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| In [[mathematics]], specifically [[order theory]], a '''well-quasi-ordering''' or '''wqo''' is a [[well-founded]] [[quasi-ordering]] with an additional restriction on sequences - that there is no infinite sequence <math>x_i</math> with <math> x_i \not \le x_j </math> for all <math> i < j </math>.
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| We can use [[well-founded induction]] 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 we use a quasi-order to obtain a new quasi-order on a set of structures derived from our original set, we find this quasiorder is 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.
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| An example of this is the power set operation. Given a quasiordering <math>\le</math> for a set <math>X</math> we 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> we can find some element of <math>B</math> which is larger than it under <math>\le</math>. We find that this quasiordering on <math>P(X)</math> needn't be well-founded but that if we took our original quasi-ordering to be a well-quasi-ordering then it is.
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| == Formal definition ==
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| 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</math>, <math>x_1</math>, <math>x_2</math>, … from <math>X</math> contains an increasing pair <math>x_i</math>≤<math>x_j</math> with <math>i</math><<math>j</math>. The set <math>X</math> is said to be '''well-quasi-ordered''', or shortly '''wqo'''.
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| A '''well partial order''', or a '''wpo''', is a wqo that is a proper ordering relation, i.e., it is [[antisymmetric relation|antisymmetric]].
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| Among other ways of defining wqo's, one is to say that they do not contain infinite ''strictly decreasing'' sequences (of the form
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| <math>x_0</math>><math>x_1</math>><math>x_2</math>>…) | |
| nor infinite sequences of ''pairwise incomparable'' elements. Hence a quasi-order (<math>X</math>,≤) is wqo if and only if it is [[well-founded relation|well-founded]] and has no infinite [[antichain]]s.
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| ==Examples== | |
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| * <math>(\mathbb{N}, \le)</math>, the set of natural numbers with standard ordering, is a well partial order. However, <math>(\mathbb{Z}, \le)</math>, the set of positive and negative integers, is '''not''' a well-quasi-order, because it is not well-founded.
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| * <math>(\mathbb{N}, \mid)</math>, the set of natural numbers ordered by divisibility, is '''not''' a well partial order: the prime numbers are an infinite antichain.
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| * <math>(\mathbb{N}^k, \le)</math>, the set of vectors of <math>k</math> natural numbers with [[product order|component-wise ordering]], is a well partial order ([[Dickson's lemma]]). 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>.
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| * 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, \dots</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
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| | last = Gasarch | first = W.
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| | contribution = A survey of recursive combinatorics
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| | doi = 10.1016/S0049-237X(98)80049-9
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| | location = Amsterdam
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| | mr = 1673598
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| | pages = 1041–1176
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| | publisher = North-Holland
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| | series = Stud. Logic Found. Math.
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| | title = Handbook of Recursive Mathematics, Vol. 2
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| | volume = 139 | |
| | year = 1998}}. See in particular page 1160.</ref> (If <math>X</math> has only one element, these three partial orders are identical.)
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| * 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 a finite unordered set, <math>u\le v</math> if and only if <math>u</math> is a subsequence of <math>v</math>.
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| * <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.
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| * Embedding between finite trees with nodes labeled by elements of a wqo <math>(X, \le)</math> is a wqo ([[Kruskal's tree theorem]]).
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| * 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).
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| * Embedding between countable [[scattered order|scattered]] [[linear order]] types is a well-quasi-order ([[Richard Laver|Laver]]'s theorem).
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| * Embedding between countable [[boolean algebras]] is a well-quasi-order. This follows from Laver's theorem and a theorem of Ketonen.
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| * Finite graphs ordered by a notion of embedding called "[[graph minor]]" is a well-quasi-order ([[Robertson–Seymour theorem]]).
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| * Graphs of finite [[tree-depth]] ordered by the [[induced subgraph]] relation form a well-quasi-order,<ref>{{citation
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| | last1 = Nešetřil | first1 = Jaroslav | author1-link = Jaroslav Nešetřil
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| | last2 = Ossona de Mendez | first2 = Patrice
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| | contribution = Lemma 6.13
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| | doi = 10.1007/978-3-642-27875-4
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| | isbn = 978-3-642-27874-7
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| | location = Heidelberg
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| | mr = 2920058
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| | pages = 137
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| | publisher = Springer
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| | series = Algorithms and Combinatorics
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| | title = Sparsity: Graphs, Structures, and Algorithms
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| | volume = 28
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| | year = 2012}}.</ref> as do the [[cograph]]s ordered by induced subgraphs.<ref>{{citation
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| | last = Damaschke | first = Peter
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| | doi = 10.1002/jgt.3190140406
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| | issue = 4
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| | journal = Journal of Graph Theory
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| | mr = 1067237
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| | pages = 427–435
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| | title = Induced subgraphs and well-quasi-ordering
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| | volume = 14
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| | year = 1990}}.</ref>
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| ==Wqo's versus well partial orders==
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| In practice, the wqo's one manipulates are almost always orderings (see examples above), but the theory is technically smoother if we do not require antisymmetry, so it is built with wqo's as the basic notion.
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| Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between
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| equivalence classes induced by the kernel of the wqo. For example, if we order <math>\mathbb{Z}</math> by divisibility, we end up with <math>n\equiv m</math>
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| if and only if <math>n=\pm m</math>, so that <math>(\mathbb{Z},\mid)\;\;\approx\;\;(\mathbb{N},\mid)</math>.
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| ==Infinite increasing subsequences==
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| If (<math>X</math>, ≤) is wqo then every infinite sequence <math>x_0</math>, <math>x_1</math>, <math>x_2</math>, … contains an '''infinite''' increasing subsequence <math>x_{n0}</math>≤<math>x_{n1}</math>≤<math>x_{n2}</math>≤…
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| (with <math>{n0}</math><<math>{n1}</math><<math>{n2}</math><…). Such a subsequence is sometimes called '''perfect'''.
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| 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.
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| The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering, leading to an equivalent notion.
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| == Properties of wqos ==
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| * 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"/>
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| * 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)
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| ==Notes==
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| {{Reflist|
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| refs=
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| <ref name="forster"> {{cite journal | last = Forster | first = Thomas | authorlink = Thomas Forster | title= Better-quasi-orderings and coinduction | journal = Theoretical Computer Science | year = 2003 | volume = 309 | pages= 111–123 | doi= 10.1016/S0304-3975(03)00131-2 | issue = 1–3}}</ref>
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| }}
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| ==References==
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| * {{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=413–422 | doi=10.2307/2370405 | jstor = 2370405 | issue = 4 }}
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| * {{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=326–336 | doi=10.1112/plms/s3-2.1.326}}
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| * {{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=297–305 | doi=10.1016/0097-3165(72)90063-5 | issue = 3}}
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| * {{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}}
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| * {{cite book | author=Milner, E. C. | 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=487–502 | publisher=D. Reidel Publishing Co. | isbn=90-277-1943-8}}
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| * {{cite journal | last = Gallier | first = Jean H. | 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}}
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| ==See also==
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| * [[Better-quasi-ordering]]
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| * [[Prewellordering]]
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| * [[Well-order]]
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| [[Category:Mathematical relations]]
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| [[Category:Order theory]]
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| [[Category:Wellfoundedness]]
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