On lovely pairs and the \((\exists y\in P)\) quantifier
type: Item
instance of: scholarly article
title: On lovely pairs and the \((\exists y\in P)\) quantifier
zbMATH Open document ID: 1097.03025
author: Evgueni Vassiliev
author: Anand Pillay
published in: Notre Dame Journal of Formal Logic
publication date: 2006-03-23T00:00:00Z
review text: Recall that a simple theory is low if for any formula \(\varphi(x,y)\) and \(z\) the condition that \(\varphi(.,y)\) forks over \(z\) is type-definable (in \(y,z\)). The authors define the theory to be weakly low if for any complete finitary type tp\((a/B)\) and formula \(\varphi(x,y,z)\) (with parameters in \(B\)) there is some formula \(\psi(y)\) with parameters in \(Ba\) true for any \(b\) independent of \(a\) over \(B\) iff \(\varphi(a,b,.)\) does not fork over \(Bb\). In particular a low theory is weakly low. An elementary pair \(P\prec M\) is said to eliminate the quantifier \(\exists y\in P\) if for every formula \(\varphi(x,y,z)\) and \(a\in M\) there is a formula \(\psi(x,w)\) and \(b\in M\) such that for all \(c\in P\) there is \(d\in P\) with \(M\models\varphi(c,d,a)\) iff \(M\models\psi(c,b)\). This condition is equivalent to quantifier-elimination of the structure \(P\) in the language where we have relations for all traces of \(M\)-definable subsets. A pair \(P\prec M\) is \(\kappa\)-lovely, if for all \(A\) of cardinality less than \(\kappa\) and all types \(p\) over \(A\) some non-forking realization of \(p\) over \(A\cup P\) is realized in \(M\), and if \(p\) does not fork over \(P\cap A\), then it is realized in \(P\). For stable theories, lovely pairs are precisely Poizat's beautiful pairs. [\textit{I. Ben-Yaacov, A. Pillay} and \textit{E. Vassiliev}, ``Lovely pairs of models'', Ann.\ Pure Appl.\ Logic 122, 235--261 (2003; Zbl 1030.03026)]. The authors show that a simple theory is weakly low iff some/any lovely pair eliminates \(\exists y\in P\); in this case \(P\) with the induced structure from \(M\) is also simple. If the theory is low, the elimination is uniform. Finally, it is shown that \textit{E. Casanovas'} example [``The number of types in simple theories'', Ann.\ Pure Appl.\ Logic 98, 69--86 (1999; Zbl 0939.03039)] of a simple non-low theory is weakly low; the authors give a variant which isn't.
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C10
zbMATH DE Number: 5014659
zbMATH Keywords: simple theory
zbMATH Keywords: lovely pair
zbMATH Keywords: quantifier elimination
zbMATH Keywords: weakly low theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1134397664
OpenAlex ID: W1991738673
cites work: Paires de structures O-minimales
cites work: Lovely pairs of models
cites work: The number of types in simple theories
cites work: Q4950412
DOI: 10.1305/NDJFL/1134397664
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description: scientific article; zbMATH DE number 5014659