 The free commutative monoid on a set is that monoid whose generators behave like letters just as for free monoids (in particular they are still atoms), but which satisfy the additional law \(uv = vu\). We make further identifications, e.g. of “dog” and “dgo”. Order of letters in a word is now immaterial, all that matters is how many copies there are of each letter. This information can be represented as an \(n\)-tuple of natural numbers where \(n\) is the size of the alphabet. Thus the free commutative monoid on \(n\) generators is \(N^n\), the algebra of \(n\)-tuples of natural numbers under addition.