Definition:Free Commutative Monoid

Definition
The free commutative monoid on an indexed set $X = \left\{{X_j: j \in J}\right\}$ is the set $M$ of all mononomials under the standard multiplication.

That is, it is the set $M$ of all finite sequences of $X$.

Also known as
Some sources refer to this as the free monoid on $X$, dropping the commutativity part.

Also see

 * Definition:Free Commutative Semigroup
 * Definition:Free Monoid