On products of ultrafilters

TL;DR

This paper characterizes when the tensor product of two ultrafilters equals their Cartesian product, providing necessary and sufficient conditions under the assumption of the Generalized Continuum Hypothesis.

Contribution

It offers a complete characterization of the equality of tensor and Cartesian products of ultrafilters, linking ultrafilter properties and ultrapower embeddings.

Findings

Tensor product equals Cartesian product iff their Cartesian product is an ultrafilter.

Ultrafilters commute in the tensor product if and only if certain conditions hold.

Ultrafilter ultrapower embeddings restrict to definable embeddings of the other ultrafilter's ultrapower.

Abstract

Assuming the Generalized Continuum Hypothesis, this paper answers the question: when is the tensor product of two ultrafilters equal to their Cartesian product? It is necessary and sufficient that their Cartesian product is an ultrafilter; that the two ultrafilters commute in the tensor product; that for all cardinals $\lambda$, one of the ultrafilters is both $\lambda$-indecomposable and $\lambda^+$-indecomposable; that the ultrapower embedding associated to each ultrafilter restricts to a definable embedding of the ultrapower of the universe associated to the other.