A Characterization of Strongly Dependent Ordered Abelian Groups

Dolich, Alfred; Goodrick, John; Dolich, Alfred; Goodrick, John

doi:10.15446/recolma.v52n2.77154

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Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.52 no.2 Bogotá Jul./Dec. 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

Alfred Dolich¹^*

John Goodrick²

^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

Abstract

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

Resumen

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

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Referencias

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[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 ]

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[10] ______, Strongly dependent theories, Israel Journal of Mathematics 204 (2014), 1-83. [ Links ]

Received: February 11, 2018; Accepted: May 27, 2018

^* Correspondencia: John Goodrick, Departamento de Matemáticas, Facultad de Ciencias, Universidad de los Andes, Carrera 1 # 18A-12 Bogotá, Colombia. Correo electrónico: jr.goodrick427@uniandes.edu.co.

This is an open-access article distributed under the terms of the Creative Commons Attribution License