A HIERARCHY OF TREE-AUTOMATIC STRUCTURES

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 >= 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 >= 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a Sigma(1)(2)-set nor a Pi(1)(2)-set. We obtain that there exist infinitely many omega(n)-automatic, hence also co-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].