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Revista Colombiana de Matemáticas

versão impressa ISSN 0034-7426

Rev.colomb.mat. vol.52 no.1 Bogotá jan./jun. 2018

https://doi.org/10.15446/recolma.v1n52.74566 

Original articles

Hilbert spaces with generic predicates

Espacios de Hilbert con predicados genéricos

Alexander Berenstein1 

Tapani Hyttinen2 

Andrés Villaveces3  * 

1 Universidad de Los Andes, Bogotá- Colombia

2 University of Helsinki, Helsinki - Finland

3 Universidad Nacional de Colombia, Bogotá - Colombia


Abstract:

We study the model theory of expansions of Hilbert spaces by generic predicates. We first prove the existence of model companions for generic expansions of Hilbert spaces in the form of a distance function to a random substructure, then a distance to a random subset. The theory obtained with the random substructure is (-stable, while the one obtained with the distance to a random subset is TP 2 and NSOP 1. That example is the first continuous structure in that class.

Keywords: Continuous logic; Random Predicates; TP 2; Stability theory

Resumen:

Estudiamos la teoría de modelos de expansiones de espacios de Hilbert mediante predicados genéricos. Primero demostramos la existencia de modelo-compañeras de expansiones genéricas de espacios de Hilbert mediante una función-distancia a una estructura aleatoria, y luego una distancia a un subconjunto aleatorio. La teoría obtenida con la subestructura aleatoria es (-estable; la obtenida mediante la distancia a subconjunto aleatorio es TP 2 y NSOP 1. Este ejemplo es la primera estructura de esta clase de complejidad en lógica continua.

Palabras clave: Lógica continua; Predicados aleatorios; TP 2; Estabilidad

Text complete end PDF

References

1. Ben-Yaacov, I., Lovely pairs of models: the non first order case, Journal of Symbolic Logic 69 (2004), 641-662. [ Links ]

2. Ben-Yaacov, I., Berenstein, A., Henson, W. C. and Usvyatsov, A., Model Theory for metric structures, in Model Theory with Applications to Algebra and Analysis, Volume 2, Cambridge University Press. [ Links ]

3. Ben-Yaacov, I., Usvyatsov, A., and Zadka, M., Generic automorphism of a Hilbert space, preprint avaliable at http://ptmat.fc.ul.pt/alexus/papers.html. [ Links ]

4. Berenstein, A., Hilbert spaces with generic groups of automorphisms, Arch. Math. Logic 46 (2007), no. 3, 289-299. [ Links ]

5. Berenstein, A. and Buechler, S., Simple stable homogeneous expansions of hilbert space, Ann. Pure Appl. Logic 128 (2004), no. 1-3, 75-101. [ Links ]

6. Chatzidakis, Z. and Pillay, A., Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998), no. 1-3. [ Links ]

7. Chernikov, A. and Ramsey, N., On model-theoretic tree properties, Journal of Mathematical Logic 16 (2016), no. 2. [ Links ]

8. Hodges, W., Model Theory, Cambridge University Press, 1993. [ Links ]

9. Poizat, B., Le carré de l'égalité, Journal of Symbolic Logic 64 (1999), 1339-1355. [ Links ]

Received: March 06, 2017; Accepted: January 06, 2018

*Correspondencia: Andrés Villaveces, Departamento de Matemáticas, Universidad Nacional de Colombia, Av. Cra 30 # 45-03, Bogotá 111321 - Colombia. Correo electrónico: avillavecesn@unal.edu.co. DOI: https://doi.org/10.15446/recolma.v1n52.74566

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