Exercise 4, § 7

Let $X$ be a set and $c$ an element not belonging to $X$. We write $Y = X \cup \{c\}$. Let $M$ be the magma with underlying set $Mo(Y)$ and law of composition given by $(u, v) \mapsto cuv$. Let $L$ be the submagma of $M$ generated by $X$ and let $\varepsilon$ be the mapping from $Y$ to $\mathbf{Z}$ equal to $-1$ on $X$ and to $1$ at $c$.

(a) Show that $L$ is the subset of $M$ consisting of the words $w = y_1 \ldots y_p$, $y_i \in Y$, satisfying the two following conditions: (i) $\varepsilon(y_1) + \cdots + \varepsilon(y_p) = -1$ (ii) $\varepsilon(y_1) + \cdots + \varepsilon(y_i) \geqslant 0$ for $1 \leqslant i \leqslant p - 1$. Show that every element of $L - X$ may be 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 least integer such that $\varepsilon(y_1) + \cdots + \varepsilon(y_k) = 0$; show that $y_1 = c$ and that the elements $u = y_2 \ldots y_k$ and $v = y_{k+1} \ldots y_p$ belong to $L'$; then $w = cuv$, whence by induction on $p$ the relation $w \in L$. Then show that the relations $u', v' \in L$ and $w = cu'v'$ imply $u' = u, v' = v$.)

(b) Show that the injection of $X$ into $L$ extends to an isomorphism of $M(X)$ onto $L$.