Arf semigroup

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.

References

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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