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The model-companion of a class of structures1

Published online by Cambridge University Press:  12 March 2014

G. L. Cherlin
G. L. Cherlin*
Affiliation:
Yale University, New Haven, Connecticut 06520
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Extract

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:

(i) Σ* is the class of models of some first order theory K*.

(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).

(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.

If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.

Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to

(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 3 , September 1972 , pp. 546 - 556
Copyright
Copyright © Association for Symbolic Logic 1972

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Footnotes

1

The material in this paper is taken from my doctoral dissertation (Yale, 1971). Research supported by an NSF Graduate Fellowship. I would like to express my gratitude for the guidance given by my advisor, Abraham Robinson. I am also indebted to Jim Schmerl for highly constructive criticism.

References

REFERENCES

[1]Coven, C., Forcing in infinitary languages, Doctoral Dissertation, Yale University, New Haven, Ct., 1971.Google Scholar
[2]Eklof, P. and Sabbagh, G., Model-completions and modules, Annals of Mathematical Logic, vol. 2 (1971), pp. 251–295.CrossRefGoogle Scholar
[3]Fisher, E., Homogeneous universal models revisited (in preparation).Google Scholar
[4]Kaiser, K., Über eine Verallgemeinerung der Robinsonschen Modellvervollständigung. I, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 37–48.CrossRefGoogle Scholar
[5]Karp, C., Languages with expressions of infinite length, North-Holland, Amsterdam, 1964.Google Scholar
[6]Robinson, A., Introduction to model theory and to the metamathematics of algebra, North-Holland, Amsterdam, 1963.Google Scholar
[7]Robinson, A., Infinite forcing in model theory, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J., Editor), North-Holland, Amsterdam, 1971, pp. 317–340.CrossRefGoogle Scholar
[8]Robinson, A., Forcing in model theory, International Congress of Mathematicians, Nice, 1970.Google Scholar
[9]Robinson, A., On the notion of algebraic closedness for noncommutative groups and fields, this Journal, vol. 36 (1971), pp. 441–444.Google Scholar
[10]Saracino, D., Infinite forcing in model theory and m-existentially closed structures, Notices of the American Mathematical Society, vol. 18 (1971), p. 668. Abstract 71T-E62.Google Scholar
[11]Scott, W., Algebraically closed groups, Proceedings of the American Mathematical Society, vol. 2 (1951), pp. 118–121.CrossRefGoogle Scholar
[12]Cherlin, G., Second order forcing, algebraic completeness, and large cardinals (in preparation).Google Scholar