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FRAÏSSÉ LIMITS OF METRIC STRUCTURES

Published online by Cambridge University Press:  13 March 2015

ITAÏ BEN YAACOV
ITAÏ BEN YAACOV*
Affiliation:
UNIVERSITÉ CLAUDE BERNARD – LYON 1, INSTITUT CAMILLE JORDAN, CNRS UMR 5208, 43 BOULEVARD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE CEDEX, FRANCEURL: http://math.univ-lyon1.fr/∼begnac/
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Abstract

We develop Fraïssé theory, namely the theory of Fraïssé classes and Fraïssé limits, in the context of metric structures. We show that a class of finitely generated structures is Fraïssé if and only if it is the age of a separable approximately homogeneous structure, and conversely, that this structure is necessarily the unique limit of the class, and is universal for it.

We do this in a somewhat new approach, in which “finite maps up to errors” are coded by approximate isometries.

Keywords

03C30,03C52Fraïssé classFraïssé limitmetric structureultrahomogeneous structureapproximate isometryKatětov functionGurarij space
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 1 , March 2015 , pp. 100 - 115
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

REFERENCES

Yaacov, Itaï Ben, Berenstein, Alexander, and Ward Henson, C., Model-theoretic independence in the Banach lattices L p(μ). Israel Journal of Mathematics, vol. 183 (2011), pp. 285–320.Google Scholar
Yaacov, Itaï Ben, Berenstein, Alexander, Ward Henson, C., and Usvyatsov, Alexander, Model theory for metric structures, Model theory with Applications to Algebra and Analysis, Volume 2 (Chatzidakis, Zoé, Macpherson, Dugald, Pillay, Anand, and Wilkie, Alex, editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315–427.Google Scholar
Yaacov, Itaï Ben, Continuous first order logic for unbounded metric structures. Journal of Mathematical Logic, vol. 8 (2008), no. 2, pp. 197–223.CrossRefGoogle Scholar
Yaacov, Itaï Ben and Ward Henson, C., Generic orbits and type isolation in the Gurarij space, research notes.Google Scholar
Yaacov, Itaï Ben and Usvyatsov, Alexander, On d-finiteness in continuous structures. Fundamenta Mathematicae, vol. 194 (2007), pp. 67–88.Google Scholar
Yaacov, Itaï Ben and Usvyatsov, Alexander, Continuous first order logic and local stability. Transactions of the American Mathematical Society, vol. 362 (2010), no. 10, pp. 5213–5259.Google Scholar
Fraïssé, Roland, Sur l’extension aux relations de quelques propriétés des ordres. Annales Scientifiques de l’École Normale Supérieure. Troisième Série, vol. 71 (1954), pp. 363–388.Google Scholar
Gurarij, Vladimir I., Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces. Sibirskii Matematicheskii Zhurnal, vol. 7 (1966), pp. 1002–1013.Google Scholar
Kubiś, Wiesław and Solecki, Sławomir, A proof of the uniqueness of the Gurariĭ space. Israel Journal of Mathematics, vol. 195 (2013), no. 1, pp. 449–456.Google Scholar
Lusky, Wolfgang, The Gurarij spaces are unique. Archiv der Mathematik, vol. 27 (1976), no. 6, pp. 627–635.Google Scholar
Meyer-Nieberg, Peter, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991.Google Scholar
Schoretsanitis, Konstantinos, Fraïssé theory for metric structures, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2007.Google Scholar
Uspenskij, Vladimir V., On subgroups of minimal topological groups. Topology and its Applications, vol. 155 (2008), no. 14, pp. 1580–1606.Google Scholar