The following is an excerpt from Serre's Lie algebras and Lie groups 1964 Lectures p.18:
Let $X$ be a set and define inductively a family of sets $X_n$ $(n\ge1)$ as follows
1) $X_1=X$
2) $X_n=\bigsqcup_{p+q=n}X_p\times X_q$ $(n\ge 2)$ ($=$ disjoint union).
Put $M_X=\bigsqcup_{n=1}^{\infty}X_n$ and define $M_X\times M_X\rightarrow M_X$ by means of $$X_p\times X_q\rightarrow X_{p+q}\subset M_X,$$ where the arrow is the canonical inclusion resulting from 2). The magma $M_X$ is called the free magma on $X$. An element $w$ of $M_X$ is called a non-associative word on $X$. Its length, $l(w)$, is the unique $n$ such that $w\in X_n$.
...
Properties of the free magma $M_X$:
- $M_X$ is generated by $X$.
- $m\in M_X - X\iff$$m=u.v$, with $u,v\in M_X$; and $u,v$ are uniquely determined by $m$.
I don't understand why he says that $M_X$ is generated by $X$. Let $$\mathfrak{P}=\{Y\subset M_X\ |\ X\subset Y\ \land\ Y.Y\subset Y\}.$$ Then $\bigcap_{Y\in\mathfrak{P}} Y$ is the magma generated by $X$. It is clear that $M_X\in\mathfrak{P}$: thus, $$\bigcap_{Y\in\mathfrak{P}} Y\subset M_X.$$ But I don't see how $M_X\subset \bigcap_{Y\in\mathfrak{P}} Y$; specifically, if $w\in M_X$, how can we show that for all $Y\in\mathfrak{P}$, it is the case that $w\in Y$?
Edit:
Suppose $w\in M_X$. Let $Y\in\mathfrak{P}$. Then there exists $n\ge 1$ such that $w\in X_n$. If $n=1$, then $w\in X_1=X\subset Y$; i.e. $w\in Y$. If $n\ge 2$, then there exists $(w',w'')\in X_p\times X_q$ such that $p+q=n$ and $w=w'w''$. I am not exactly sure what the next step is, here, to prove that $w\in Y$. Must I induct on $n$?