Scott Complexity of Reduced Abelian $p$-Groups

TL;DR

This paper establishes an explicit upper bound on the Scott complexity of reduced abelian p-groups based on their Ulm invariants, demonstrating tightness for limit ordinals and providing a sequence with arbitrarily high complexity.

Contribution

It introduces a new algebraic characterization of back-and-forth relations on reduced abelian p-groups and links Scott complexity to Ulm invariants, advancing understanding of their model-theoretic properties.

Findings

Upper bound on Scott complexity in terms of Ulm invariants

Tightness of the bound for limit ordinals

Construction of groups with arbitrarily high Scott complexity

Abstract

Given a reduced abelian $p$-group, we give an upper bound on the Scott complexity of the group in terms of its Ulm invariants. For limit ordinals, we show that this upper bound is tight. This gives an explicit sequence of such groups with arbitrarily high Scott complexity below $\omega_1$. Along the way, we give a largely algebraic characterization of the back-and-forth relations on reduced abelian $p$-groups, making progress on an open problem from the book by Ash and Knight.