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Directed free pseudospaces. (English) Zbl 1245.03052

The paper under review studies model-theoretic properties of certain directed free pseudospaces. The (undirected) free pseudospace of A. Baudisch and A. Pillay [J. Symb. Log. 65, No. 1, 443–460 (2000; Zbl 0947.03050)], given by a certain point-line-plane configuration, provides a non-CM-trivial theory (denoted by \(\Sigma\)) of infinite Morley rank which does not interpret an infinite field. The notion of CM-triviality was introduced by E. Hrushovski [Ann. Pure Appl. Logic 62, No. 2, 147–166 (1993; Zbl 0804.03020)]. It is a property of geometric complexity of stable theories which is incompatible with the existence of an infinite interpretable field and which is more general than one-basedness. The new strongly minimal set (a counter-example to Zilber’s trichotomy conjecture) constructed by Hrushovski in [loc. cit.] is CM-trivial.
In the same way as the free pseudoplane of Lachlan (based on a certain point-line configuration) is the prototype of a stable structure which is not one-based, the free pseudospace is the prototype of a non-CM-trivial stable theory. Hodges constructed a directed version of the free pseudoplane which is \(\omega\)-stable and one-based. D. M. Evans [J. Symb. Log. 68, No. 4, 1385–1402 (2003; Zbl 1067.03045)] asked whether the free pseudospace is also a reduct of a one-based theory.
In the present paper, the author first reviews the construction of the directed and undirected free pseudoplanes, and also of the free pseudospace of Baudisch and Pillay. Then, a natural expansion \(\Theta\) of \(\Sigma\) is considered, describing an (incomplete) theory of directed free pseudospaces. It is shown that \(\Theta\) has no \(\omega\)-stable completion. For a carefully chosen completion \(\Theta'\) of \(\Theta\), it is then proved that one gets quantifier elimination in a language where predecessor functions are named. From this, the main results of the paper are derived: \(\Theta'\) is superstable (of infinite \(\mathrm{U}\)-rank), one-based and trivial. In particular, \(\Sigma\) is thus shown to be the reduct of a one-based theory, answering the aforementioned question of Evans.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
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