On the number of countable models of complete theories with finite Rudin–Keisler preorders

The aim of this article is to generalize the classification of complete theories with finitely many countable models with respect to two principal characteristics, Rudin–Keisler preorders and the distribution functions of the number of limit models, to an arbitrary case with a finite Rudin–Keisler preorder. We establish that the same characteristics play a crucial role in the case we consider. We prove the compatibility of arbitrary finite Rudin–Keisler preorders with arbitrary distribution functions $f$ satisfying the condition rang $\operatorname{rang}f\subseteq\omega\cup\{\omega,2^\omega\}$.