Structures without Scattered-Automatic Presentation

Abstract

Bruyère and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order \(\mathfrak{L}\). Automata on finite words can be used to represent infinite structures, the so-called word-automatic structures. Analogously, for a linear order \(\mathfrak{L}\) there is the class of \(\mathfrak{L}\)-automatic structures. In this paper we prove the following limitations on the class of \(\mathfrak{L}\)-automatic structures for a fixed \(\mathfrak{L}\) of finite condensation rank 1 + α.

Firstly, no scattered linear order with finite condensation rank above ω ^α + 1 is \(\mathfrak{L}\)-automatic. In particular, every \(\mathfrak{L}\)-automatic ordinal is below \(\omega^{\omega^{\alpha+1}}\). Secondly, we provide bounds on the (ordinal) height of well-founded order trees that are \(\mathfrak{L}\)-automatic. If α is finite or \(\mathfrak{L}\) is an ordinal, the height of such a tree is bounded by ω ^α + 1. Finally, we separate the class of tree-automatic structures from that of \(\mathfrak{L}\)-automatic structures for any ordinal \(\mathfrak{L}\): the countable atomless boolean algebra is known to be tree-automatic, but we show that it is not \(\mathfrak{L}\)-automatic.