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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.