An analogue of U-rank for atomic classes

TL;DR

This paper introduces an analogue of U-rank for atomic models in countable theories, establishing conditions for the number of models and their structural properties based on type ranks.

Contribution

It develops a U-rank analogue for atomic classes and proves results relating type rank finiteness to model multiplicity and structure.

Findings

Non-ranked types imply 2^{} non-isomorphic models of size .

Finite rank types lead to additive rank and domination by independent pseudo-minimal types.

Rank finiteness correlates with structural simplicity and model count.

Abstract

For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types.