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On well-quasi-ordering transfinite sequences

Published online by Cambridge University Press:  24 October 2008

C. St. J. A. Nash-Williams
Affiliation:
University of Aberdeen

Abstract

. Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q1,q2,… of elements of Q, there exist i and j such that i < j and qiqi. A restricted transfinite sequence on Q is a function from a well-ordered set onto a finite subset of Q. If f, g are restricted transfinite sequences on Q with domains A, B respectively and there exists a one-to-one order-preserving mapping μ of A into B such that f(α) ≤ h(μ(α)) for every α ∈ A, we write fg. It is proved that this rule well-quasi-orders the set of restricted transfinite sequences on Q. The proof uses the following subsidiary theorem, which is a generalization of a classical theorem of Ramsey (4). Let P be the set of positive integers, and A(I) denote the set of ascending finite sequences of elements of a subset I of P. If s, tA(P), write st if, for some m, the terms of s are the first m terms of t. Let T1,…,Tn be disjoint subsets of A(P) whose union T does not include two distinct sequences s, t such that st. Then there exists an infinite subset I of P such that TA(I)is contained in a single Tj.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

(1)Erdős, P. and Rado, R.A theorem on partial well-ordering of sets of vectors. J. London Math. Soc. 34 (1959), 222224.CrossRefGoogle Scholar
(2)Higman, G.Ordering by divisibility in abstract algebras. Proc. London Math. Soc. (3), 2 (1952), 326336.CrossRefGoogle Scholar
(3)Rado, R.Partial well-ordering of sets of vectors. Mathemalika 1 (1954), 8995.CrossRefGoogle Scholar
(4)Ramsey, F. P.On a problem of formal logic. Proc. London Math. Soc. (2), 30 (1930), 264286.CrossRefGoogle Scholar