# Definition:Free Commutative Monoid

## Definition

The free commutative monoid on an indexed set $X = \family {X_j: j \in J}$ is the set $M$ of all monomials 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.