Ultrafilters maximal for finite embeddability

Abstract: In [1] the authors showed some basic properties of a pre-order that arose in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and they presented its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-\v{C}ech compactification of the discrete space of natural numbers. In this present paper we continue the study of these pre-orders. In particular, we prove that there exist ultrafilters maximal for finite embeddability, and we show that the set of such ultrafilters is the closure of the minimal bilateral ideal in the semigroup $(\bN,\oplus)$, namely $\overline{K(\bN,\oplus)}$. As a consequence, we easily derive many combinatorial properties of ultrafilters in $\overline{K(\bN,\oplus)}$. We also give an alternative proof of our main result based on nonstandard models of arithmetic.