# Definition:Free Commutative Monoid

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## 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.

## Also see

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids