Model companion of ordered theories with an automorphism

Abstract

Kikyo and Shelah showed that if T T is a theory with the Strict Order Property in some first-order language L \mathcal {L} , then in the expanded language L σ := L ∪ { σ } \mathcal {L}_\sigma := \mathcal {L}\cup \{\sigma \} with a new unary function symbol σ \sigma , the bigger theory T σ := T ∪ { “ σ is an L -automorphism } 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 σ T_\sigma as the base theory T T together with a “restricted” class of automorphisms, then T σ T_\sigma can have a model companion in L σ \mathcal {L}_\sigma . We show this in the context of linear orders and ordered abelian groups.