free algebra

Let $\mathcal{K}$ be a class of algebraic systems (of the same type $\tau$ ). Consider an algebra $A\in\mathcal{K}$ generated by (http://planetmath.org/SubalgebraOfAnAlgebraicSystem) a set $X=\{x_{i}\}$ indexed by $i\in I$ . $A$ is said to be a free algebra over $\mathcal{K}$ , with free generating set $X$ , if for any algebra $B\in\mathcal{K}$ with any subset $\{y_{i}\mid i\in I\}\subseteq B$ , there is a homomorphism $\phi:A\to B$ such that $\phi(x_{i})=y_{i}$ .

If we define $f:I\to A$ to be $f(i)=x_{i}$ and $g:I\to B$ to be $g(i)=y_{i}$ , then freeness of $A$ means the existence of $\phi:A\to B$ such that $\phi\circ f=g$ .

Note that $\phi$ above is necessarily unique, since $\{x_{i}\}$ generates $A$ . For any $n$ -ary polynomial $p$ over $A$ , any $z_{1},\ldots,z_{n}\in\{x_{i}\mid i\in I\}$ , $\phi(p(z_{1},\ldots,z_{n}))=p(\phi(z_{1}),\ldots,\phi(z_{n}))$ .

For example, any free group is a free algebra in the class of groups. In general, however, free algebras do not always exist in an arbitrary class of algebras.

Remarks.

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$A$ is free over itself (meaning $\mathcal{K}$ consists of $A$ only) iff $A$ is free over some equational class.
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If $\mathcal{K}$ is an equational class, then free algebras exist in $\mathcal{K}$ .
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Any term algebra of a given structure $\tau$ over some set $X$ of variables is a free algebra with free generating set $X$ .

Title	free algebra
Canonical name	FreeAlgebra
Date of creation	2013-03-22 16:51:05
Last modified on	2013-03-22 16:51:05
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 08B20
Synonym	free algebraic system
Related topic	TermAlgebra
Defines	free generating set