# Product-free sets in the free semigroup

## TL;DR

This paper investigates the maximum density of product-free subsets within the free semigroup over a finite alphabet, establishing that the maximum possible density is exactly 1/2 under a natural measure.

## Contribution

It proves that the maximum density of product-free subsets in the free semigroup over a finite alphabet is exactly 1/2, providing a precise measure of their size.

## Key findings

- Maximum density of product-free subsets is 1/2.
- The natural measure assigns weight |A|^{-n} to words of length n.
- The result characterizes the largest possible product-free sets.

## Abstract

In this paper, we study product-free subsets of the free semigroup over a finite alphabet $A$. We prove that the maximum density of a product-free subset of the free semigroup over $A$, with respect to the natural measure that assigns a weight of $|A|^{-n}$ to each word of length $n$, is precisely $1/2$.