Reduction property and dimensional order property

Abstract

Question. Which properties of T ― Th(N) are also possessed by T (under certain conditions)? There are a few papers treating the question. In [HP] Hodges and Pillay have shown that if T is minimal over P (definition 2.3) and every automorphism of N can be extended to an automorphism of M (they call Mis a symmetric extension of N), then N is Ko-categorical iff M is Ko-categorical. In [KT] Kikyo and Tsuboi defined the 0-reduction property, the reduction property, the strong reduction property, and the uniform reduction property. These reduction properties are model theoretical rephrasing of symmetry. They have shown that if T is minimal over P and has the uniform reduction property (i.e., for each L-formula (p{xy), there is an L~ -formula {//(xz) such that (Vy)(3fe P)(Vjc e P)[<p(xy) <-+i/fp(xz)] holds), then T~ is A-stable iff T is A-stable and T~ is unidimensional iff T is unidimensional. In this paper, we mainly deal with the 0-reduction property (definition 2.1). The 0-reduction property together with the minimality condition ensures that T is not far from T~ if T is stable. But the 0-reduction property is not so strong for an unstable theory. In fact there is a theory T such that T has the 0-reduction property over P, is minimal over P, and the number of models of T is more than that of T~.