that is closed under the monoid operation and contains the identity element of . Symbolically, is a submonoid of if , and whenever . In this case, is a monoid under the binary operation inherited from .

On the other hand, if is a subset of a monoid that is closed under the monoid operation, and is a monoid for this inherited operation, then is not always a submonoid, since the identity elements may differ. For example, the singleton set is