Constructing \(\omega\)-stable structures: Computing rank (Q2773235)

In a previous paper [J. Symb. Log. 65, 371-391 (2000; Zbl 0957.03044)], the authors set up a general framework for constructing \(\omega\)-stable expansions of strongly minimal sets. In the present paper they consider the difficulties of calculating Morley and \(U\)-rank of the infinite rank \(\omega\)-stable theories constructed by variants of \textit{E. Hrushovski}'s methods [Isr. J. Math. 79, 129-151 (1992; Zbl 0773.12005); Ann. Pure Appl. Log. 62, 147-166 (1993; Zbl 0804.03020)]. The existence of an expansion of an algebraically closed field with Morley rank \(\omega \times 2\), proved by \textit{B. Poizat} [J. Symb. Log. 64, 1339-1355 (1999; Zbl 0938.03058)], is extended to rank \(\omega \times k\) for \(k > 2\) with the price of further complexities for proving the lower bound. A corrected proof of the lemma in the authors' paper mentioned above, establishing that the generic model is \(\omega\)-saturated in the rank \(2\) case, is also included.