# Definition:Submonoid

## Definition

A submonoid of a monoid $\struct {S, \circ}$ is a monoid $\struct {T, \circ}$ such that $T \subseteq S$.

We write $\struct {T, \circ} \subseteq \struct {S, \circ}$.

## Also defined as

Some sources insist that the identity element $e_T$ of a submonoid $\struct {T, \circ}$ of a monoid $\struct {S, \circ}$ must be the same element as the identity element $e_S$ of $\struct {S, \circ}$.

However, the more general definition as given here on $\mathsf{Pr} \infty \mathsf{fWiki}$ is such that it is not necessarily the case that they coincide.

## Also see

• Results about submonoids can be found here.