A note on a problem of \textit{Hua} (Q1239272)

Let \(X^{+}\) denote the free semigroup on the finite alphabet \(X\). A mapping \(f:X^{+}\rightarrow X{+}\) is said to satisfy \textit{Hua's} condition if for every \(x,y\in X^{+}\) either \(f(xy) =f(x)f(y)\) or \(f(xy)=f(y)f(x)\). It is shown that such a mapping is either a homomorphism or an anti-homomorphism. The methods used to prove this result are of a combinatorial nature and quite elementary. Some simple facts about codes and primitive words are used together with several applications of induction.