Free Monoid - The Free Commutative Monoid

The Free Commutative Monoid

Given a set A, the free commutative monoid on A is the set of all finite multisets with elements drawn from A, with the monoid operation being multiset sum and the monoid unit being the empty multiset.

For example, if A = {a, b, c}, elements of the free commutative monoid on A are of the form

{ε, a, ab, a2b, ab3c4, ...}

The fundamental theorem of arithmetic states that the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the prime numbers.

The free commutative semigroup is the subset of the free commutative monoid which contains all multisets with elements drawn from A except the empty multiset.