Metric abstract elementary classes with perturbations

Åsa Hirvonen ¹   ; Tapani Hyttinen ¹

¹ Department of Mathematics and Statistics University of Helsinki P.O. Box 68 00014 University of Helsinki, Finland

Fundamenta Mathematicae, Tome 217 (2012) no. 2, pp. 123-170

Résumé

We define an abstract setting suitable for investigating perturbations of metric structures generalizing the notion of a metric abstract elementary class. We show how perturbation of Hilbert spaces with an automorphism and atomic Nakano spaces with bounded exponent fit into this framework, where the perturbations are built into the definition of the class being investigated. Further, assuming homogeneity and some other properties true in the example classes, we develop a notion of independence for this setting and show that it satisfies the usual independence axioms. Finally we define an isolation notion. Although it remains open whether this isolation gives any reasonable form of primeness, we prove that dominance works.

Détail

DOI : 10.4064/fm217-2-2

Keywords: define abstract setting suitable investigating perturbations metric structures generalizing notion metric abstract elementary class perturbation hilbert spaces automorphism atomic nakano spaces bounded exponent fit framework where perturbations built definition class being investigated further assuming homogeneity other properties example classes develop notion independence setting satisfies usual independence axioms finally define isolation notion although remains whether isolation gives reasonable form primeness prove dominance works