A Hierarchy of TreeAutomatic 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 $\omega$treeautomatic 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 $\omega$treeautomatic, 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].
 Publication:

arXiv eprints
 Pub Date:
 November 2011
 arXiv:
 arXiv:1111.1504
 Bibcode:
 2011arXiv1111.1504F
 Keywords:

 Mathematics  Logic;
 Computer Science  Logic in Computer Science
 EPrint:
 To appear in The Journal of Symbolic Logic. arXiv admin note: substantial text overlap with arXiv:1007.0822