The magic of tensor products of ultrafilters

TL;DR

This paper explores the fundamental and new properties of tensor products of ultrafilters, highlighting their combinatorial significance and applications to additive number theory and semigroup structures.

Contribution

It introduces new characterizations of limits and combinatorial structures of tensor products of ultrafilters, expanding their theoretical understanding and applications.

Findings

Characterization of limit superior and inferior via tensor products

New combinatorial structure results for sets in tensor products

Representation of tensor products as idempotents in semigroups

Abstract

Tensor products of ultrafilters have special combinatorial features closely related to Ramsey's Theorem, making them useful tools in applications. Here we first review their fundamental properties and isolate some new ones, including a characterisation of the limit superior and inferior of sequences as limits along tensor products, and an application to the Banach asymptotic density. We then prove a general result on the combinatorial structure of sets belonging to tensor products and, as a result, we obtain several characterisations of the additive properties of sets of natural numbers. Finally, we show that tensor products can be described as idempotents of an appropriate semigroup.