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Large and small existentially closed structures

Published online by Cambridge University Press:  12 March 2014

H. Simmons
H. Simmons*
Affiliation:
University of Aberdeen, King's college, Aberdeen, Scotland AB9 2UB
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Extract

The generic structures introduced by Abraham Robinson are now well established in model theory. Anyone who works with these structures soon begins to feel that, in some sense, the infinite generic structures are large and the finite generic structures are small. Here ‘large’ and ‘small’ do not refer to the cardinality of the structures but are used in the way we describe saturated structures as large and atomic structures as small. In this paper I isolate what I consider to be the large and small existentially closed (e.c.) structures and attempt to determine their role in the class of all e.c. structures.

In the usual context of model theory we are concerned with elementary embeddings and all formulas. E.c. structures are concerned with all embeddings and ∃1-formulas. Thus we need to look at ∃1-analogues of saturated and atomic structures. These ∃1-saturated structures have been around for some time; they are just the existentially universal structures. The corresponding ∃1-atomic structures are not new here (they appear in [8]) but I believe that this paper will add much to the understanding of them.

The bulk of this paper is in §§2 and 4. §2 deals with ∃1-atomic structures, and §4 is concerned with the ∃1-analogues of several results in Vaught's paper [12].

A similar program has been carried out by Pouzet in [6], [7], [8]; however, there the significance of e.c. and e.c. structures is not realized and consequently some of the simplicity of the situation is lost.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 379 - 390
Copyright
Copyright © Association for Symbolic Logic 1976

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References

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