A hierarchy of tree-automatic structures

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 Σ₂¹-set nor a Π₂¹-set. We obtain that there exist infinitely many ωⁿ-automatic, hence also ω-tree-automatic, atomless boolean algebras ℬ_n, n≥1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].