free algebra

Let $𝒦$ be a class of algebraic systems (of the same type $\tau$). Consider an algebra $A\in 𝒦$ 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 $𝒦$, with free generating set $X$, if for any algebra $B\in 𝒦$ with any subset $\{y_i\mid i\in I\}\subseteq B$, there is a homomorphism $\varphi :A\to B$ such that $\varphi (x_i)=y_i$.

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 $\varphi :A\to B$ such that $\varphi \circ f=g$.

Note that $\varphi$ above is necessarily unique, since $\{x_i\}$ generates $A$. For any $n$-ary polynomial $p$ over $A$, any $z_1,\mathrm{\dots },z_n\in \{x_i\mid i\in I\}$, $\varphi (p(z_1,\mathrm{\dots },z_n))=p(\varphi (z_1),\mathrm{\dots },\varphi (z_n))$.

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.

• $A$ is free over itself (meaning $𝒦$ consists of $A$ only) iff $A$ is free over some equational class.

• If $𝒦$ is an equational class, then free algebras exist in $𝒦$.

• Any term algebra of a given structure $\tau$ over some set $X$ of variables is a free algebra with free generating set $X$.