Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-22T20:56:09.196Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite satisfiability and ℵ0-categorical structures with trivial dependence

Published online by Cambridge University Press:  12 March 2014

Marko Djordjević
Marko Djordjević*
Affiliation:
Department of Mathematics, Uppsala University, Box 480 75106 Uppsala, Sweden.E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

The main subject of the article is the finite submodel property for ℵ0-categorical structures, in particular under the additional assumptions that the structure is simple, 1-based and has trivial dependence. Here, a structure has the finite submodel property if every sentence which is true in the structure is true in a finite substructure of it. It will be useful to consider a couple of other finiteness properties, related to the finite submodel property, which are variants of the usual concept of saturation.

For the rest of the introduction we will assume that M is an ℵ0-categorical (infinite) structure with a countable language. We also assume that there is an upper bound to the arity of the function symbols in M:s language and that, for every 0 < n < ℵ0 and R ⊆ Mn which is definable in M without parameters, there exists a relation symbol, in the language of M, which is interpreted as R; these assumptions are not necessary for most results to be presented, but it simplifies the statement of a result which I mention in this introduction.

First we will consider ‘canonically embedded’ substructures of Meq. Here, a structure N is canonically embedded in Meq if N's universe is a subset of Meq which is definable without parameters and, for every 0 < n < ℵ0 and R ⊆ Nn which is ε-definable in Meq there is a relation symbol in the language of N which is interpreted as R; we also assume that the language of N has no other relation (or function or constant) symbols.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 810 - 830
Copyright
Copyright © Association for Symbolic Logic 2006

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]Cherlin, G., Harrington, L., and Lachlan, A. H., ℵ0-categorical, ℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 103–135.CrossRefGoogle Scholar
[2]Cherlin, G. and Hrushovski, E., Finite structures with few types, Princeton University Press, 2003.Google Scholar
[3]Piro, T. De and Kim, B., The geometry of 1-based minimal types, Transactions of The American Mathematical Society, vol. 355 (2003), pp. 4241–4263.CrossRefGoogle Scholar
[4]Djordjević, M., The finite submodel property and ω-categorical expansions of pregeometries, Annals of Pure and Applied Logic, vol. 139 (2006), pp. 201–229.CrossRefGoogle Scholar
[5]Hart, B., Kim, B., and Pillay, A., Coordinatisation and canonical bases in simple theories, this Journal, vol. 65 (2000), pp. 293–309.Google Scholar
[6]Hodges, W., Model theory, Cambridge University Press, 1993.CrossRefGoogle Scholar
[7]Shelah, S., Classification theory, Elsevier Science Publishers B.V., 1990.Google Scholar
[8]Wagner, F. O., Simple theories, Kluwer Academic Publishers, 2000.CrossRefGoogle Scholar