Authors: Finkel, Olivier
Todorčević, Stevo 
Title: A hierarchy of tree-automatic structures
Journal: Journal of Symbolic Logic
Volume: 77
Issue: 1
First page: 350
Last page: 368
Issue Date: 1-Mar-2012
Rank: M21
ISSN: 0022-4812
DOI: 10.2178/jsl/1327068708
We consider ω n-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ω n for some integer n ≥ 1. We show that all these structures are co-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω 2- automatic (resp. ω n-automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ω n-automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a ∑ 12-set nor a π 12-set. We obtain that there exist infinitely many ω 2 -automatic, hence also ω-tree-automatic, atomless boolean algebras ℬ n, n ≥ 1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].
Keywords: ω-tree-automatic structures | ω -automatic structures n | Automata reading ordinal words | Boolean algebras | Groups | Independence results | Isomorphism relation | Models of set theory | Partial orders | Rings
Publisher: Cambridge University Press

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