Mathematics – Logic
Scientific paper
2011-11-07
Journal of Symbolic Logic 77, 1 (2012) 350-368
Mathematics
Logic
To appear in The Journal of Symbolic Logic. arXiv admin note: substantial text overlap with arXiv:1007.0822
Scientific paper
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].
Finkel Olivier
Todorcevic Stevo
No associations
LandOfFree
A Hierarchy of Tree-Automatic Structures does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A Hierarchy of Tree-Automatic Structures, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Hierarchy of Tree-Automatic Structures will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-690091