Structures Without Scattered-Automatic Presentation

Abstract

Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L. Automata on finite words can be used to represent infinite structures, the so-called word-automatic structures. Analogously, for a linear order L there is the class of L-automatic structures. In this paper we prove the following limitations on the class of L-automatic structures for a fixed L of finite condensation rank 1+\alpha. Firstly, no scattered linear order with finite condensation rank above \omega^(\alpha+1) is L-\alpha-automatic. In particular, every L-automatic ordinal is below \omega^\omega^(\alpha+1). Secondly, we provide bounds on the (ordinal) height of well-founded order trees that are L-automatic. If \alpha is finite or L is an ordinal, the height of such a tree is bounded by \omega^{\alpha+1}. Finally, we separate the class of tree-automatic structures from that of L-automatic structures for any ordinal L: the countable atomless boolean algebra is known to be tree-automatic, but we show that it is not L-automatic.