Exercise 6, § 7

Let $X$ be a set.

(a) Let $N$ be a submagma of $M(X)$ and $Y = N - (N.N)$. The injection $Y \to N$ extends to a homomorphism $u : M(Y) \to N$. Show that $u$ is an isomorphism. (In other words, every submagma of a free magma is free.)

(b) If $x \in M(X)$, let $M_x$ denote the submagma of $M(X)$ generated by $x$; by (a), $M_x$ is identified with the free magma constructed on $x$. Show that, if $x, y \in M(X)$, then 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; show that either $z = x$ or $z = y$.)