Ordering MAD families a la Katětov

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.