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A hierarchy of tree-automatic structures

Published online by Cambridge University Press:  12 March 2014

Olivier Finkel  and
Stevo Todorčević
Olivier Finkel
Affiliation:
Equipe De Logique Mathématique, Institut De Mathématiques De Jussieu, CNRSand Université Paris 7, France, E-mail: [email protected]
Stevo Todorčević
Affiliation:
Equipe De Logique Mathématique, Institut De Mathématiques De Jussieu, CNRSand Université Paris 7, France Department of Mathematics, University of Toronto,Toronto, M5S 2E4, CanadaE-mail: [email protected], E-mail: [email protected]
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Abstract

We consider ωn-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ωn for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω2-automatic (resp. ωn-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 ωn-automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a -set nor a -set. We obtain that there exist infinitely many ωn-automatic, hence also ω-tree-automatic, atomless boolean algebras , which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 350 - 368
Copyright
Copyright © Association for Symbolic Logic 2012

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