Proposition 3

Let M be a monoid. Every mapping f of X into M extends uniquely to a homomorphism of Mo(X) into M.

Let $g$ be a homomorphism of $\mathrm{Mo}(\mathbf{X})$ into $M$ extending $f$. If $w = (x_i)_{1 \leq i \leq n}$ is a word, then $w = \prod_{i=1}^n x_i$ in the monoid $\mathrm{Mo}(\mathbf{X})$, whence

in the monoid $M$ (§ 1, no. 2, formula (2)). This proves the uniqueness of $g$.

Let $h(w) = \prod_{i=1}^n f(x_i)$ for every word $w = (x_i)_{1 \leq i \leq n}$. The associativity theorem (§ 1, no. 3, Theorem 1) and the definition of product in $\mathrm{Mo}(\mathbf{X})$ imply $h(ww') = h(w)h(w')$. By convention the empty product $h(e)$ is the identity element of $M$ and $h(x) = f(x)$ for $x \in \mathbf{X}$. Hence $h$ is a homomorphism of $\mathrm{Mo}(x)$ into $M$ extending $f$.

Let $u : \mathbf{X} \to \mathbf{Y}$ be a mapping. By Proposition 3, there exists one and only one homomorphism of Mo(X) into Mo(Y) which coincides with u on X; it is denoted by Mo(u). It maps a word $(x_i)_{1 \leq i \leq n}$ to the word $(u(x_i))_{1 \leq i \leq n}$. As in the case of magmas (no. 1), the equation Mo(v \circ u) = Mo(v) \circ Mo(u) is established for every mapping $v : Y \to Z$ and it can be shown that Mo(u) is injective (resp. surjective, bijective) if u is. For every subset S of X, Mo(S) is identified with the submonoid of Mo(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 Mo(X). Let R be the equivalence relation on Mo(X) compatible with the law on Mo(X) and generated by the $(u_\alpha, v_\alpha)$ (§ 1, no. 6). The monoid Mo(X)/R is called the monoid defined by X and the relators $(u_\alpha, v_\alpha)_{\alpha \in I}$. Let h be the canonical morphism of Mo(X) onto Mo(X)/R. Then Mo(X) is generated by h(X).

Let N be a monoid and $(n_x)_{x \in X}$ a family of elements of N. Let k be the morphism of Mo(X) into N such that $k(x) = n_x$ for all $x \in X$ (Proposition 3). If $k(u_\alpha) = k(v_\alpha)$ for all $\alpha \in I$, there exists one and only one magma morphism $f : Mo(X)/R \to N$ such that $f(h(x)) = n_x$ for all $x \in X$ (§ 1, no. 6, Proposition 9); as k is unital, f is a monoid morphism.