Multiplicative finite embeddability vs divisibility of ultrafilters

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.