# Definition:Free Monoid

## Definition

Let $S$ be a set.

A free monoid over $S$ is a monoid $M$ together with a mapping $i: S \to M$, subject to:

For all monoids $N$, for all mappings $f: S \to N$, there is a unique monoid homomorphism $\bar f: M \to N$, such that:
$\bar f \circ i = f$

This condition is called the universal (mapping) property or UMP of the free monoid over $S$.

### Category-Theoretic Formulation

Let $\mathbf{Mon}$ be the category of monoids, and let $\mathbf{Set}$ be the category of sets.

Let $\left\vert{\cdot}\right\vert$ be the underlying set functor on $\mathbf{Mon}$.

Let $M \in \mathbf{Mon}_0$ be a monoid, and let $i: S \to \left\vert{M}\right\vert$ be a mapping.

Then $\left({M, i}\right)$ is said to be a free monoid over $S$ if and only if:

For all $N \in \mathbf{Mon}_0$ and $f: S \to \left\vert{N}\right\vert \in \mathbf{Set}_1$, a unique $\bar f \in \mathbf{Mon}_1$ makes the following diagram commute:
$\begin{xy} <0em,4em>*{\mathbf{Mon} :}, <0em,1em>*{\mathbf{Set} :}, <4em,4em>*+{M} = "M", <8em,4em>*+{N} = "N", "M";"N" **@{.} ?>*@{>} ?*!/_1em/{\bar f}, <4em,1em>*+{\left\vert{M}\right\vert} = "MM", <8em,1em>*+{\left\vert{N}\right\vert} = "NN", <4em,-3em>*+{S} = "S", "MM";"NN" **@{-} ?>*@{>} ?*!/_1em/{\left\vert{\bar f}\right\vert}, "S";"MM" **@{-} ?>*@{>} ?*!/_1em/{i}, "S";"NN" **@{-} ?>*@{>} ?*!/^1em/{f} \end{xy}$

This condition is called the universal (mapping) property or UMP of the free monoid over $S$.

## Also see

• Results about free monoids can be found here.