Well-quasi-ordering

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. (Here, by abuse of terminology, a quasiorder \le is said to be well-founded if the corresponding strict order x\le y\land y\nleq x is a well-founded relation.) 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 \le for a set X one can define a quasiorder \le^{+} on X's power set P(X) by setting A \le^{+} B if and only if for each element of A one can find some element of B that is larger than it with respect to \le. One can show that this quasiordering on P(X) need not 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, \ldots from X contains an increasing pair x_i \le x_j with i. 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 \cdots) nor infinite sequences of pairwise incomparable elements. Hence a quasi-order (X, ≤) is wqo if and only if (X,

Ordinal type

Let X be well partially ordered. A (necessarily finite) sequence (x_1, x_2, \ldots, x_n) of elements of X that contains no pair x_i \le x_j with i is usually called a bad sequence. The tree of bad sequences T_X is the tree that contains a vertex for each bad sequence, and an edge joining each nonempty bad sequence (x_1, \ldots, x_{n-1}, x_n) to its parent (x_1, \ldots, x_{n-1}). The root of T_X corresponds to the empty sequence. Since X contains no infinite bad sequence, the tree T_X contains no infinite path starting at the root.{{cite journal

A linearization of X is an extension of the partial order into a total order. It is easy to verify that o(X) is an upper bound on the ordinal type of every linearization of X. De Jongh and Parikh proved that in fact there always exists a linearization of X that achieves the maximal ordinal type o(X).

Examples

• (\N, \le), the set of natural numbers with standard ordering, is a well partial order (in fact, a well-order). However, (\Z, \le), the set of positive and negative integers (see Pic.1), is not a well-quasi-order, because it is not well-founded. The infinite sequence -1, -2,... contains no increasing pair.
• (\N, |), the set of natural numbers ordered by divisibility, is not a well-quasi-order: the prime numbers are an infinite antichain (see Pic.2).
• (\N^k, \le), the set of vectors of k natural numbers (where k is finite) with component-wise ordering, is a well partial order (Dickson's lemma; see Pic.3). More generally, if (X, \le) is well-quasi-order, then (X^k,\le^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, \ldots. 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. (If X has only one element, these three partial orders are identical.)
• More generally, (X^*,\le), the set of finite X-sequences ordered by embedding is a well-quasi-order if and only if (X, \le) 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 an unordered set, u\le v if and only if u is a subsequence of v.
• (X^\omega,\le), the set of infinite sequences over a well-quasi-order (X, \le), 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, \le) is a wqo (Kruskal's tree theorem).
• Embedding between infinite trees with nodes labeled by elements of a wqo (X, \le) 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,{{cite book | last1 = Nešetřil | first1 = Jaroslav | author1-link = Jaroslav Nešetřil | last2 = Ossona de Mendez | first2 = Patrice | author2-link = Patrice Ossona de Mendez | contribution = Lemma 6.13 | doi = 10.1007/978-3-642-27875-4 | isbn = 978-3-642-27874-7 | location = Heidelberg | mr = 2920058 | pages = 137 | publisher = Springer

Constructing new wpo's from given ones

Let X_1 and X_2 be two disjoint wpo sets. Let Y=X_1\cup X_2, and define a partial order on Y by letting y_1\le_Y y_2 if and only if y_1,y_2\in X_i for the same i\in{1,2} and y_1 \le_{X_i} y_2. Then Y is wpo, and o(Y) = o(X_1) \oplus o(X_2), where \oplus denotes natural sum of ordinals.

Given wpo sets X_1 and X_2, define a partial order on the Cartesian product Y=X_1\times X_2, by letting (a_1,a_2)\le_Y (b_1,b_2) if and only if a_1\le_{X_1} b_1 and a_2\le_{X_2} b_2. Then Y is wpo (this is a generalization of Dickson's lemma), and o(Y) = o(X_1)\otimes o(X_2), where \otimes denotes natural product of ordinals.

Given a wpo set X, let X^* be the set of finite sequences of elements of X, partially ordered by the subsequence relation. Meaning, let (x_1,\ldots,x_n)\le_{X^} (y_1,\ldots,y_m) if and only if there exist indices 1\le i_1 such that x_j \le_X y_{i_j} for each 1\le j\le n. By Higman's lemma, X^ is wpo. The ordinal type of X^* is o(X^*)=\begin{cases}\omega^{\omega^{o(X)-1}},&o(X) \text{ finite};\ \omega^{\omega^{o(X)+1}},&o(X)=\varepsilon_\alpha+n \text{ for some }\alpha\text{ and some finite }n;\ \omega^{\omega^{o(X)}},&\text{otherwise}.\end{cases}

Given a wpo set X, let T(X) be the set of all finite rooted trees whose vertices are labeled by elements of X. Partially order T(X) by the tree embedding relation. By Kruskal's tree theorem, T(X) is wpo. This result is nontrivial even for the case |X|=1 (which corresponds to unlabeled trees), in which case o(T(X)) equals the small Veblen ordinal. In general, for o(X) countable, we have the upper bound o(T(X))\le\vartheta(\Omega^\omega o(X)) in terms of the \vartheta ordinal collapsing function. (The small Veblen ordinal equals \vartheta(\Omega^\omega) in this ordinal notation.)

Wqo's versus well partial orders

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, if we order \Z by divisibility, we end up with n\equiv m if and only if n=\pm m, so that (\Z,|)\approx(\N,|).

Infinite increasing subsequences

If (X, \le) is wqo then every infinite sequence x_0, x_1, x_2, \ldots, contains an infinite increasing subsequence x_{n_0} \le x_{n_1}\le x_{n_2} \le \cdots (with n_0). 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. 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,\le) the quasiordering (P(X), \le^+) defined by A \le^+ B \iff \forall a \in A, \exists b \in B, a \le b is well-founded if and only if (X,\le) is a wqo.
• A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by x \sim y \iff x\le y \land y \le x) has no infinite descending sequences or antichains. (This can be proved using a Ramsey argument as above.)
• Given a well-quasi-ordering (X,\le), any sequence of upward-closed subsets S_0 \subseteq S_1 \subseteq \cdots \subseteq X eventually stabilises (meaning there exists n \in \N such that S_n = S_{n+1} = \cdots; a subset S \subseteq X is called upward-closed if \forall x,y \in X, x \le y \wedge x \in S \Rightarrow y \in S): assuming the contrary \forall i \in \N, \exists j \in \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,\le), any subset S of X has a finite number of minimal elements with respect to \le, for otherwise the minimal elements of S would constitute an infinite antichain.

Notes

References

References

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2. Gasarch, W.. (1998). "Handbook of Recursive Mathematics, Vol. 2". North-Holland.
3. Damaschke, Peter. (1990). "Induced subgraphs and well-quasi-ordering". [[Journal of Graph Theory]].
4. Schmidt, Diana. (1979). "Well-partial orderings and their maximal order types". Heidelberg}} Republished in: {{cite book.
5. (1993). "Proof-theoretic investigations on Kruskal's theorem". Annals of Pure and Applied Logic.
6. Milner, E. C.. (1985). "Graphs and Order. The Role of Graphs in the Theory of Ordered Sets and Its Applications". D. Reidel Publishing Co..
7. Forster, Thomas. (2003). "Better-quasi-orderings and coinduction". Theoretical Computer Science.