#
nLab

submonoid

# Contents

## Definition

A **submonoid** of a monoid $M$ with unit $1$ is a subset $N$ of $M$ containing $1$ which is also a monoid with respect to the inherited multiplication.

## Examples

- Given a group $G$, any subgroup is a submonoid.
- Given a category $C$ and a subcategory $D$, an object $X$ in both, the monoid $\Hom_D(X,X)$ is a submonoid of $\Hom_C(X,X)$.
- A commutative? and cancellative? monoid $M$ is a submonoid of its Grothendieck group $G(M)$.

Last revised on May 11, 2016 at 14:50:29.
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