Model companion of ordered theories with an automorphism

Laskowski, Michael; Pal, Koushik

doi:10.1090/S0002-9947-2015-06496-4

Model companion of ordered theories with an automorphism
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by Michael C. Laskowski and Koushik Pal PDF

Trans. Amer. Math. Soc. 367 (2015), 6877-6902 Request permission

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 $T_\sigma := T\cup \{“\sigma \mbox { is an } \mathcal {L}\mbox {-automorphism''}\}$ 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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