Proposition 2

Let $u : X \to Y$ be a mapping. If $u$ is injective (resp. surjective, bijective), so is $M(u)$.

Suppose $u$ is injective. When $X$ is empty, $M(X)$ is empty, hence $M(u)$ is injective. If $X$ is non-empty, there exists a mapping $v$ of $Y$ into $X$ such that $v \circ u$ is the identity mapping of $X$ (Set Theory, II, § 3, no. 8, Proposition 8); the mapping $M(v) \circ M(u) = M(v \circ u)$ is the identity mapping of $M(X)$ and hence $M(u)$ is injective.

When $u$ is surjective, there exists a mapping $w$ of $Y$ into $X$ such that $u \circ w$ is the identity mapping of $Y$ (Set Theory, II, § 3, no. 8, Proposition 8). Then $M(u) \circ M(w) = M(u \circ w)$ is the identity mapping of $M(Y)$ and hence $M(u)$ is surjective.

Finally, if $u$ is bijective, it is injective and surjective and hence $M(u)$ has the same properties.

Let $S$ be a subset of $X$. By Proposition 2 the injection of $S$ into $X$ can be extended to an isomorphism of $M(S)$ onto a submagma $M'(S)$ of $M(X)$. The magmas $M(S)$ and $M'(S)$ are identified by means of this isomorphism. Then $M(S)$ is the submagma of $M(X)$ generated by $S$.

Let $X$ be a set and $(u_\alpha, v_\alpha)_{\alpha \in I}$ be a family of ordered pairs of elements of $M(X)$. Let $R$ be the equivalence relation on $M(X)$ compatible with the law of $M(X)$ and generated by the $(u_\alpha, v_\alpha)$ (§ 1, no. 6). The magma $M(X)/R$ is called the magma defined by $X$ and the relators $(u_\alpha, v_\alpha)_{\alpha \in I}$. Let $h$ be the canonical morphism of $M(X)$ onto $M(X)/R$. Then $M(X)/R$ is generated by $h(X)$.

Let $N$ be a magma and $(n_x)_{x \in X}$ a family of elements of $N$. Let $k$ be the morphism from $M(X)$ to $N$ such that $k(x) = n_x$ for all $x \in X$ (Proposition 1). If $k(u_\alpha) = k(v_\alpha)$ for all $\alpha \in I$, there exists one and only one morphism $f : M(X)/R \to N$ such that $f(h(x)) = n_x$ for all $x \in X$ (§ 1, no. 6, Proposition 9).