A submonoid is a subset of the elements of a monoid that are themselves a monoid under the same monoid operation.
For example, consider the monoid formed by the nonnegative
integers under the operation 
.
Then restricting 
and 
from all the integers to the set of elements
forms a submonoid
of the monoid 
under the operation 
.
Submonoid
See also
MonoidExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Submonoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Submonoid.html