 This monoid is an example of a free algebra, namely the free commutative monoid on two generators \(x\) and \(y\). What makes it free is that it satisfies no laws other than those of a commutative monoid. It is not however a free monoid because it satisfies the commutative law. The free monoid on two generators \(x\) and \(y\) is instead the set of all finite strings over the two-letter alphabet \(\{x,y\}\).