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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101901.png" /> be a set. Define sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101903.png" />, inductively as follows:
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{{TEX|done}}{{MSC|08B20}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101904.png" /></td> </tr></table>
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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:
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$$
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X_1 = X
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$$
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$$
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X_{n+1} = \coprod_{p+q=n} X_p \times X_q
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$$
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where $\coprod$ denotes the disjoint union (see [[Union of sets|Union of sets]]). Let
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$$
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M_X = \coprod_n X_n
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101905.png" /></td> </tr></table>
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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 $g$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101906.png" /> denotes the disjoint union (see [[Union of sets|Union of sets]]). Let
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Certain special subsets of $M_X$, called [[Hall set]]s (also [[Lazard set]]s), are important in combinatorics and the theory of Lie algebras.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101907.png" /></td> </tr></table>
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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]].
 
 
There is an obvious [[Binary relation|binary relation]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101908.png" />: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f1101909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019010.png" />, then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019011.png" /> goes to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019013.png" />. This is the free magma on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019014.png" />. It has the obvious freeness property: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019015.png" /> is any [[Magma|magma]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019016.png" /> is a function, then there is a unique morphism of magmas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019017.png" /> extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019018.png" />.
 
 
 
Certain special subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019019.png" />, called Hall sets (cf. [[Hall set|Hall set]]), are important in combinatorics and the theory of Lie algebras.
 
 
 
The free magma over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019020.png" /> can be identified with the set of binary complete, planar, rooted trees with leaves labelled by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110190/f11019021.png" />. See [[Binary tree|Binary tree]].
 
  
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Reutenauer,  "Free Lie algebras" , Oxford Univ. Press  (1993)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)</TD></TR>
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</table>

Latest revision as of 16:25, 8 April 2016

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://encyclopediaofmath.org/index.php?title=Free_magma&oldid=12862
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article