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In [[mathematics]], specifically [[order theory]], a '''well-quasi-ordering''' or '''wqo''' on a set <math>X</math> is a [[quasi-ordering]] of <math>X</math> for which every [[Infinity|infinite]] sequence of elements <math>x_0, x_1, x_2, \ldots</math> from <math>X</math> contains an increasing pair <math>x_i \leq x_j</math> with <math>i < j.</math>

==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 <math>\le</math> is said to be well-founded if the corresponding strict order <math>x\le y\land y\nleq x</math> 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 <math>\le</math> for a set <math>X</math> one can define a quasiorder <math>\le^{+}</math> on <math>X</math>'s power set <math>P(X)</math> by setting <math>A \le^{+} B</math> if and only if for each element of <math>A</math> one can find some element of <math>B</math> that is larger than it with respect to <math>\le</math>. One can show that this quasiordering on <math>P(X)</math> 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 <math>X</math> is a [[quasi-ordering]] (i.e., a [[reflexive relation|reflexive]], [[transitive relation|transitive]] [[binary relation]]) such that any [[Infinity|infinite]] sequence of elements <math>x_0, x_1, x_2, \ldots</math> from <math>X</math> contains an increasing pair <math>x_i \le x_j</math> with <math>i< j</math>. The set <math>X</math> 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 relation|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 <math>x_0> x_1> x_2> \cdots</math>){{ref|a}} nor infinite sequences of ''pairwise incomparable'' elements. Hence a quasi-order (''X'', ≤) is wqo if and only if (''X'', <) is [[well-founded relation|well-founded]] and has no infinite [[antichain]]s.

==Ordinal type==

Let <math>X</math> be well partially ordered. A (necessarily finite) sequence <math>(x_1, x_2, \ldots, x_n)</math> of elements of <math>X</math> that contains no pair <math>x_i \le x_j</math> with <math>i< j</math> is usually called a ''bad sequence''. The ''tree of bad sequences'' <math>T_X</math> is the tree that contains a vertex for each bad sequence, and an edge joining each nonempty bad sequence <math>(x_1, \ldots, x_{n-1}, x_n)</math> to its parent <math>(x_1, \ldots, x_{n-1})</math>. The root of <math>T_X</math> corresponds to the empty sequence. Since <math>X</math> contains no infinite bad sequence, the tree <math>T_X</math> contains no infinite path starting at the root.{{cn|reason=This is far from being obvious, and requires a (reference to a) proof.|date=February 2024}} Therefore, each vertex <math>v</math> of <math>T_X</math> has an ordinal height <math>o(v)</math>, which is defined by transfinite induction as <math>o(v) = \lim_{w \mathrm{\ child\ of\ } v} (o(w)+1)</math>. The ''ordinal type'' of <math>X</math>, denoted <math>o(X)</math>, is the ordinal height of the root of <math>T_X</math>.

A ''linearization'' of <math>X</math> is an extension of the partial order into a total order. It is easy to verify that <math>o(X)</math> is an upper bound on the ordinal type of every linearization of <math>X</math>. De Jongh and Parikh<ref name="dejongh_parikh">{{cite journal |last1=de Jongh |first1=Dick H. G. |last2=Parikh |first2=Rohit |title=Well-partial orderings and hierarchies |journal=Indagationes Mathematicae (Proceedings) |date=1977 |volume=80 |issue=3 |pages=195-207 |doi=10.1016/1385-7258(77)90067-1|doi-access=free }}</ref> proved that in fact there always exists a linearization of <math>X</math> that achieves the maximal ordinal type <math>o(X)</math>.

==Examples==

[[File:Integers-line.svg|thumb|200px|'''Pic.1:''' Integer numbers with the usual order]]
[[File:Infinite lattice of divisors.svg|thumb|200px|'''Pic.2:''' [[Hasse diagram]] of the natural numbers ordered by divisibility]]
[[File:N-Quadrat, gedreht.svg|thumb|200px|'''Pic.3:''' Hasse diagram of <math>\N^2</math> with componentwise order]]

* <math>(\N, \le)</math>, the set of natural numbers with standard ordering, is a well partial order (in fact, a [[well-order]]). However, <math>(\Z, \le)</math>, the set of positive and negative integers, is '''not''' a well-quasi-order, because it is not well-founded (see Pic.1).
* <math>(\N, |)</math>, the set of natural numbers ordered by divisibility, is '''not''' a well-quasi-order: the prime numbers are an infinite antichain (see Pic.2).
* <math>(\N^k, \le)</math>, the set of vectors of <math>k</math> natural numbers (where <math>k</math> is finite) with [[product order|component-wise ordering]], is a well partial order ([[Dickson's lemma]]; see Pic.3). More generally, if <math>(X, \le)</math> is well-quasi-order, then <math>(X^k,\le^k)</math> is also a well-quasi-order for all <math>k</math>.
* Let <math>X</math> be an arbitrary finite set with at least two elements. The set [[Kleene star|<math>X^*</math>]] of [[Formal language#Words_over_an_alphabet|words over]] <math>X</math> ordered [[lexicographical order|lexicographically]] (as in a dictionary) is '''not''' a well-quasi-order because it contains the infinite decreasing sequence <math>b, ab, aab, aaab, \ldots</math>. Similarly, <math>X^*</math> ordered by the [[prefix (computer science)#Prefix|prefix]] relation is '''not''' a well-quasi-order, because the previous sequence is an infinite antichain of this partial order. However, <math>X^*</math> ordered by the [[subsequence]] relation is a well partial order.<ref>{{citation | last = Gasarch | first = W. | author1-link=William Gasarch  | contribution = A survey of recursive combinatorics | doi = 10.1016/S0049-237X(98)80049-9 | location = Amsterdam | mr = 1673598 | pages = 1041–1176 | publisher = North-Holland | series = Stud. Logic Found. Math. | title = Handbook of Recursive Mathematics, Vol. 2 | volume = 139 | year = 1998}}. See in particular page 1160.</ref>  (If <math>X</math> has only one element, these three partial orders are identical.)
* More generally, <math>(X^*,\le)</math>, the set of finite <math>X</math>-sequences ordered by [[embedding]] is a well-quasi-order if and only if <math>(X, \le)</math> is a well-quasi-order ([[Higman's lemma]]). Recall that one embeds a sequence <math>u</math> into a sequence <math>v</math> by finding a subsequence of <math>v</math> that has the same length as <math>u</math> and that dominates it term by term. When <math>(X,=)</math> is an unordered set, <math>u\le v</math> if and only if <math>u</math> is a subsequence of <math>v</math>.
* <math>(X^\omega,\le)</math>, the set of infinite sequences over a well-quasi-order <math>(X, \le)</math>, 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-ordering]]s 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 <math>(X, \le)</math> is a wqo ([[Kruskal's tree theorem]]).
* Embedding between infinite trees with nodes labeled by elements of a wqo <math>(X, \le)</math> is a wqo ([[Crispin St. J. A. Nash-Williams|Nash-Williams]]' theorem).
* Embedding between countable [[scattered order|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,<ref>{{citation | 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
 | series = Algorithms and Combinatorics | title = Sparsity: Graphs, Structures, and Algorithms | volume = 28 | year = 2012 }}.</ref> as do the [[cograph]]s ordered by induced subgraphs.<ref>{{citation | last = Damaschke | first = Peter | doi = 10.1002/jgt.3190140406 | issue = 4 | journal = [[Journal of Graph Theory]] | mr = 1067237 | pages = 427–435 | title = Induced subgraphs and well-quasi-ordering | volume = 14 | year = 1990}}.</ref>

==Constructing new wpo's from given ones==

Let <math>X_1</math> and <math>X_2</math> be two disjoint wpo sets. Let <math>Y=X_1\cup X_2</math>, and define a partial order on <math>Y</math> by letting <math>y_1\le_Y y_2</math> if and only if <math>y_1,y_2\in X_i</math> for the same <math>i\in\{1,2\}</math> and <math>y_1 \le_{X_i} y_2</math>. Then <math>Y</math> is wpo, and <math>o(Y) = o(X_1) \oplus o(X_2)</math>, where <math>\oplus</math>  denotes [[Ordinal arithmetic|natural sum]] of ordinals.<ref name="dejongh_parikh" />

Given wpo sets <math>X_1</math> and <math>X_2</math>, define a partial order on the Cartesian product <math>Y=X_1\times X_2</math>, by letting <math>(a_1,a_2)\le_Y (b_1,b_2)</math> if and only if <math>a_1\le_{X_1} b_1</math> and <math>a_2\le_{X_2} b_2</math>. Then <math>Y</math> is wpo (this is a generalization of [[Dickson's lemma]]), and <math>o(Y) = o(X_1)\otimes o(X_2)</math>, where <math>\otimes</math> denotes [[Ordinal arithmetic|natural product]] of ordinals.<ref name="dejongh_parikh" />

Given a wpo set <math>X</math>, let <math>X^*</math> be the set of finite sequences of elements of <math>X</math>, partially ordered by the subsequence relation. Meaning, let <math>(x_1,\ldots,x_n)\le_{X^*} (y_1,\ldots,y_m)</math> if and only if there exist indices <math>1\le i_1<\cdots<i_n\le m</math> such that <math>x_j \le_X y_{i_j}</math> for each <math>1\le j\le n</math>. By [[Higman's lemma]], <math>X^*</math> is wpo. The ordinal type of <math>X^*</math> is<ref name="dejongh_parikh" /><ref name="schmidt">{{cite thesis |last=Schmidt |first=Diana |type=Habilitationsschrift |date=1979 |title=Well-partial orderings and their maximal order types |publisher=Heidelberg}} Republished in: {{cite book |last=Schmidt |first=Diana |date=2020 |chapter=Well-partial orderings and their maximal order types |title=Well-Quasi Orders in Computation, Logic, Language and Reasoning |publisher=Springer |pages=351-391 |editor-last1=Schuster |editor-first1=Peter M. |editor-last2=Seisenberger |editor-first2=Monika |editor-last3=Weiermann |editor-first3=Andreas |series=Trends in Logic |volume=53 |doi=10.1007/978-3-030-30229-0_13}}</ref> <math display="block">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}</math>

Given a wpo set <math>X</math>, let <math>T(X)</math> be the set of all finite rooted trees whose vertices are labeled by elements of <math>X</math>. Partially order <math>T(X)</math> by the [[Kruskal's tree theorem|tree embedding relation]]. By [[Kruskal's tree theorem]], <math>T(X)</math> is wpo. This result is nontrivial even for the case <math>|X|=1</math> (which corresponds to unlabeled trees), in which case <math>o(T(X))</math> equals the [[small Veblen ordinal]]. In general, for <math>o(X)</math> countable, we have the upper bound <math>o(T(X))\le\vartheta(\Omega^\omega o(X))</math> in terms of the <math>\vartheta</math> [[ordinal collapsing function]]. (The small Veblen ordinal equals <math>\vartheta(\Omega^\omega)</math> in this ordinal notation.)<ref>{{cite journal |last1=Rathjen |first1=Michael |last2=Weiermann |first2=Andreas |title=Proof-theoretic investigations on Kruskal's theorem |journal=Annals of Pure and Applied Logic |date=1993 |volume=60 |pages=49-88 |doi=10.1016/0168-0072(93)90192-G|doi-access=free }}</ref>

==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{{fact|date=March 2017}} 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, if we order <math>\Z</math> by divisibility, we end up with <math>n\equiv m</math> if and only if <math>n=\pm m</math>, so that <math>(\Z,|)\approx(\N,|)</math>.

==Infinite increasing subsequences==
If <math>(X, \le)</math> is wqo then every infinite sequence <math>x_0, x_1, x_2, \ldots,</math> contains an '''infinite''' increasing subsequence <math>x_{n_0} \le x_{n_1}\le x_{n_2} \le \cdots</math> (with <math>n_0< n_1< n_2< \cdots</math>). Such a subsequence is sometimes called '''perfect'''.
This can be proved by a [[Ramsey theory|Ramsey argument]]: given some sequence <math>(x_i)_i</math>, consider the set <math>I</math> of indexes <math>i</math> such that <math>x_i</math> has no larger or equal <math>x_j</math> to its right, i.e., with <math>i<j</math>. If <math>I</math> is infinite, then the <math>I</math>-extracted subsequence contradicts the assumption that <math>X</math> is wqo. So <math>I</math> is finite, and any <math>x_n</math> with <math>n</math> larger than any index in <math>I</math> 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 <math>(X,\le)</math> the quasiordering <math>(P(X), \le^+)</math> defined by  <math> A \le^+ B \iff \forall a \in A, \exists b \in B, a \le b</math> is well-founded if and only if <math>(X,\le)</math> is a wqo.<ref name="forster"/>
* A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by <math>x \sim y \iff x\le y \land y \le x</math>) has no infinite descending sequences or [[antichain]]s. (This can be proved using a [[Ramsey theory|Ramsey argument]] as above.)
* Given a well-quasi-ordering <math>(X,\le)</math>, any sequence of upward-closed subsets <math>S_0 \subseteq S_1 \subseteq \cdots \subseteq X</math> eventually stabilises (meaning there exists <math>n \in \N</math> such that <math>S_n = S_{n+1} = \cdots</math>; a subset <math>S \subseteq X</math> is called ''upward-[[closure (mathematics)|closed]]'' if <math>\forall x,y \in X, x \le y \wedge x \in S \Rightarrow y \in S</math>): assuming the contrary <math>\forall i \in \N, \exists j \in \N, j > i, \exists x \in S_j \setminus S_i</math>, a contradiction is reached by extracting an infinite non-ascending subsequence.
* Given a well-quasi-ordering <math>(X,\le)</math>, any subset <math>S</math> of <math>X</math> has a finite number of minimal elements with respect to <math>\le</math>, for otherwise the minimal elements of <math>S</math> would constitute an infinite antichain.

==See also==

* {{annotated link|Better-quasi-ordering}}
* {{annotated link|Prewellordering}}
* {{annotated link|Well-order}}

==Notes==

{{note|a}}Here ''x'' < ''y'' means: <math>x\le y</math> and <math>x \neq y.</math>

==References==

{{Reflist|
refs=
<ref name="forster"> {{cite journal | last = Forster | first = Thomas | authorlink = Thomas Forster (mathematician)| title= Better-quasi-orderings and coinduction | journal = Theoretical Computer Science | year = 2003 | volume = 309 | pages= 111&ndash;123 | doi= 10.1016/S0304-3975(03)00131-2 | issue = 1&ndash;3| doi-access =  }}</ref>
}}

* {{cite journal | last = Dickson | first = L. E. | authorlink = Leonard Dickson | title=Finiteness of the odd perfect and primitive abundant numbers with ''r'' distinct prime factors | journal=[[American Journal of Mathematics]] | year = 1913 | volume = 35 | pages=413&ndash;422 | doi=10.2307/2370405 | jstor = 2370405 | issue = 4 }}
* {{cite journal | last = Higman | first = G. |authorlink = Graham Higman | title=Ordering by divisibility in abstract algebras | journal=Proceedings of the London Mathematical Society | year=1952 | volume=2 | pages=326&ndash;336 | doi=10.1112/plms/s3-2.1.326}}
* {{cite journal | authorlink = Joseph Kruskal | last=Kruskal |first= J. B. | title=The theory of well-quasi-ordering: A frequently discovered concept | journal=[[Journal of Combinatorial Theory]] | series = Series A | year=1972 | volume=13 | pages=297&ndash;305 | doi=10.1016/0097-3165(72)90063-5 | issue = 3| doi-access=free }}
* {{cite journal | last = Ketonen | first = Jussi | title = The structure of countable Boolean algebras | journal = [[Annals of Mathematics]] | volume = 108 | pages = 41–89 | year = 1978  | doi = 10.2307/1970929 | jstor = 1970929 | issue = 1}}
* {{cite book | last=Milner | first = E. C. | authorlink = Eric Charles Milner | year=1985 | chapter = Basic WQO- and BQO-theory | editor-first=I.|editor-last=Rival|editor-link=Ivan Rival | title = Graphs and Order. The Role of Graphs in the Theory of Ordered Sets and Its Applications | pages=487&ndash;502 | publisher=D. Reidel Publishing Co. | isbn=90-277-1943-8}}
* {{cite journal | last = Gallier | first = Jean H. |authorlink= Jean Gallier | title= What's so special about Kruskal's theorem and the ordinal Γo? A survey of some results in proof theory | journal = Annals of Pure and Applied Logic | year = 1991 | volume = 53 | pages= 199–260 | doi= 10.1016/0168-0072(91)90022-E | issue = 3| doi-access =  }}

{{Order theory}}

[[Category:Order theory]]
[[Category:Wellfoundedness]]