Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-14T06:25:07.188Z Has data issue: false hasContentIssue false
  • English
  • Français

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 08903Institute of Advanced Study, Princeton, NJ 08540
Get accessRights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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