Arf semigroup

In mathematics, Arf semigroups are certain subsets of the non-negative integers closed under addition, that were studied by . They appeared as the semigroups of values of Arf rings. 
A subset of the integers forms a monoid if it includes zero, and if every two elements in the subset have a sum that also belongs to the subset. In this case, it is called a "numerical semigroup".
A numerical semigroup is called an Arf semigroup if, for every three elements "x", "y", and "z" with "z" = min("x", "y", and "z"), the semigroup also contains the element .
For instance, the set containing zero and all even numbers greater than 10 is an Arf semigroup.