Difference between revisions of "Free magma"
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− | + | 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 [[word]]s 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|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 set]]s (also [[Lazard set]]s), 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 tree]]s with leaves labelled by $X$. See [[Binary tree]]. | |
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− | The free magma over | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , '''2: Algèbres de Lie libres''' , Hermann (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Reutenauer, "Free Lie algebras" , Oxford Univ. Press (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , '''2: Algèbres de Lie libres''' , Hermann (1972)</TD></TR> | ||
+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Reutenauer, "Free Lie algebras" , Oxford Univ. Press (1993)</TD></TR> | ||
+ | <TR><TD valign="top">[a3]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965)</TD></TR> | ||
+ | </table> |
Revision as of 16:25, 8 April 2016
2020 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) |
Free magma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_magma&oldid=34706