The theory of ordered abelian groups does not have the independence property

Authors:
Y. Gurevich and P. H. Schmitt

Journal:
Trans. Amer. Math. Soc. **284** (1984), 171-182

MSC:
Primary 03C60; Secondary 06F20

DOI:
https://doi.org/10.1090/S0002-9947-1984-0742419-0

MathSciNet review:
742419

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Abstract: We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmitt's *Elementary properties of ordered abelian groups*, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every -type in the theory of coloured chains has at most coheirs, thereby strengthening a result by B. Poizat.

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0742419-0

Article copyright:
© Copyright 1984
American Mathematical Society