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Chains of theories and companionability


Authors: Özcan Kasal and David Pierce
Journal: Proc. Amer. Math. Soc. 143 (2015), 4937-4949
MSC (2010): Primary 03C10, 03C60, 12H05, 13N15
Published electronically: July 15, 2015
MathSciNet review: 3391051
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Abstract: The theory of fields that are equipped with a countably infinite family of commuting derivations is not companionable, but if the axiom is added whereby the characteristic of the fields is zero, then the resulting theory is companionable. Each of these two theories is the union of a chain of companionable theories. In the case of characteristic 0, the model-companions of the theories in the chain form another chain, whose union is therefore the model-companion of the union of the original chain. However, in a signature with predicates, in all finite numbers of arguments, for linear dependence of vectors, the two-sorted theory of vector-spaces with their scalar-fields is companionable, and it is the union of a chain of companionable theories, but the model-companions of the theories in the chain are mutually inconsistent. Finally, the union of a chain of non-companionable theories may be companionable.


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Additional Information

Özcan Kasal
Affiliation: Middle East Technical University, Northern Cyprus Campus, Turkey
Email: kasal@metu.edu.tr

David Pierce
Affiliation: Mimar Sinan Fine Arts University, Istanbul, Turkey
Email: dpierce@msgsu.edu.tr

DOI: https://doi.org/10.1090/proc12789
Received by editor(s): May 15, 2013
Received by editor(s) in revised form: June 2, 2014
Published electronically: July 15, 2015
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society