Model companion of ordered theories with an automorphism

Abstract

Abstract. Kikyo and Shelah showed that if T is a theory with the Strict Order Property in some first-order language L, then in the expanded language Lσ := L ∪ {σ} with a new unary function symbol σ, the bigger theory Tσ := T ∪ {"σ is an L-automorphism"} does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory Tσ as the base theory T together with a "restricted" class of automorphisms, then Tσ can have a model companion in Lσ. We show this in the context of linear orders and ordered abelian groups.

Linearly ordered sets with only one operator have the amalgamation property

“…In the special case of a single strictly increasing automorphism AP in Theorem 3.1 is a consequence of [LP,Theorem 2.2]. Recall that a theory with model completion has AP.…”