TITLE

# Slender classes

AUTHOR(S)
Downey, Rod; Montalbán, Antonio
PUB. DATE
August 2008
SOURCE
Journal of the London Mathematical Society;Aug2008, Vol. 78 Issue 1, p36
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
A Î 10 class P is called thin if, given a subclass Pâ€² of P, there is a clopen C with î”¯ â€² = Pâˆ©C. Cholak, Coles, Downey and Herrmann [Trans. Amer. Math. Soc. 353 (2001) 4899â€“4924] proved that a Î 10 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such thin classes are automorphic in the lattice of Î 10 classes under inclusion. From this it follows that if the boolean algebra has a finite number n of atoms, then the resulting classes are all automorphic. We prove a conjecture of Cholak and Downey [J. London Math. Soc. 70 (2004) 735â€“749] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class.
ACCESSION #
44398880

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