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Slender classes
Author(s) -
Downey Rod,
Montalbán Antonio
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn008
Subject(s) - stone's representation theorem for boolean algebras , mathematics , free boolean algebra , boolean algebras canonically defined , conjecture , complete boolean algebra , boolean algebra , two element boolean algebra , class (philosophy) , automorphism , lattice (music) , combinatorics , type (biology) , discrete mathematics , algebra over a field , pure mathematics , algebra representation , computer science , physics , ecology , artificial intelligence , biology , acoustics
A Π 1 0 class P is called thin if, given a subclass P ′ of P , there is a clopen C with ′ = P ∩ C . Cholak, Coles, Downey and Herrmann [ Trans. Amer. Math. Soc. 353 (2001) 4899–4924] proved that a Π 1 0 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 Π 1 0 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.