Factoring a minimal ultrafilter into a thick part and a syndetic part

Abstract

Let S be an infinite discrete semigroup. The operation on S extends uniquely to the Stone-Čech compactification βS making βS a compact right topological semigroup with S contained in its topological center. As such, βS has a smallest two sided ideal, K(βS). An ultrafilter p on S is minimal if and only if p ∈ K(βS).We show that any minimal ultrafilter p factors into a thick part and a syndetic part. That is, there exist filters F and G such that F consists only of thick sets, G consists only of syndetic sets, and p is the unique ultrafilter containing F ∪ G.Letting L = F and C = G, the sets of ultrafilters containing F and G respectively, we have that L is a minimal left ideal of βS, C meets every minimal left ideal of βS in exactly one point, and L ∩ C = {p}. We show further that K(βS) can be partitioned into relatively closed sets, each of which meets each minimal left ideal in exactly one point.With some weak cancellation assumptions on S, one has also that for each minimal ultrafilter p, S * \ {p} is not normal. In particular, if p is a member of either of the disjoint sets K(βN, +) or K(βN, •), then N * \ {p} is not normal.