Lovely pairs of models

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 characterise when a saturated model of the theory TP of lovely pairs is a lovely pair (that is when the notion of a lovely pair is “axiomatizable”), finding an analogue of the non finite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we study forking and canonical bases in TP. 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 TP, are lovely pairs, and “simple Robinson universal domains”. ∗Supported under a CNRS-UIUC collaboration agreement †Partially supported by an NSF grant