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Finitely generic models of TUH, for certain model companionable theories T

Published online by Cambridge University Press:  12 March 2014

Francoise Point
Francoise Point*
Affiliation:
Faculté des Sciences, Université de l'état de Mons7000 Mons, Belgium
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Extract

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].

We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).

We determine the finitely generic elements of ∑UH via the three following conditions on T:

(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.

(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.

(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.

(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 3 , September 1985 , pp. 604 - 610
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

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