Dimensional order property and pairs of models (Q1115856)

The author considers the relation between DOP and model-theoretic properties of complete theories of pairs of models in the language with a new unary predicate for the small model of a pair. It is proved in the paper that for a superstable theory T the following three conditions are equivalent: (a) T does not have DOP, (b) all complete theories of pairs of T are stable, (c) all complete theories of pairs of T are superstable and do not have DOP. The author proves also that for an \(\omega\)-stable non-multidimensional theory T (i.e. NDOP, shallow of depth 1) all complete theories of pairs of T are \(\omega\)-stable. An example of an \(\omega\)-stable theory of depth 2, where this is not longer true, is given, too.