Arf semigroup
In mathematics, Arf semigroups are certain subsets of the non-negative integers closed under addition, that were studied by {{harvs|txt|last=Arf|first=Cahit|authorlink=Cahit Arf|year= 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 {{nowrap|x + y − z}}.
For instance, the set containing zero and all even numbers greater than 10 is an Arf semigroup.
References
- {{Citation | last1=Arf | first1=Cahit | title=Une interprétation algébrique de la suite des ordres de multiplicité d'une branche algébrique | doi=10.1112/plms/s2-50.4.256 | mr=0031785 | year=1948 | journal=Proceedings of the London Mathematical Society | series = Second series | issn=0024-6115 | volume=50 | issue=4 | pages=256–287}}
- {{citation
| last1 = Rosales | first1 = J. C.
| last2 = García-Sánchez | first2 = P. A.
| contribution = 2.2 Arf numerical semigroups
| doi = 10.1007/978-1-4419-0160-6
| isbn = 978-1-4419-0159-0
| location = New York
| mr = 2549780
| pages = 23–27
| publisher = Springer
| series = Developments in Mathematics
| title = Numerical semigroups
| volume = 20
| year = 2009}}.
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