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ℵ 0 ‐Categorical Finite Extensions of Boolean Powers
Author(s) -
Apps A. B.
Publication year - 1983
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-47.3.385
Subject(s) - mathematics , categorical variable , stone's representation theorem for boolean algebras , complete boolean algebra , two element boolean algebra , permutation (music) , boolean algebras canonically defined , automorphism , partition (number theory) , boolean ring , discrete mathematics , combinatorics , pure mathematics , algebra over a field , algebra representation , principal ideal ring , statistics , physics , commutative ring , commutative property , acoustics
We consider when a finite extension of a Boolean power of a finite perfect group by a Boolean ring R is ℵ 0 ‐categorical. We obtain necessary and sufficient conditions for this, initially in terms of a certain induced finite group of automorphisms of R , and subsequently in terms of the topological behaviour of an induced finite partition of the Stone space of R . In so doing, we introduce the notions of a ‘topological permutation system’ and a ‘Boolean representation’ thereof, and show that any topological permutation system has a unique Boolean representation, and that this representation is ℵ 0 ‐categorical. Lastly, it is shown that a subgroup of index 2 in an ℵ 0 ‐categorical group need not be ℵ 0 ‐categorical.