Free magma
2020 Mathematics Subject Classification: Primary: 08B20 [MSN][ZBL]
A free algebra in the variety of magma. The free magma on a set 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) Zbl 0798.17001 |
[a3] | J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) Zbl 0132.27803 |
Free magma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_magma&oldid=54370