The Rank of Tree-Automatic Linear Orderings

A tree-automatic structure is a structure whose domain can be encoded by a regular tree language such that each relation is recognisable by a finite automaton processing tuples of trees synchronously. The finite condensation rank (FC-rank) of a linear ordering measures how far it is away from being dense. We prove that the FC-rank of every tree-automatic linear ordering is below ω. This generalises Delhommé’s result that each tree-automatic ordinal is less than ωωω . Furthermore, we show an analogue for tree-automatic linear orderings where the branching complexity of the trees involved is bounded.