# A Characterization of Infinite LSP Words

## TL;DR

This paper characterizes infinite LSP words using $S$-adicity, providing a finite set of morphisms and an automaton to recognize all such words based on their directive words.

## Contribution

It introduces a new characterization of infinite LSP words through $S$-adicity and automata, extending previous finite word results to the infinite case.

## Key findings

- Infinite LSP words are characterized by $S$-adicity and automaton recognition.
- A finite set of morphisms and an automaton are sufficient for recognition.
- The characterization bridges finite and infinite word properties.

## Abstract

G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of $S$-adicity. More precisely we provide a finite set of morphisms $S$ and an automaton ${\cal A}$ such that an infinite word is LSP if and only if it is $S$-adic and all its directive words are recognizable by ${\cal A}$.