On automatic infinite permutations

TL;DR

This paper explores the concept of automaticity in infinite permutations, comparing it to infinite words, and introduces automata for generating specific permutations like Thue-Morse.

Contribution

It extends the notion of automaticity from infinite words to permutations and analyzes the relationships between different definitions.

Findings

Standard definitions of automaticity differ between permutations and words.

A chain of inclusions among various automaticity definitions is established.

An automaton generating the Thue-Morse permutation is constructed and studied.

Abstract

An infinite permutation $\alpha$ is a linear ordering of $\mathbb N$. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.