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Lovely pairs of models. (English) Zbl 1030.03026

Summary: We introduce the notion of a lovely pair of models of a simple theory \(T\), generalizing Poizat’s “belles paires” of models of a stable theory and the third author’s “generic pairs” of models of an \(SU\)-rank 1 theory. We characterize when a saturated model of the theory \(T_P\) of lovely pairs is a lovely pair (that is when the notion of a lovely pair is “axiomatizable”), finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, \(T_P\) is also simple, and we study forking and canonical bases in \(T_P\). We also prove that assuming only that \(T\) is low, the existentially universal models of the universal part of a natural expansion \(T^+_P\) of \(T_P\), are lovely pairs, and “simple Robinson universal domains”.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
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References:

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