Congruence of ultrafilters

Abstract

We continue the research of the relation | on the set βN of ultrafilters on N, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of =∼-equivalence classes, where F =∼ G means that F and G are mutually | -divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that =∼-equivalent ultrafilters do not necessarily have the same residue modulo m ∈ N. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we also introduce a strengthening of | and show that it also behaves well in relation to the congruence relation.