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 min(x + y, 10). Then restricting x and y from all the integers to the set of elements S = {0, 3, 5, 6, 8, 9, 10} forms a submonoid of the monoid {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} under the operation min(x + y, 10).