Mrَwka 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 | β ψ ( ω , R ) ∖ ψ ( ω , R ) | = 1 . In other words there is a unique free z-ultrafilter p 0 on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for κ ⩽ c , Mrówkaʹs MADF R can be used to produce a MADF M ⊂ [ κ ] ω such that | β ψ ( κ , M ) ∖ ψ ( κ , M ) | = 1 . For κ > c , and every M ⊂ [ κ ] ω , it is always the case that | β ψ ( 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 p 0 . In particular both p and p 0 have a clopen local base in βψ (although β ψ ( κ , M ) need not be zero-dimensional). A result for κ > c , that does not apply to p 0 , is that for certain κ > c , p is a P-point in βψ.