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).

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Weisstein, Eric W. "Submonoid." From MathWorld--A Wolfram Web Resource.

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