Product-free sets in the free semigroup

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 $\mathrm{\mid }A{\mathrm{\mid }}^{-n}$ to each word of length $n$, is precisely $1\mathrm{/}2$.