# Free magma

2010 Mathematics Subject Classification: Primary: 08B20 [MSN][ZBL]

A free algebra in the variety of magma. The free magma on a set $X$ of free generators coincides with the set of all bracketed words in the elements of $X$. 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)
How to Cite This Entry:
Free magma. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Free_magma&oldid=38554
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article