Trivial Isomorphisms between Reduced Products

TL;DR

This paper develops a method under weak forcing axioms to demonstrate that reduced products of certain countable models have only trivial automorphisms, with implications for structures like fields, orders, graphs, and Boolean algebras.

Contribution

It introduces a general technique linking forcing axioms to automorphism triviality in reduced products of various countable models.

Findings

Reduced products of finite or countably infinite structures have only trivial automorphisms under certain axioms.

Forcing axioms imply trivial automorphisms for reduced products of fields, orders, trees, and graphs.

OCA_T implies all automorphisms of P(N)/Fin are trivial.

Abstract

We introduce a general method for showing under weak forcing axioms that reduced products of countable models of a theory $T$ have as few automorphisms as possible. We show that such forcing axioms imply that reduced products of countably infinite or finite fields, linear orders, trees, or random graphs have only trivial automorphisms. We also show that Todor\v{c}evi\'c's Open Colouring Axiom, $\mathsf{OCA}_{\mathrm{T}}$, implies that all automorphisms of $\mathcal{P}(\mathbb{N})/{\mathrm{Fin}}$ are trivial.