nLab submonoid

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 $G(M)$.

See also

• GCD ring