Kruskal's tree theorem).
Embedding between infinite trees with nodes labeled by elements of a wqo $(X,\leq )$ 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,^[2] as do the cographs ordered by induced subgraphs.^[3]

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.

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, if we order $\mathbb {Z}$ by divisibility, we end up with $n\equiv m$ if and only if $n=\pm m$ , so that $(\mathbb {Z} ,\mid )\;\;\approx \;\;(\mathbb {N} ,\mid )$ .

Infinite increasing subsequences

If ( $X$ , ≤) is wqo then every infinite sequence $x_{0}$ , $x_{1}$ , $x_{2}$ , … contains an infinite increasing subsequence $x_{n0}$ ≤ $x_{n1}$ ≤ $x_{n2}$ ≤… (with ${n0}$ < ${n1}$ < ${n2}$ <…). Such a subsequence is sometimes called perfect. This can be proved by a Ramsey argument: given some sequence $(x_{i})_{i}$ , consider the set $I$ of indexes $i$ such that $x_{i}$ has no larger or equal $x_{j}$ to its right, i.e., with $i<j$ . If $I$ is infinite, then the $I$ -extracted subsequence contradicts the assumption that $X$ is wqo. So $I$ is finite, and any $x_{n}$ with $n$ larger than any index in $I$ can be used as the starting point of an infinite increasing subsequence.

The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering, leading to an equivalent notion.

Properties of wqos

Given a quasiordering $(X,\leq )$ the quasiordering $(P(X),\leq ^{+})$ defined by $A\leq ^{+}B\iff \forall a\in A\exists b\in B(a\leq b)$ is well-founded if and only if $(X,\leq )$ is a wqo.^[4]
A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by $x\sim y\iff x\leq y\land y\leq x$ ) has no infinite descending sequences or antichains. (This can be proved using a Ramsey argument as above.)
Given a well-quasi-ordering $(X,\leq )$ , any sequence of subsets $S_{0}\subseteq S_{1}\subseteq ...\subseteq X$ such that $\forall i\in \mathbb {N} ,\forall x,y\in X,x\leq y\wedge x\in S_{i}\Rightarrow y\in S_{i}$ eventually stabilises (meaning there is an index $n\in \mathbb {N}$ such that $S_{n}=S_{n+1}=...$ ; subsets $S\subseteq X$ with the property $\forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S$ are usually called upward-closed): assuming the contrary $\forall i\in \mathbb {N} \exists j\in \mathbb {N} ,j>i,\exists x\in S_{j}\setminus S_{i}$ , a contradiction is reached by extracting an infinite non-ascending subsequence.
Given a well-quasi-ordering $(X,\leq )$ , any subset $S\subseteq X$ which is upward-closed with respect to $\leq$ has a finite number of minimal elements w.r.t. $\leq$ , for otherwise the minimal elements of $S$ would constitute an infinite antichain.

Notes

^ Gasarch, W. (1998), "A survey of recursive combinatorics", Handbook of Recursive Mathematics, Vol. 2, Stud. Logic Found. Math., vol. 139, Amsterdam: North-Holland, pp. 1041–1176, doi:10.1016/S0049-237X(98)80049-9, MR 1673598. See in particular page 1160.
^ Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), "Lemma 6.13", Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Heidelberg: Springer, p. 137, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058.
^ Damaschke, Peter (1990), "Induced subgraphs and well-quasi-ordering", Journal of Graph Theory, 14 (4): 427–435, doi:10.1002/jgt.3190140406, MR 1067237.
^ Forster, Thomas (2003). "Better-quasi-orderings and coinduction". Theoretical Computer Science. 309 (1–3): 111–123. doi:10.1016/S0304-3975(03)00131-2.

References

Dickson, L. E. (1913). "Finiteness of the odd perfect and primitive abundant numbers with r distinct prime factors". American Journal of Mathematics. 35 (4): 413–422. doi:10.2307/2370405. JSTOR 2370405.
Higman, G. (1952). "Ordering by divisibility in abstract algebras". Proceedings of the London Mathematical Society. 2: 326–336. doi:10.1112/plms/s3-2.1.326.
Kruskal, J. B. (1972). "The theory of well-quasi-ordering: A frequently discovered concept". Journal of Combinatorial Theory. Series A. 13 (3): 297–305. doi:10.1016/0097-3165(72)90063-5.
Ketonen, Jussi (1978). "The structure of countable Boolean algebras". Annals of Mathematics. 108 (1): 41–89. doi:10.2307/1970929. JSTOR 1970929.
Milner, E. C. (1985). "Basic WQO- and BQO-theory". In Rival, I. (ed.). Graphs and Order. The Role of Graphs in the Theory of Ordered Sets and Its Applications. D. Reidel Publishing Co. pp. 487–502. ISBN 90-277-1943-8.
Gallier, Jean H. (1991). "What's so special about Kruskal's theorem and the ordinal Γo? A survey of some results in proof theory". Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E.