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Model companion of ordered theories with an automorphism


Authors: Michael C. Laskowski and Koushik Pal
Journal: Trans. Amer. Math. Soc. 367 (2015), 6877-6902
MSC (2010): Primary 03C10, 03C64; Secondary 20K30, 20A05
DOI: https://doi.org/10.1090/S0002-9947-2015-06496-4
Published electronically: February 12, 2015
MathSciNet review: 3378817
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Abstract: Kikyo and Shelah showed that if $ T$ is a theory with the Strict Order Property in some first-order language $ \mathcal {L}$, then in the expanded language $ \mathcal {L}_\sigma := \mathcal {L}\cup \{\sigma \}$ with a new unary function symbol $ \sigma $, the bigger theory $ T_\sigma := T\cup \{\lq\lq \sigma$$ \mbox { is an } \mathcal {L}\mbox {-automorphism''}\}$ does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory $ T_\sigma $ as the base theory $ T$ together with a ``restricted'' class of automorphisms, then $ T_\sigma $ can have a model companion in $ \mathcal {L}_\sigma $. We show this in the context of linear orders and ordered abelian groups.


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Additional Information

Michael C. Laskowski
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: mcl@math.umd.edu

Koushik Pal
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: koushik@math.umd.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06496-4
Received by editor(s): May 31, 2013
Published electronically: February 12, 2015
Additional Notes: The authors were partially supported by the first author’s NSF grant DMS-0901336.
Article copyright: © Copyright 2015 American Mathematical Society