Countably-categorical Boolean rings with distinguished ideals

TL;DR

This paper classifies countable Boolean rings with finitely many distinguished ideals that are countably categorical, extending previous classifications of Boolean algebras by employing topological and order-theoretic methods.

Contribution

It provides two new classifications of such Boolean rings using invariants based on finite PO systems and finite posets, broadening the understanding of their structure.

Findings

Two classifications via invariants using finite PO systems and finite posets.

Extension of Macintyre and Rosenstein's work to rings with distinguished ideals.

Self-contained discussion linking to previous results.

Abstract

We describe and classify countable Boolean rings (which may or may not have a multiplicative identity) with finitely many distinguished ideals whose elementary theory is countably categorical. This extends the description by Macintyre and Rosenstein and subsequent authors of countably categorical Boolean algebras with finitely many distinguished ideals. Following Pierce, we take a topological approach using the language of PO systems (partially ordered sets with a distinguished subset) and topological Boolean algebras. We provide two different classifications via invariants that uniquely determine the isomorphism type: one using finite PO systems and the other using finite posets. We discuss how our findings link with previous results, but the paper is otherwise self-contained.