An algebraic approach to MSO-definability on countable linear orderings

TL;DR

This paper introduces an algebraic framework for recognizing languages over countable linear orderings, showing its equivalence to MSO logic definability and applying it to solve several open problems in logic and order theory.

Contribution

It develops an algebraic notion of recognizability for countable linear orderings and proves its equivalence to MSO definability, leading to new logical and structural results.

Findings

Established the first collapse result for MSO quantifier alternation over countable linear orderings.

Solved an open problem on MSO-definability of sets of rational numbers using reals.

Proved MSO-definability of yields from MSO-definable sets of trees.

Abstract

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruy{\`e}re, Carton, and S{\'e}nizergues.