Model companion of ordered theories with an automorphism

TL;DR

This paper investigates conditions under which theories with automorphisms, specifically in ordered structures, admit a model companion, contrasting with prior results that showed non-existence in more general settings.

Contribution

The paper demonstrates that restricting automorphisms allows the existence of a model companion for theories of ordered structures, such as linear orders and ordered abelian groups.

Findings

Model companion exists for restricted automorphisms in linear orders.

Model companion exists for restricted automorphisms in ordered abelian groups.

Non-existence results for full automorphisms are contrasted with positive results for restrictions.

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.