# Pseudo-solutions of word equations

## TL;DR

This paper introduces a unified framework for pseudo-solutions of word equations, simplifying the analysis of pseudo-repetitions and pseudo-periodicity in morphic words, and showing that classical periodicity results extend to pseudo-periodicity.

## Contribution

It presents a more general and simpler framework for pseudo-solutions and pseudo-rank, enabling broader analysis of word equations without case-by-case investigation.

## Key findings

- Classical equations forcing periodicity also force pseudo-periodicity.
- The framework simplifies the study of pseudo-repetitions in morphic words.
- No need for separate analysis of generalizations of key equations.

## Abstract

We present a framework which allows a uniform approach to the recently introduced concept of pseudo-repetitions on words in the morphic case. This framework is at the same time more general and simpler. We introduce the concept of a pseudo-solution and a pseudo-rank of an equation. In particular, this allows to prove that if a classical equation forces periodicity then it also forces pseudo-periodicity. Consequently, there is no need to investigate generalizations of important equations one by one.