A hierarchy of tree-automatic structures

Abstract

AbstractWe considerωn-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of lengthωnfor some integern≥ 1. We show that all these structures areω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation forω2-automatic (resp.ωn-automatic forn> 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 forωn-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 manyωn-automatic, hence alsoω-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].