 A free \(C\)-algebra is an algebra that lives at the frontier of syntax and semantics. On the semantic side it is a member of \(C\). On the syntactic side its elements behave like terms subject to the laws of \(C\), but no other laws expressible with its generators. Commutativity \(xy = yx\) is expressible with two generators and so a free monoid on two or more generators cannot be commutative, though the free monoid on one generator, namely the set of all finite strings over a one-letter alphabet does form a commutative monoid on one generator.