Exercise 6, § 7

Let $X$ be a set. (a) Let $N$ be a submagma of $M(X)$, and let $Y = N - (N.N)$. The injection $Y \to N$ extends to a homomorphism $u : M(Y) \to N$. Prove that $u$ is an isomorphism. (In other words, every submagma of a free magma is free.) (b) If $x \in M(X)$, we denote by $M_x$ the submagma of M(X) generated by x; after (a), $M_x$ is identified with the free magma constructed on x. Prove that, if $x \in M(X), y \in M(X)$, we have either $M_x \subset M_y$, or $M_y \subset M_x$, or $M_x \cap M_y = \varnothing$. (If $M_x \cap M_y \neq \varnothing$, let z be an element of $M_x \cap M_y$ of minimum length; we shall prove that we have either $z = x$, or $z = y$.)