Free magma

Let $X$ be a set. Define sets $X_n$, $N \ge 1$, inductively as follows: $$ X_1 = X $$ $$ X_{n+1} = \coprod_{p+q=n} X_p \times X_q $$ where $\coprod$ denotes the disjoint union (see Union of sets). Let $$ M_X = \coprod_n X_n $$

There is an obvious binary operation on $M_X$: if $v \in X_p$, $w \in X_q$, then the pair $(v,w)$ goes to the element $(v,w)$ of $X_{p+q}$. This is the free magma on $X$. It has the obvious freeness property: if $N$ is any magma and $g : X \rightarrow N$ is a function, then there is a unique morphism of magmas $\tilde g : M_X \rightarrow N$ extending $g$.

Certain special subsets of $M_X$, called Hall sets (also Lazard sets), are important in combinatorics and the theory of Lie algebras.

The free magma over $X$ can be identified with the set of binary complete, planar, rooted trees with leaves labelled by $X$. See Binary tree.

References

[a1]	N. Bourbaki, "Groupes et algèbres de Lie" , 2: Algèbres de Lie libres , Hermann (1972)
[a2]	C. Reutenauer, "Free Lie algebras" , Oxford Univ. Press (1993)
[a3]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965)