Mrówka maximal almost disjoint families for uncountable cardinals

We consider generalizations of a well-known class of spaces, called by S. Mrówka, N∪R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers N. We denote these generalizations by ψ=ψ(κ,R) for κ⩾ω. Mrówka proved the interesting theorem that there exists an R such that |βψ(ω,R)∖ψ(ω,R)|=1. In other words there is a unique free z-ultrafilter p0 on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for κ⩽c, Mrówka's MADF R can be used to produce a MADF M⊂ω[κ] such that |βψ(κ,M)∖ψ(κ,M)|=1. For κ>c, and every M⊂ω[κ], it is always the case that |βψ(κ,M)∖ψ(κ,M)|≠1, yet there exists a special free z-ultrafilter p on ψ(κ,M) retaining some of the properties of p0. In particular both p and p0 have a clopen local base in βψ (although βψ(κ,M) need not be zero-dimensional). A result for κ>c, that does not apply to p0, is that for certain κ>c, p is a P-point in βψ.