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Journal of the London Mathematical Society Advance Access originally published online on March 27, 2008
Journal of the London Mathematical Society 2008 78(1):36-50; doi:10.1112/jlms/jdn008
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© 2008 London Mathematical Society

Slender classes

Rod Downey

School of Mathematics,
Statistics and Computer Science
Victoria University
PO Box 600
Wellington
New Zealand
Rod.Downey@mcs.vuw.ac.nz
http://www.mcs.vuw.ac.nz/~downey

Antonio Montalbán

Department of Mathematics
University of Chicago
5734 S, University Avenue
Chicago, IL 60637
USA
www.math.uchicago.edu/~antonio

A {Pi}10 class P is called thin if, given a subclass P' of P, there is a clopen C with P ' = P{cap}C. Cholak, Coles, Downey and Herrmann [Trans. Amer. Math. Soc. 353 (2001) 4899–4924] proved that a {Pi}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 {Pi}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) 735–749] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class.

2000 Mathematics Subject Classification 03D25, 03D28,03D45.

The first author's research was partially supported by the Marsden Fund of New Zealand. The second author was partially supported by NSF grant DMS-0600824 and by the Marsden Fund of New Zealand.

Received November 20, 2006; revised January 8, 2008; published online March 27, 2008.


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