Reduction property and dimensional order property (Q1375777)

Let \(T\) be a theory in a relational language \(L\) including a 1-ary predicate \(P\). Given a model \(M\) of \(T\), \(P(M)\) can be viewed as a structure in \(L^- = L - \{ P \}\). Let \(T^-\) denote its theory. The paper under review is concerned with the problem of determining the common model theoretic properties of \(T\) and \(T^-\). A useful assumption on \(T\) in this setting is the minimality over \(P\): this means that, for every \(M\), \(M\) is minimal over \(P(M)\). Previous contributions on this matter include a theorem of Hodges and Pillay stating that, if \(T\) is minimal over \(P\) and every \(L^-\)-automorphism of \(P(M)\) can be extended to an \(L\)-automorphism of \(M\), then \(P(M)\) is \(\omega\)-categorical if and only if \(M\) is. Furthermore Kikyo and Tsuboi showed that also \(\lambda\)-stability and unidimensionality are preserved under passing from \(T\) to \(T^-\) and vice-versa, provided that \(T\) is minimal over \(P\) and \(L\)-formulas can be reduced to \(L^-\)-formulas in a suitable sense. The paper under review explores the consequences in this setting of another reduction property, called \(\emptyset\)-reduction property and requiring that, for some (equivalently for every) model \(M\) of \(T\), for any \(L(\emptyset)\)-formula \(\varphi(\vec{x})\), there is a \(L^-(\emptyset)\)-formula \(\psi(\vec{x})\) such that \( M \models \forall \vec{x} \in P (\varphi(\vec{x}) \leftrightarrow \psi^P(\vec{x})) \) (here the exponent \(P\) means relativization). It is shown that, if \(T\) is superstable, is minimal over \(P\) and satisfies the \(\emptyset\)-reduction property, then both dimensional order property DOP and depth are preserved under passing from \(T\) to \(T^-\) and from \(T^-\) to \(T\). Some counterexamples underline that the minimality assumption and the \(\emptyset\)-reduction property cannot be omitted in the statement concerning DOP. When \(T\) is stable and still satisfies the \(\emptyset\)-reduction property and minimality over \(P\) (and \(k_r (T) = k_r (T^-)\)), then it is proved that the \(a\)-models of \(T\) are just the expansions of \(a\)-models of \(T^-\). It is also remarked that, in the unstable case, this relationship between models of \(T\) and \(T^-\) is not so strong. For instance, the author gives an example of an unstable \(T\) being minimal over \(P\) and satisfying the \(\emptyset\)-reduction property such that \(T\) has \(2^{\omega}\) countable models and \(T^-\) is \(\omega\)-categorical.