Free Commutative Monoids
The free commutative monoid on a set X (see free object) can be taken to be the set of finite multisets with elements drawn from X, with the monoid operation being multiset sum and the empty multiset as identity element. Such monoids are also known as (finite) formal sums of elements of X with natural coefficents. The free commutative semigroup is the subset of the free commutative monoid which contains all multisets with elements drawn from X except the empty multiset.
Free abelian groups are formal sums (i.e. linear combinations) of elements of X with integer coefficients. Equivalently, they may be seen as signed finite multisets with elements drawn from X.
Read more about this topic: Multiset
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