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A generalized model companion for a theory of partially ordered fields

Published online by Cambridge University Press:  12 March 2014

Werner Stegbauer
Werner Stegbauer*
Affiliation:
University of Heidelberg, Heidelberg, Federal Republic of Germany
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Extract

The notion of a model companion for a first-order theory T was introduced and discussed in [1] and [2] as a generalization of the concept of a model completion of a theory. Both concepts reflect, on a general model theoretic level, properties of the theory of algebraically closed fields. The literature provides many examples of first-order theories with and without model companions—see [3] for a survey of these results. In this paper, we give a further generalization of the notion of a model companion.

Roughly speaking, we allow instead of embeddings more general classes of maps (e.g. homomorphisms) and we allow any set of formulas which is preserved by these maps instead of existential formulas. This plan is worked out in detail in [5], where we discuss also several examples. One of these examples is given in this paper.

In order to clarify the model theoretic background, we now introduce the relevant concepts and theorems from [5].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 4 , December 1979 , pp. 643 - 652
Copyright
Copyright © Association for Symbolic Logic 1979

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References

REFERENCES

[1]Barwise, J. and Robinson, A., Completing theories by forcing, Annals of Mathematical Logic, vol. 2(1970), pp. 119–142.CrossRefGoogle Scholar
[2]Robinson, A., Infinite forcing in model theory, Proceedings of the Second Scandinavian Logic Symposium, Oslo, 1970, North-Holland, Amsterdam, 1971.Google Scholar
[3]Hirschfeld, J. and Wheeler, W., Forcing, arithmetic, and division rings, Lecture Notes in Mathematics, vol. 454(1975), Springer-Verlag, Berlin and New York, pp. 317–340.Google Scholar
[4]Fuchs, L., Teilweise geordnete algebraische Strukturen, Vandenhoeck & Ruprecht, Göttingen, 1966.Google Scholar
[5]Stegbauer, W., Verallgemeinert existentiell vollständige Strukturen, Thesis, Universität Heidelberg, 1977.Google Scholar