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.
ReferencesEdit
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, ISBN978-1-4419-0159-0, MR 2549780.
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semigroup, mathematics, certain, subsets, negative, integers, closed, under, addition, that, were, studied, cahit, 1948, they, appeared, semigroups, values, rings, subset, integers, forms, monoid, includes, zero, every, elements, subset, have, that, also, belo. 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 EditArf Cahit 1948 Une interpretation algebrique de la suite des ordres de multiplicite d une branche algebrique 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 Garcia Sanchez 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 nbsp This abstract algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Arf semigroup amp oldid 1072495774, wikipedia, wiki, book, books, library,
