This paper generalizes results of F. Körner from [4] where she established the existence of maximal automorphisms (i.e. automorphisms moving all non-algebraic elements). An ω-maximal automorphism is an automorphism whose powers are maximal automorphisms. We prove that any structure has an elementary extension with an ω-maximal automorphism. We also show the existence of ω-maximal automorphisms in all countable arithmetically saturated structures. Further we describe the pairs of tuples (¯a,¯b) for which there is an ω-maximal automorphism mapping ¯a to ¯b.

We expand the notion of resplendency to theories of the kind T + p", where T is a fi rst-order theory and p" expresses that the type p is omitted. We investigate two di erent formulations and prove necessary and sucient conditions for countable recursively saturated models of PA. Some of the results in this paper can be found in one of the author's doctoral thesis [3].

We present a novel, perspicuous framework for building iterated ultrapowers. Furthermore, our framework naturally lends itself to the construction of a certain type of order indiscernibles, here dubbed tight indiscernibles, which are shown to provide smooth proofs of several results in general model theory.