Corollary

Under hypothesis (A) the homomorphisms $\phi_{i}$ and $h$ are injective. For $i \neq j$ in $I$, $\phi_{i}(M_{i}) \cap \phi_{j}(M_{j}) = h(\mathbf{A})$.

First $h$ is injective: if $h(a) = h(a')$, then $(a)$ and $(a')$ are two reduced decompositions of the same element of $M$, whence $a = a'$. Let $i \in I$; then $h(\mathbf{A}) = \phi_{i}(h_{i}(\mathbf{A})) \subset \phi_{i}(M_{i})$; the uniqueness of reduced decompositions implies

whence $\phi_{i}(M_{i} - h_{i}(\mathbf{A})) = \phi_{i}(M_{i}) - h(\mathbf{A})$.

The injectivity of the homomorphisms $\phi_{i}$ and the relation

for $i \neq j$ are then consequences of the following fact: for $i, j$ in $I$, $x$ in $M_{i} - h_{i}(\mathbf{A})$ and $y$ in $M_{j} - h_{j}(\mathbf{A})$, the relation $\phi_{i}(x) = \phi_{j}(y)$ implies $i = j$ and $x = y$. Let $x = h_{i}(a) \cdot p$ and $y = h_{j}(b) \cdot q$ with $a, b$ in $\mathbf{A}$, $p$ in $P_{i} - \{e_{i}\}$ and $y$ in $P_{j} - \{e_{j}\}$. Then $\phi_{i}(x) = h(a)\phi_{i}(p)$ and $\phi_{j}(y) = h(b)\phi_{j}(q)$ and hence $(a; i; p)$ and $(b; j; q)$ are two reduced decompositions of the same element of $M$. It follows that $i = j, a = b$ and $p = q$, whence $x = h_{i}(a)p = h_{j}(b)q = y$.

When hypothesis (A) is fulfilled, we shall identify each monoid $M_{i}$ with a submonoid of $M$ by means of $\phi_{i}$; similarly, we shall identify $\mathbf{A}$ with a submonoid of $M$ by $h$. Then $M$ is generated by $\bigcup_{i \in I} M_{i}$ and $M_{i} \cap M_{j} = \mathbf{A}$ for $i \neq j$.

Every element of $M$ can be written uniquely in the form $a \prod_{\alpha = 1}^{n} p_{\alpha}$ with $a \in \mathbf{A}, p_{1} \in P_{i_{1}} - \{e\}, \ldots, p_{n} \in P_{i_{n}} - \{e\}$ and $i_{\alpha} \neq i_{\alpha + 1}$ for $1 \leq \alpha < n$. Finally, if $M'$ is a monoid and $(f_{i}: M_{i} \to M')$ (for $i \in I$) a family of homomorphisms whose restrictions to $\mathbf{A}$ are the same homomorphism of $\mathbf{A}$ into $M'$, there exists one and only one homomorphism $f: M \to M'$ inducing $f_{i}$ on $M_{i}$ for all $i \in I$.

Hypothesis (A) is satisfied in two important cases:

(a) $A = \{e\}$. In this case, there is a family $(M_i)_{i \in I}$ of monoids and $M$ is called the monoidal sum of this family. Each $M_i$ is identified with a submonoid of $M$ and $M$ is generated by $\bigcup_{i \in I} M_i$; further, $M_i \cap M_j = \{e\}$ for $i \neq j$. Every element of $M$ may be written uniquely in the form $x_1 \ldots x_n$ with

and $i_\alpha \neq i_{\alpha+1}$ for $1 \leq \alpha \leq n$. Finally, for every family of homomorphisms $(f_i : M_i \to M')$, there exists a unique homomorphism $f : M \to M'$ whose restriction to $M_i$ is $f_i$ for all $i \in I$.

(b) There is a family of groups $(G_i)_{i \in I}$ containing as subgroup the same group $A$ and $h_i$ is the injection of $A$ into $G_i$. The sum of the family $(G_i)_{i \in I}$ amalgamated by $A$ is then a group $G$ : the monoid $G$ is generated by $\bigcup_{i \in I} \phi_i(G_i)$ and every element of $\bigcup_{i \in I} \phi_i(G_i)$ admits an inverse in $G$ (cf. § 2, no. 3, Corollary 1 to Proposition 4); it is denoted by $*_A G_i$ or $G_1 *_A G_2$ when $I = \{1, 2\}$. When $A$ consists of the identity element, it is also said that $G$ is the free product of the family $(G_i)_{i \in I}$ of groups and it is denoted by $* G_i$ (or $G_1 * G_2$ if $I = \{1, 2\}$).†