Characterizing Borel Isomorphism Among Some Weakly Minimal Trivial Theories

TL;DR

This paper explores the Borel isomorphism relations among certain weakly minimal trivial theories, providing characterizations and a complexity dichotomy for specific classes of theories.

Contribution

It offers a new characterization of Borel isomorphism among weakly minimal trivial theories and establishes a complexity dichotomy for tame expansions.

Findings

Characterization of Borel isomorphism among certain theories

Dichotomy in Borel complexity for tame expansions

Application to families of finite equivalence relations

Abstract

We characterize having Borel isomorphism relation among some weakly minimal trivial theories, namely the examples of families of finite equivalence relations from recent joint work with Laskowski, and tame expansions of crosscutting-equivalence relations. We also prove a dichotomy in Borel complexity for the latter.