Exercise 4, § 7

Let $X$ be a set and $c$ an element not belonging to $X$. Put $Y = X \cup \{c\}$. Let $M$ be the magma having $Mo(Y)$ as underlying set, the composition law being given by $(u, v) \mapsto cuv$. Let $L$ be the submagma of $M$ generated by $X$, and let $\varepsilon$ be the mapping of $Y$ into $\mathbf{Z}$ equal to $-1$ on $X$ and to $1$ at $c$. a) Prove that $L$ is the subset of $M$ formed by the words $w = y_1 \ldots y_p$, where $y_i \in Y$, satisfying the following two conditions: (i) $\varepsilon(y_1) + \cdots + \varepsilon(y_p) = -1$. (ii) $\varepsilon(y_1) + \cdots + \varepsilon(y_i) \geq 0$ for $1 \leq i \leq p - 1$. Prove that every element of $L - X$ is written uniquely in the form $cuv$, with $u, v$ in $L$. (The elements $w$ satisfying (i) and (ii) form a submagma $L'$ of $M$ containing $X$. If $w = y_1 \ldots y_p$ belongs to $L'$, let $k$ be the smallest integer such that $\varepsilon(y_1) + \cdots + \varepsilon(y_k) = 0$; prove that one has $y_1 = c$ and that the elements $u = y_2 \ldots y_k$ and $v = y_{k+1} \ldots y_p$ belong to $L'$; one has $w = cuv$, whence by induction on $p$ the relation $w \in L$. Then prove that the relations $u' \in L, v' \in L$ and $w = cu'v'$ entail $u' = u, v' = v.$) b) Prove that the injection of $X$ into $L$ extends to an isomorphism of $M(X)$ onto $L$.