An unclassifiable unidimensional theory without OTOP (Q1377557)

A stable (first order) theory \(T\) is said to admit prime models over pairs PMOP if, whenever \(M_0 < M_1\), \(M_2 <N \models T\) and \(M_1\) is independent from \(M_2\) over \(M_0\), then there is a model over \(T\) prime over \(M_1 \cup M_2\). It is known that if a unidimensional theory of arbitrary cardinality has PMOP, then the class of models of \(T\) is well behaved, in particular \(T\) is classifiable. On the other hand, for a countable \(T\), if \(T\) fails to have the Omitting Types Order Property OTOP, then \(T\) has PMOP and hence is classifiable. In the paper under review, the authors show that the countability assumption cannot be removed in the previous statement. For there does exist a unidimensional theory \(T\) of cardinality \(2^{\omega}\) such that \(T\) does not satisfy OTOP but \(T\) is unclassifiable. Let us sketch briefly the example proposed in this paper. The language of \(T\) is multisorted. There are a sort concerning a vectorspace \(V\) over the field with 2 elements, and further sorts \(S_{\eta}\) for every \(\eta \in 2^{\omega}\). Suitable predicates pick out subspaces \(U_n\) of \(V\) (\(n \in \omega\)) in a descending chain satisfying \([U_n: U_{n+1}]= 2\) for every \(n\), as well as the corresponding cosets. For each \(\eta \in 2^{\omega}\), \(S_{\eta} = V \times V\) is a cover of \(V\), and the fiber of \(S_{\eta}\) over any \(v \in V\) is acted on regularly by \(V\). Given \(\eta\), there is no interaction between the fibers in \(S_{\eta}\) of two different elements of \(V\); but, given \(v\), a suitable interaction is defined between the fibers above \(v\) in two different \(S_{\eta}\). \(T\) is just the theory of this structure. It is shown that \(T\) has quantifier elimination in the given language, is superstable and unidimensional, and does not have any \(L_{\infty, \omega}\)-definable order, in particular cannot satisfy OTOP. The unclassifiability of \(T\) is proved by coding some suitable graphs into models of \(T\). Finally the authors explore some possible variations of \(T\) where \(2^{\omega}\) is replaced by some other uncountable sets of ordinals.