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On the Weak Non-Finite Cover Property and the n-Tuples of Simple Structures

Evgueni Vassiliev
The Journal of Symbolic Logic
Vol. 70, No. 1 (Mar., 2005), pp. 235-251
Stable URL: http://www.jstor.org/stable/27588356
Page Count: 17
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On the Weak Non-Finite Cover Property and the n-Tuples of Simple Structures
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Abstract

The weak non-finite cover property (wnfcp) was introduced in [1] in connection with "axiomatizability" of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory TP of lovely pairs of models of T (true in the stable case). While the question remains open, we show, among other things, that if (for a T with the wnfcp) TP is low, then TP has the wnfcp. To study this question, we describe "double lovely pairs", and, along the way, we develop the notion of a "lovely n-tuple" of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models of stable theories [2].

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