Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-25T04:39:36.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Partially ordered sets and the independence property

Published online by Cambridge University Press:  12 March 2014

James H. Schmerl
James H. Schmerl*
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
Get access Rights & Permissions [Opens in a new window]

Abstract

No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional iff its theory has the independence property iff its theory has the multi-order property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 396 - 401
Copyright
Copyright © Association for Symbolic Logic 1989

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.)

References

REFERENCES

[1]Dilworth, R. P., A decomposition theorem for partially ordered sets, Annals of Mathematics, ser. 2, vol. 51 (1950), pp. 161–166.CrossRefGoogle Scholar
[2]Gurevich, Y. and Schmitt, P. H., The theory of ordered abelian groups does not have the independence property, Transactions of the American Mathematical Society, vol. 284 (1984), pp. 171–182.CrossRefGoogle Scholar
[3]Keisler, H. Jerome, The stability function of a theory, this Journal, vol. 43 (1978), pp. 481–486.Google Scholar
[4]Parigot, M., Théories d'arbres, this Journal, vol. 47 (1982), pp. 841–853.Google Scholar
[5]Poizat, B., Théories instables, this Journal, vol. 46 (1981), pp. 513–522.Google Scholar
[6]Rabin, M. O., Decidability of second-order theories and automata on finite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 1–35.Google Scholar
[7]Schmerl, J. H., Decidability and ℵ0-categoricity of theories of partially ordered sets, this Journal, vol. 45 (1980), pp. 585–611.Google Scholar
[8]Schmerl, J. H., Decidability and finite axiomatizability of theories of ℵ0-categorical partially ordered sets, this Journal, vol. 46 (1981), pp. 101–120.Google Scholar
[9]Schmerl, J. H., ℵ0-categorical partially ordered sets, Orders: description and roles (proceedings, L'Arbresle, 1982; Pouzet, M. and Richard, D., editors), North-Holland, Amsterdam, 1984, pp. 269–285.Google Scholar
[10]Schmerl, J. H., ℵ0-categorical distributive lattices of finite breadth, Proceedings of the American Mathematical Society, vol. 87 (1983), pp. 707–713.Google Scholar
[11]Shelah, S., Stability, the f.c.p. and superstability: model theoretic properties of formulas in the first order theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 271–362.CrossRefGoogle Scholar