# A Hierarchy of Tree-Automatic Structures

Abstract : We consider $\omega^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $\omega^n$ for some integer $n\geq 1$. We show that all these structures are $\omega$-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for $\omega^2$-automatic (resp. $\omega^n$-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 $\omega^n$-automatic boolean algebras, $n > 1$, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a $\Sigma_2^1$-set nor a $\Pi_2^1$-set. We obtain that there exist infinitely many $\omega^n$-automatic, hence also $\omega$-tree-automatic, atomless boolean algebras $B_n$, $n\geq 1$, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [FT10].
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https://hal.archives-ouvertes.fr/hal-00638581
Contributeur : Olivier Finkel <>
Soumis le : dimanche 6 novembre 2011 - 10:31:59
Dernière modification le : vendredi 4 janvier 2019 - 17:32:32
Document(s) archivé(s) le : mardi 7 février 2012 - 02:20:49

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• HAL Id : hal-00638581, version 1
• ARXIV : 1111.1504

### Citation

Olivier Finkel, Stevo Todorcevic. A Hierarchy of Tree-Automatic Structures. The Journal of Symbolic Logic, Association for Symbolic Logic, 2012, 77 (1), pp.350-368. ⟨hal-00638581⟩

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