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.
Publié le : 2003-12-14
Classification:
Maximal almost disjoint family; Rudin-Keisler ordering of filters,
Katětov order,
cardinal invariants of the continuum,
indestructibility of MAD families,
03E05,
03E17,
54B20
@article{1067620190,
author = {Hru\v s\'ak, Michael and Garc\'\i a Ferreira, Salvador},
title = {Ordering MAD families a la Kat\v etov},
journal = {J. Symbolic Logic},
volume = {68},
number = {1},
year = {2003},
pages = { 1337-1353},
language = {en},
url = {http://dml.mathdoc.fr/item/1067620190}
}
Hrušák, Michael; García Ferreira, Salvador. Ordering MAD families a la Katětov. J. Symbolic Logic, Tome 68 (2003) no. 1, pp. 1337-1353. http://gdmltest.u-ga.fr/item/1067620190/