In mathematics, Arf semigroups are certain subsets of the non-negative integers closed under addition, that were studied by Cahit Arf (1948). 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 x + y − z.

For instance, the set containing zero and all even numbers greater than 10 is an Arf semigroup.

• Arf, Cahit (1948), "Une interprétation algébrique de la suite des ordres de multiplicité d'une branche algébrique", Proceedings of the London Mathematical Society, Second series, 50 (4): 256–287, doi:10.1112/plms/s2-50.4.256, ISSN 0024-6115, MR 0031785
• Rosales, J. C.; García-Sánchez, P. A. (2009), "2.2 Arf numerical semigroups", Numerical semigroups, Developments in Mathematics, vol. 20, New York: Springer, pp. 23–27, doi:10.1007/978-1-4419-0160-6, ISBN 978-1-4419-0159-0, MR 2549780.

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