Equivalents of NOTOP
Logic
2025-09-24 v1
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 -dominated by an independent triple is isolated over the triple. If has NOTOP, then every model 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).
Cite
@article{arxiv.2505.00844,
title = {Equivalents of NOTOP},
author = {Michael C. Laskowski and Danielle S. Ulrich},
journal= {arXiv preprint arXiv:2505.00844},
year = {2025}
}