Notes on ultrafilter extensions of almost bounded structures

TL;DR

This paper investigates ultrafilter extensions of almost bounded relational structures, showing they retain many properties of bounded structures and exploring their implications for ultrapowers and modal logics.

Contribution

It introduces almost bounded structures, a relaxation of boundedness, and demonstrates their ultrafilter extensions preserve elementary substructure and embedding properties.

Findings

Ultrafilter extensions of almost bounded structures are elementary substructures.

Elementary embeddings can be lifted to ultrafilter extensions.

Countable extensions are isomorphic to certain ultrapowers.

Abstract

We extend some of our earlier results on the interconnection between ultrafilter extensions, and ultrapowers. Throughout we restrict ourselves to relational structures with one binary relation. Recently it was shown that for bounded structures, where a universal finite bound on the maximal in-and out-degree is given, ultrafilter extensions are elementary extensions of the original structures. Comparing the constructions, it seems that the real challenge is when the degree has no global finite bound, or there are elements with infinite degree. This is a first step towards this direction by slightly relaxing the notion of boundedness, called almost bounded structures. Among others, we show that members of this class are still elementary substructures of their extensions, elementary embeddings can be lifted up to the extensions, moreover for the countable case, the extensions are isomorphic to certain ultrapowers. We also comment on the modal logics they generate.