Properties preserved by classes of Chu transforms

TL;DR

This paper characterizes the properties preserved by dense Chu transforms, especially focusing on inconsistency-flow formulas, and explores their applications in various mathematical structures like topological spaces, graphs, and ultrafilters.

Contribution

It provides a precise characterization of properties preserved by dense Chu transforms through the concept of inconsistency-flow formulas, extending the understanding of Chu transforms in model theory and category theory.

Findings

Inconsistency-flow formulas are exactly those preserved by dense Chu transforms.

Chu transforms between ultrafilters correspond to the Rudin-Keisler ordering.

The paper extends the application of Chu transforms to topological spaces, graphs, and ultrafilters.

Abstract

Chu spaces and Chu transforms were first investigated in category theory by Barr and Chu in 1979. In 2000 van Benthem shifted to the model-theoretic point of view by isolating a class of infinitary two-sorted properties, the flow formulas, which are preserved by all Chu transforms. D\v{z}amonja and V\"a\"an\"anen in 2021 considered a special kind of Chu transforms, satisfying a density condition. These authors used dense Chu transforms to compare abstract logics, in particular showing that such transforms preserve compactness. This motivates the problem of characterizing which properties are preserved by dense Chu transforms. We solve this problem by isolating the inconsistency-flow formulas, which are flow formulas with an added predicate to express inconsistency. Our main result characterizes the first-order inconsistency-flow formulas exactly as those first-order properties preserved by dense Chu transforms. Furthermore, we consider several instances of Chu transforms in concrete situations such as topological spaces, graphs and ultrafilters. In particular, we prove that Chu transforms between ultrafilters correspond to the Rudin-Keisler ordering.