Model companion of ordered theories with an automorphism

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 $\mathcal{L}_σ:= \mathcal{L}\cup\{σ\}$ with a new unary function symbol $σ$, the bigger theory $T_σ:= T\cup\{``σ\mbox{is an} \mathcal{L}\mbox{-automorphism''}\}$ does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory $T_σ$ as the base theory $T$ together with a ``restricted'' class of automorphisms, then $T_σ$ can have a model companion in $\mathcal{L}_σ$. We show this in the context of linear orders and ordered abelian groups.

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