On generic structures with a strong amalgamation property. (English) Zbl 1178.03042

Ikeda, Koichiro; Kikyo, Hirotaka; Tsuboi, Akito

J. Symb. Log. 74, No. 3, 721-733 (2009).

Summary: Let \({\mathcal L}\) be a finite relational language and \(\alpha= (\alpha_R: R\in{\mathcal L})\) a tuple with \(0<\alpha_R\leq 1\) for each \(R\in{\mathcal L}\). Consider a dimension function
\[ \delta_\alpha(A)= |A|- \sum_{R\in{\mathcal L}}\alpha_R e_R(A) \]
where each \(e_R(A)\) is the number of realizations of \(R\) in \(A\). Let \(K_\alpha\) be the class of finite structures \(A\) such that \(\delta_\alpha(X)\geq 0\) for any substructure \(X\) of \(A\). We show that the theory of the generic model of \(K_\alpha\) is AE-axiomatizable for any \(\alpha\).

MSC:

03C13	Model theory of finite structures
03C45	Classification theory, stability, and related concepts in model theory
03C50	Models with special properties (saturated, rigid, etc.)

Keywords:

dimension function; finite structure; generic model; AE-axiomatization

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