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DOP and FCP in generic structures. (English) Zbl 0914.03042

This paper deals with a finite relational language \(L\) and it is built on several former papers of the authors and others. Basic definitions, properties and axioms are formulated in the first section.
The second section is concerned with independence and orthogonality. The following result is proved:
Let \(K_0\) be a class of finite structures closed under substructure and isomorphisms, containing the empty structure, and fulfilling several conditions, especially having the amalgamation property. Let \(T\) be the theory of the generic model for \((K_0,\leq_s)\). Suppose further that there is a pair of independent points and a nonalgebraic type such that some conditions related to the dimension are filfilled. Then: The theory \(T\) has the dimensional order property, moreover \(T\) has the eventually non-isolated dimensional order property (ENI-DOP) if the given type is not isolated. \(T\) has the dimensional discontinuity property.
In Section 3 the authors construct a nonalgebraic type over a two-element set with \(d(p)= 0\).
In the last part, the authors show that for several classes with the so-called full amalgamation property, the theory of the generic does not have the finite cover property.

MSC:

03C52 Properties of classes of models
03C45 Classification theory, stability, and related concepts in model theory
03C65 Models of other mathematical theories
08A02 Relational systems, laws of composition
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References:

[1] DOI: 10.1090/S0894-0347-1988-0924703-8 · doi:10.1090/S0894-0347-1988-0924703-8
[2] Fundamentals of stability theory (1988) · Zbl 0685.03024
[3] DOI: 10.1016/0168-0072(95)00027-5 · Zbl 0857.03020 · doi:10.1016/0168-0072(95)00027-5
[4] Classification theory and the number of nonisomorphic models (1991)
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