The Rank of Tree-Automatic Linear Orderings

TL;DR

This paper extends the understanding of the complexity of tree-automatic linear orderings by establishing upper bounds on their FC-rank based on their branching complexity, generalizing previous results on ordinals.

Contribution

It generalizes Delhommé's result by showing that all tree-automatic linear orderings have FC-rank below , and further relates FC-rank bounds to branching complexity.

Findings

Tree-automatic linear orderings have FC-rank below .

Linear orderings with branching complexity at most k have FC-rank below ^k.

Generalization of previous ordinal bounds to broader classes of linear orderings.

Abstract

We generalise Delhomm\'e's result that each tree-automatic ordinal is strictly below \omega^\omega^\omega{} by showing that any tree-automatic linear ordering has FC-rank strictly below \omega^\omega. We further investigate a restricted form of tree-automaticity and prove that every linear ordering which admits a tree-automatic presentation of branching complexity at most k has FC-rank strictly below \omega^k.