Ordering MAD families a la Katětov

Abstract

Abstract An ordering (≤ K ) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size c and decreasing chains of length c + bellow every element. Assuming t = c a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every ω ω -bounding forcing ℙ of size c there is a Cohen-destructible, ℙ-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.