Proposition 4

(a) For all $i \in I$, the mapping $\phi_i$ is a homomorphism of $M_i$ into M and $\phi_i \circ h_i = h$ for all $i \in I$. Further, M is generated by $\bigcup_{i \in I} \phi_i(M_i)$.

(b) Let $M'$ be a monoid and $f' : M_i \to M'$ (for $i \in I$) homomorphisms such that $f_i \circ h_i$ is independent of $i \in I$. There exists one and only one homomorphism $f : M \to M'$ such that $f_i = f \circ \phi_i$ for all $i \in I$.

In what follows we shall make the following hypothesis:

(A) For all $i \in I$, there exists a subset $P_i$ of $M_i$ containing $e_i$ such that the mapping $(a, p) \mapsto h_i(a).p$ of $A \times P_i$ into $M_i$ is bijective.

It implies that the homomorphisms $h_i$ are injective. Let $x \in M$; every finite sequence $\sigma = (a; i_1, \ldots, i_n; p_1, \ldots, p_n)$ with $a \in A, i_\alpha \in I$ and $p_\alpha \in P_{i_\alpha}$ for $1 \leq \alpha \leq n$, satisfying

is called a decomposition of $x$. The integer $n \geq 0$ is called the length of the decomposition $\sigma$ and is denoted by $l(\sigma)$; the sequence $(e)$ is a decomposition of length 0 of the identity element of $M$. The decomposition $\sigma$ is called reduced if $i_\alpha \neq i_{\alpha+1}$ for $1 \leq \alpha < n$ and $p_\alpha \neq e_{i_\alpha}$ for $1 \leq \alpha \leq n$.