Multiplicative finite embeddability vs divisibility of ultrafilters

TL;DR

This paper investigates the concept of multiplicative finite embeddability among ultrafilters, positioning it within existing divisibility relations, and explores the structure and hierarchy of ultrafilters with new notions of largeness in subsets of natural numbers.

Contribution

It introduces multiplicative finite embeddability as a new relation between ultrafilters, analyzes its minimal and maximal elements, and compares new largeness notions with existing ones.

Findings

Multiplicative finite embeddability lies between divisibility relations $igm|_M$ and $ ilde{igm|}$.

The set of minimal elements under this relation is very rich.

New notions of largeness of subsets of $ $ are introduced and compared to existing concepts.

Abstract

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations $\mid_M$ and $\widetilde{\mid}$. The set of its minimal elements proves to be very rich, and the $\widetilde{\mid}$-hierarchy is used to get a better intuition of this richness. We find the place of the set of $\widetilde{\mid}$-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets of $\mathbb{N}$, and compare it to other such notions, important for infinite combinatorics and topological dynamics.