Equivalents of NOTOP

TL;DR

This paper explores multiple equivalent conditions for a countable, superstable theory to have NOTOP, linking it to V-DI, atomic models over independent trees, and Shelah's PMOP, thereby deepening understanding of stability theory.

Contribution

It provides new equivalences for NOTOP in superstable theories, including V-DI and atomic model characterizations, and confirms Shelah's assertion relating NOTOP to PMOP.

Findings

NOTOP is equivalent to V-DI in superstable theories

Models with NOTOP are atomic over independent trees

NOTOP implies PMOP without NDOP

Abstract

Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type $V$-dominated by an independent triple is isolated over the triple. If $T$ has NOTOP, then every model $N$ is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah's assertion from Chapter XII of \cite{Shc} that NOTOP implies PMOP (without using NDOP).