 From the semantic side, a \(C\)-algebra \(B\) together with a subset \(X\) of \(B\) thought of as variables is said to be a free \(C\)-algebra on \(X\), or is freely generated by \(X\), when, given any \(C\)-algebra \(A\), any valuation in \(A\) of the variables in \(X\) (that is, any function \(f: X\rightarrow A\)) uniquely extends to a homomorphism \(h: B\rightarrow A\). (We say that \(h: B\rightarrow A\) extends \(f: X\rightarrow A\) when the restriction of \(h\) to \(X\) is \(f\).)