# The linear nature of pseudowords

## TL;DR

This paper explores how pseudowords over certain algebraic structures can be uniquely reconstructed from associated labeled linear orders, specifically focusing on aperiodic finite semigroups.

## Contribution

It demonstrates that pseudowords over the pseudovariety of aperiodic finite semigroups are uniquely determined by their associated labeled linear orders.

## Key findings

- Pseudowords can be recovered from their linear orders in the aperiodic case.
- The structure of labeled linear orders encodes all information about the pseudoword.
- The method provides a new way to analyze pseudowords via order-theoretic properties.

## Abstract

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.