Generic Automorphisms and the DMP

Hirotaka Kikyo

This is a joint work with Anand Pillay. Given a complete theory T with quantifier elimination in a language L, we consider the (incomplete) theory T ∪{“σ is an L-automorphism”} in the language L∪{σ} and denote its model companion by T A if it exists. Similarly, we denote the model companions for T with two automorphisms and for T with countably many automorphisms by T A_{2} and T A_{ω} respectively.

When T is the theory ACF of algebraically closed fields, T A exists and called ACF A. One axiomatization of ACF A is based on the fact that ACF has the DMP (definable multiplicity property).

Theorem 1. Let T be a strongly minimal theory with quantifier elimination. The following are equivalent: (1) T A_{2} exists; (2) T A_{ω} exists; (3) T has the DMP.

Chatzidakis pointed out to us that (1) implies (2).

Theorem 2. Let T be a strongly minimal theory with quantifier elimination which is a finite cover of a strongly minimal theory which does have the DMP. Then T A exists if and only if T has the DMP.

In general, there are many unstable theories T with no T A. For example, if T is unstable without independence property then there is no T A. Also, if T has the fcp then T_{σ} has no model completion ([1]). As far as I know, every known theory T which has T A is stable.

[1] H. Kikyo, Model companions of theories with an automorphism, MSRI Preprint #1998-039 (1998) (to appear in Journal of Symbolic Logic).

[2] H. Kikyo and A. Pillay, The definable multiplicity property and generic automorphisms, to appear in Annals of Pure and Applied Logic.

Address: Department of Mathematical Sciences, Tokai University, 1117 Kitakaname, Hiratsuka 259-1292, Japan

E-mail: kikyo@ss.u-tokai.ac.jp