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Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. vol.52 no.2 Bogotá jul./dez. 2018
https://doi.org/10.15446/recolma.v52n2.77154
Original articles
A Characterization of Strongly Dependent Ordered Abelian Groups
Una caracterización de los grupos abelianos fuertemente dependientes
1 Kingsborough Community College. Department of Mathematics and Computer Science, 2001 Oriental Boulevard, Brooklyn, NY 11235-2398. e-mail: alfredo.dolich@kbcc.cuny.edu
2 Universidad de los Andes. Departamento de Matemáticas, Facultad de Ciencias, Carrera 1 # 18A-12, Bogotá, Colombia. e-mail: jr.goodrick427@uniandes.edu.co
We characterize all ordered Abelian groups whose first-order theory in the language {+, <, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].
Keywords: Strongly dependent theories; NIP; ordered Abelian groups
Damos una caracterización completa de los grupos abelianos ordenados cuyas teorías completas en el lenguaje {+; <; 0} son fuertemente dependientes. El resultado principal de este artículo fue obtenido de manera independiente por Halevi y Hasson [7] y Farré [5].
Palabras clave: Teorías dependientes; grupos abelianos ordenados
Referencias
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[2] A. Chernikov, I. Kaplan, and P. Simon, Groups and fields with NTP 2, Proceedings of the American Mathematical Society 143 (2015), no. 1, 395-406. [ Links ]
[3] R. Cluckers and I. Halupczok, Quantifier elimination in ordered abelian groups, Confluentes Mathematici 3 (2011), no. 4, 587-615. [ Links ]
[4] A. Dolich, J. Goodrick, and D. Lippel, Dp-minimal theories: basic facts and examples, Notre Dame Journal of Formal Logic 52 (2011), no. 3, 267-288. [ Links ]
[5] R. Farré, Strong ordered abelian groups and dp-rank, arXiv: 1706.05471, 2017. [ Links ]
[6] Y. Gurevich and P. H. Schmitt, The theory of ordered abelian groups does not have the independence property, Transactions of the American Mathematical Society 284 (1984), no. 1, 171-182. [ Links ]
[7] Y. Halevi and A. Hasson, Strongly dependent ordered abelian groups and henselian fields, arXiv: 1706.03376, 2017. [ Links ]
[8] F. Jahnke, P. Simon , and E. Walsberg, Dp-minimal valued fields, Journal of Symbolic Logic 82 (2015), 151-165. [ Links ]
[9] S. Shelah, Classification theory, second ed., North-Holland, 1990. [ Links ]
[10] ______, Strongly dependent theories, Israel Journal of Mathematics 204 (2014), 1-83. [ Links ]
Received: February 11, 2018; Accepted: May 27, 2018
This is an open-access article distributed under the terms of the Creative Commons Attribution License
