Exercise 3, § 7

Let $X$ be a set and $M_a$ the free magma constructed on a set $\{a\}$ with one element. Let $\rho$ denote the homomorphism of $M(X)$ into $M_a$ which maps every element of $X$ onto $a$; on the other hand let $\lambda : M(X) \to Mo(X)$ be the homomorphism defined in no. 9. Show that the mapping $w \mapsto (\lambda(w), \rho(w))$ is an isomorphism of $M(X)$ onto the submagma of $Mo(X) \times M_a$ consisting of the ordered pairs $(u, v)$ where $u$ and $v$ have the same length.