Free magma

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Let be a set. Define sets , , inductively as follows:

where denotes the disjoint union (see Union of sets). Let

There is an obvious binary operation on : if , , then the pair goes to the element of . This is the free magma on . It has the obvious freeness property: if is any magma and is a function, then there is a unique morphism of magmas extending .

Certain special subsets of , called Hall sets (cf. Hall set), are important in combinatorics and the theory of Lie algebras.

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


[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:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article