Morphisms on infinite alphabets, countable states automata and regular sequences

TL;DR

This paper explores the relationship between regular sequences, uniform morphisms on countable alphabets, and countable automata, showing invariance properties and providing new characterizations of such sequences.

Contribution

It establishes that certain regular sequences can be represented as projections of fixed points of uniform morphisms and generated by countable automata, revealing new structural insights.

Findings

Regular sequences can be viewed as projections of fixed points of uniform morphisms.

Regular sequences can be generated by countable states automata.

Regularity of sequences is invariant under certain codings.

Abstract

In this paper, we prove that a class of regular sequences can be viewed as projections of fixed points of uniform morphisms on a countable alphabet, and also can be generated by countable states automata. Moreover, we prove that the regularity of some regular sequences is invariant under some codings.