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Two-dimensional partial orderings: Recursive model theory

Published online by Cambridge University Press:  12 March 2014

Alfred B. Manaster  and
Joseph G. Rosenstein
Alfred B. Manaster
Affiliation:
University of California at San Diego, La Jolla, CA 92037
Joseph G. Rosenstein
Affiliation:
Rutgers University, New Brunswick, NJ 08903 Institute of Advanced Study, Princeton, NJ 08540
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Extract

In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings.

The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering 〈A, R〉 to be n-dimensional if there are n linear orderings of A, 〈A, L1〉, 〈A, L2〉 …, 〈A, Ln〉 such that R = L1 ∩ L2 ∩ … ∩ Ln. Thus, for example, if Q is the linear ordering of the rationals, then the (rational) plane Q × Q with the product ordering (〈x1, y1〉 ≤Q×Q 〈x2, y2, if and only if x1 ≤ x2 and y1 ≤ y2) is 2-dimensional, since ≤Q×Q is the intersection of the two lexicographic orderings of Q × Q. In fact, as shown by Dushnik and Miller, a countable partial ordering is n-dimensional if and only if it can be embedded as a subordering of Qn.

Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes explicitly a collection of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of as a subordering.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 1 , March 1980 , pp. 121 - 132
Copyright
Copyright © Association for Symbolic Logic 1980

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References

BIBLIOGRAPHY

[1]Baker, K. A., Fishburn, P. C. and Roberts, F. S., Partial orders of dimension 2, Networks, vol. 2 (1971), pp. 11–28.CrossRefGoogle Scholar
[2]Crossley, J. N. and Nerode, A., Combinatorial functors, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 81, Springer, New York, 1974.Google Scholar
[3]Dushnik, B. and Miller, E. W., Partially ordered sets, American Journal of Mathematics, vol. 63 (1941), pp. 600–610.CrossRefGoogle Scholar
[4]Harzheim, E., Ein Endlichkeitssatz über die Dimension teilweise geordneter Mengen, Mathematische Nachrichten, vol. 46 (1970), pp. 183–188, MR 43 #113.CrossRefGoogle Scholar
[5]Jockusch, C. G. Jr.,, Ramsey's theorem and recursion theory, this Journal, vol. 37 (1972), pp. 268–280.Google Scholar
[6]Jockusch, C. G. Jr., and Soare, R., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[7]Kelly, D., The 3-irreducible partially ordered sets, Canadian Journal of Mathematics, vol. 29 (1977), pp. 367–383.CrossRefGoogle Scholar
[8]Manaster, A. B. and Rosenstein, J. G., Effective matchmaking (Recursion-theoretic aspects of a theorem of Philip Hall), Proceedings of the London Mathematical Society, vol. 25 (1972), pp. 615–654.CrossRefGoogle Scholar
[9]Manaster, A. B. and Rosenstein, J. G., Two-dimensional partial orderings; Undecidability, this Journal, vol. 45 (1980), pp. 133–143.Google Scholar
[10]Trotter, W. T. Jr., and Moore, J. I. Jr.,, Characterization problems for graphs, partially ordered sets, lattices, and families of sets, Discrete Mathematics, vol. 16 (1976), pp. 361–381.CrossRefGoogle Scholar