In order theory a better-quasi-ordering or bqo is a quasi-ordering that does not admit a certain type of bad array. Every better-quasi-ordering is a well-quasi-ordering.


Though well-quasi-ordering is an appealing notion, many important infinitary operations do not preserve well-quasi-orderedness. An example due to Richard Rado illustrates this.[1] In a 1965 paper Crispin Nash-Williams formulated the stronger notion of better-quasi-ordering in order to prove that the class of trees of height ω is well-quasi-ordered under the topological minor relation.[2] Since then, many quasi-orderings have been proven to be well-quasi-orderings by proving them to be better-quasi-orderings. For instance, Richard Laver established Laver's theorem (previously a conjecture of Roland Fraïssé) by proving that the class of scattered linear order types is better-quasi-ordered.[3] More recently, Carlos Martinez-Ranero has proven that, under the proper forcing axiom, the class of Aronszajn lines is better-quasi-ordered under the embeddability relation.[4]


It is common in better-quasi-ordering theory to write for the sequence with the first term omitted. Write for the set of finite, strictly increasing sequences with terms in , and define a relation on as follows: if there is such that is a strict initial segment of and . The relation is not transitive.

A block is an infinite subset of that contains an initial segment[clarification needed] of every infinite subset of . For a quasi-order , a -pattern is a function from some block into . A -pattern is said to be bad if [clarification needed] for every pair such that ; otherwise is good. A quasi-ordering is called a better-quasi-ordering if there is no bad -pattern.

In order to make this definition easier to work with, Nash-Williams defines a barrier to be a block whose elements are pairwise incomparable under the inclusion relation . A -array is a -pattern whose domain is a barrier. By observing that every block contains a barrier, one sees that is a better-quasi-ordering if and only if there is no bad -array.

Simpson's alternative definition

Simpson introduced an alternative definition of better-quasi-ordering in terms of Borel functions , where , the set of infinite subsets of , is given the usual product topology.[5]

Let be a quasi-ordering and endow with the discrete topology. A -array is a Borel function for some infinite subset of . A -array is bad if for every ; is good otherwise. The quasi-ordering is a better-quasi-ordering if there is no bad -array in this sense.

Major theorems

Many major results in better-quasi-ordering theory are consequences of the Minimal Bad Array Lemma, which appears in Simpson's paper[5] as follows. See also Laver's paper,[6] where the Minimal Bad Array Lemma was first stated as a result. The technique was present in Nash-Williams' original 1965 paper.

Suppose is a quasi-order.[clarification needed] A partial ranking of is a well-founded partial ordering of such that . For bad -arrays (in the sense of Simpson) and , define:

We say a bad -array is minimal bad (with respect to the partial ranking ) if there is no bad -array such that . The definitions of and depend on a partial ranking of . The relation is not the strict part of the relation .

Theorem (Minimal Bad Array Lemma). Let be a quasi-order equipped with a partial ranking and suppose is a bad -array. Then there is a minimal bad -array such that .

See also


  1. ^ Rado, Richard (1954). "Partial well-ordering of sets of vectors". Mathematika. 1 (2): 89–95. doi:10.1112/S0025579300000565. MR 0066441.
  2. ^ Nash-Williams, C. St. J. A. (1965). "On well-quasi-ordering infinite trees". Mathematical Proceedings of the Cambridge Philosophical Society. 61 (3): 697–720. Bibcode:1965PCPS...61..697N. doi:10.1017/S0305004100039062. ISSN 0305-0041. MR 0175814. S2CID 227358387.
  3. ^ Laver, Richard (1971). "On Fraïssé's Order Type Conjecture". The Annals of Mathematics. 93 (1): 89–111. doi:10.2307/1970754. JSTOR 1970754.
  4. ^ Martinez-Ranero, Carlos (2011). "Well-quasi-ordering Aronszajn lines". Fundamenta Mathematicae. 213 (3): 197–211. doi:10.4064/fm213-3-1. ISSN 0016-2736. MR 2822417.
  5. ^ a b Simpson, Stephen G. (1985). "BQO Theory and Fraïssé's Conjecture". In Mansfield, Richard; Weitkamp, Galen (eds.). Recursive Aspects of Descriptive Set Theory. The Clarendon Press, Oxford University Press. pp. 124–38. ISBN 978-0-19-503602-2. MR 0786122.
  6. ^ Laver, Richard (1978). "Better-quasi-orderings and a class of trees". In Rota, Gian-Carlo (ed.). Studies in foundations and combinatorics. Academic Press. pp. 31–48. ISBN 978-0-12-599101-8. MR 0520553.