Imaginaries in beautiful pairs
type: Item
instance of: scholarly article
title: Imaginaries in beautiful pairs
zbMATH Open document ID: 1060.03053
author: Anand Pillay
author: Evgueni Vassiliev
published in: Illinois Journal of Mathematics
publication date: 2005-03-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 2140247
zbMATH Keywords: beautiful pair
zbMATH Keywords: lovely pair
zbMATH Keywords: elimination of imaginaries
zbMATH Keywords: finite cover property
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2140247