Ultrafilter extensions of bounded graphs are elementary

TL;DR

This paper investigates the relationship between ultrafilter extensions and bounded graphs, demonstrating that certain structures are elementary substructures of their ultrafilter extensions and share the same modal logic.

Contribution

It establishes that ultrafilter extensions of bounded graphs are elementary and have identical modal logics, linking model theory and modal logic in a novel way.

Findings

Bounded graphs are elementary substructures of their ultrafilter extensions.

Ultrafilter extensions preserve modal logic properties.

The study connects ultrafilter extensions with ultrapowers in model theory.

Abstract

The main motivation of this paper is the study of first-order model theoretic properties of structures having their roots in modal logic. We will focus on the connections between ultrafilter extensions and ultrapowers. We show that certain structures (called bounded graphs) are elementary substructures of their ultrafilter extensions, moreover their modal logics coincide.