Imaginaries in beautiful pairs
A beautiful pair (belle paire) is an \({\mathcal L}\cup\{P\}\)-structure \((M,P)\), where the unary predicate \(P\) is interpreted as an \(\mathcal L\)-elementary \(| T| ^+\)-saturated proper substructure of \(M\) such that any \(\mathcal L\)-type over \(P(M)\) together with a finite tuple from \(M\) is realized in \(M\) [\textit{B. Poizat}, J. Symb. Log. 48, 239--249 (1983; Zbl 0525.03023)]. These were generalized by \textit{E. Vassiliev} [Ann. Pure Appl. Logic 120, 103--149 (2003; Zbl 1010.03023)] and \textit{I. Ben-Yaacov} and the authors [Ann. Pure Appl. Logic 122, 235--261 (2003; Zbl 1030.03026)] to lovely pairs: \(P(M)\) is \(\mathcal L\)-elementary in \(M\), and for any \(A\) of cardinality \(| T| \) and every \(\mathcal L\)-type \(p\) over \(A\) (i) \(p\) has a non-forking extension to \(A\cup P(M)\) realized in \(M\), and (ii) if \(p\) does not fork over \(A\cap P(M)\), then \(p\) is realized in \(P(M)\). The authors use the characterisation of canonical bases given in the paper with Ben-Yaacov, together with standard binding group techniques, in order to show that the following are equivalent, provided the \(\mathcal L\)-theory eliminates imaginaries: (i) The beautiful pairs theory \(T_P\) eliminates imaginaries. (ii) \(T_P\) eliminates imaginaries geometrically (any imaginary is interalgebraic with a real tuple). (iii) No infinite group is definable in \(T\). However, no explicit description of new imaginaries arising in \(T_P\) is given.
scientific article; zbMATH DE number 2140247