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In [[order theory]] a '''better-quasi-ordering''' or '''bqo''' is a [[quasi-ordering]] that does not admit a certain type of bad array. Every bqo is [[well-quasi-order]]ed.
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== Motivation ==
 
Though [[wqo]] is an appealing notion, many important infinitary operations do not preserve [[wqo]]ness.
An example due to [[Richard Rado]] illustrates this.<ref name="Rado54"/>
In a 1965 paper [[Nash-Williams|Crispin Nash-Williams]] formulated the stronger notion of bqo in order to prove that the class of [[Tree (set theory)|tree]]s of height ω is [[wqo]] under the [[topological minor]] relation.<ref name="Nash-Williams65"/> Since then, many [[quasi-order]]s have been proven to be [[wqo]] by proving them to be bqo. For instance, [[Richard Laver]] established [[Roland Fraïssé|Fraïssé's]] conjecture by proving that the class of [[scattered]] [[linear order]] types is bqo.<ref name="Laver71"/> More recently, Carlos Martinez-Ranero has proven that, under the [[Proper Forcing Axiom]], the class of [[Aronszajn line]]s is bqo under the embeddability relation.<ref name="Martinez-Ranero2011"/>
 
== Definition ==
 
It is common in bqo theory to write <math> {_*}x </math> for the sequence <math>x</math> with the first term omitted. Write <math>[\omega]^{<\omega}</math> for the set of finite, strictly increasing sequences with terms in <math>\omega</math>, and define a relation <math>\triangleleft</math> on <math>[\omega]^{<\omega}</math> as follows: <math>s\triangleleft t</math> if and only if there is <math>u</math> such that <math>s</math> is a strict initial segment of <math>u</math> and <math>t={}_*u</math>. Note that the relation <math>\triangleleft</math> is not [[Transitive relation|transitive]].
 
A ''block'' is a subset ''B'' of <math>[\omega]^{<\omega}</math> that contains an initial segment of every
infinite subset of <math>\bigcup B</math>. For a [[quasi-order]] <math>Q</math> a ''<math>Q</math>-pattern'' is a function from a block ''B'' into <math>Q</math>. A <math>Q</math>-pattern <math>f\colon B\to Q</math> is said to be ''bad'' if <math>f(s)\not \le_Q f(t)</math> for every pair <math>s,t\in B</math> such that <math>s\triangleleft t</math>; otherwise <math>f</math> is ''good''. A [[quasi-order]] <math>Q</math> is ''better-quasi-ordered'' (''bqo'') if there is no bad <math>Q</math>-pattern.
 
In order to make this definition easier to work with, Nash-Williams defines a ''barrier'' to be a block whose elements are pairwise [[Comparability|incomparable]] under the inclusion relation <math>\subset</math>. A ''<math>Q</math>-array'' is a <math>Q</math>-pattern whose domain is a barrier. By observing that every block contains a barrier, one sees that <math>Q</math> is bqo if and only if there is no bad <math>Q</math>-array.
 
== Simpson's alternative definition ==
 
Simpson introduced an alternative definition of bqo in terms of Borel maps <math>[\omega]^{\omega}\to Q</math>, where <math>[\omega]^{\omega}</math>, the set of infinite subsets of <math>\omega</math>, is given the usual (product) topology.<ref name="Simpson85"/>
 
Let ''<math>Q</math>'' be a quasi-order and endow <math>Q</math> with the [[discrete topology]]. A ''<math>Q</math>-array'' is a [[Borel function]] <math>[A]^{\omega}\to Q</math> for some infinite subset <math>A</math> of <math>\omega</math>. A <math>Q</math>-array <math>f</math> is ''bad'' if <math>f(X)\not\le_Q f({_*}X)</math> for every <math>X\in[A]^{\omega}</math>;
<math>f</math> is ''good'' otherwise. The quasi-order <math>Q</math> is ''bqo'' if there is no bad <math>Q</math>-array in this sense.
 
== Major theorems ==
 
Many major results in bqo theory are consequences of the Minimal Bad Array Lemma, which appears in Simpson's paper<ref name="Simpson85"/> as follows. See also Laver's paper,<ref name="Laver78"/> where the Minimal Bad Array Lemma was first stated as a result. The technique was present in Nash-Williams' original 1965 paper.
 
Suppose <math>(Q,\le_Q)</math> is a [[quasi-order]]. A ''partial ranking'' <math>\le'</math> of <math>Q</math> is a [[well-founded]] [[partial order]]ing of <math>Q</math> such that <math>q\le'r \to q \le_Q r</math>. For bad <math>Q</math>-arrays (in the sense of Simpson) <math>f\colon [A]^{\omega}\to Q</math> and <math>g\colon [B]^{\omega}\to Q</math>, define:
: <math>g\le^* f \text{ if } B\subseteq A \text{ and } g(X)\le' f(X) \text{ for every } X\in[B]^{\omega}</math>
: <math>g <^* f \text{ if } B\subseteq A \text{ and } g(X) <' f(X) \text{ for every } X\in[B]^{\omega}</math>
We say a bad <math>Q</math>-array <math>g</math> is ''minimal bad'' (with respect to the partial ranking <math>\le'</math>) if there is no bad <math>Q</math>-array <math>f</math> such that <math>f <^* g</math>.
Note that the definitions of <math>\le^*</math> and <math><'</math> depend on a partial ranking <math>\le'</math> of <math>Q</math>. Note also that the relation <math><^*</math> is not the strict part of the relation <math>\le^*</math>.
 
'''Theorem''' (Minimal Bad Array Lemma)'''.''' Let <math>Q</math> be a [[quasi-order]] equipped with a partial ranking and suppose <math>f</math> is a bad <math>Q</math>-array. Then there is a minimal bad <math>Q</math>-array <math>g</math> such that <math>g \le^* f</math>.
 
==See also==
 
* [[Well-quasi-ordering]]
* [[Well-order]]
 
==References==
 
{{Reflist|
refs=
<ref name = "Laver71">{{
cite journal
| last1 = Laver
| first1 = Richard
| title = On Fraisse's Order Type Conjecture
| journal = The Annals of Mathematics
| volume = 93
|issue = 1
| year = 1971
| pages = 89–111
| jstor = 1970754
| doi = 10.2307/1970754
}}</ref>
<ref name="Martinez-Ranero2011">{{
cite journal
| last1 = Martinez-Ranero
| first1 = Carlos
| title = Well-quasi-ordering Aronszajn lines
| journal = Fundamenta Mathematicae
| volume = 213
| issue = 3
| year = 2011
| pages = 197–211
| issn = 0016-2736
| doi = 10.4064/fm213-3-1
| mr = 2822417
}}</ref>
<ref name = "Nash-Williams65">{{
cite journal
| last1 = Nash-Williams
| first1 = C. St. J. A.
| authorlink1 = Crispin Nash-Williams
| title = On well-quasi-ordering infinite trees
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| volume = 61
| issue = 3
| year = 1965
| pages = 697–720
| issn = 0305-0041
| doi = 10.1017/S0305004100039062
| mr = 0175814
| bibcode = 1965PCPS...61..697N
}}</ref>
<ref name = "Rado54">{{
cite journal
| last = Rado
| first = Richard
| authorlink = Richard Rado
| title = Partial well-ordering of sets of vectors
| journal = Mathematika
| year = 1954
| volume = 1
| pages = 89–95
| doi = 10.1112/S0025579300000565
| mr = 0066441
| issue = 2
}}</ref>
<ref name = "Simpson85">{{
cite book
| last = Simpson
| first = Stephen G.
| chapter = BQO Theory and Fraïssé's Conjecture
| title = Recursive Aspects of Descriptive Set Theory
| editor1-last = Mansfield
| editor1-first = Richard
| editor2-last = Weitkamp
| editor2-first = Galen
| publisher = The Clarendon Press, Oxford University Press
| year = 1985
| pages = 124–38
| mr = 786122
| isbn = 978-0-19-503602-2
}}</ref>
<ref name = "Laver78">{{
cite book
| last = Laver
| first = Richard
| chapter = Better-quasi-orderings and a class of trees
| title = Studies in foundations and combinatorics
| editor1-last = Rota
| editor1-first = Gian-Carlo
| publisher = Academic Press
| year = 1978
| pages = 31–48
| mr = 0520553
| isbn = 978-0-12-599101-8
}}</ref>
}}
 
[[Category:Mathematical relations]]
[[Category:Order theory]]
[[Category:Wellfoundedness]]

Latest revision as of 12:36, 5 May 2014

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