Editing Free monoid (section)

==The free commutative monoid==
Given a set ''A'', the '''free [[commutative monoid]]''' on ''A'' is the set of all finite [[multiset]]s 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'', ''a''<sup>2</sup>''b'', ''ab''<sup>3</sup>''c''<sup>4</sup>, ...}.

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 number]]s.

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

The [[free partially commutative monoid]], or ''[[trace monoid]]'', is a generalization that encompasses both the free and free commutative monoids as instances. This generalization finds applications in [[combinatorics]] and in the study of [[Parallel computing|parallelism]] in [[computer science]].