On a class of Arf numerical semigroups
Citation
Süer, M., İlhan, S. (2016). On a class of Arf numerical semigroups. International Conference on Mathematics and Mathematics Education (ICMME-2016), 12-14 Mayıs 2016, ElazığAbstract
A subset S of N is called a numerical semigroup if S is closed under addition
and S has element 0 and N\S is finite where N denotes the set of nonnegative
integers. In this study, we are interested two subclass of maximal embedding
dimension numerical semigroups, which are those semigroups having the Arf
property and saturated numerical semigroups. We introduce a new class of both Arf
property and saturated numerical semigroups with multiplicity four. We consider
numerical semigroups minimally generated by {4, k, k+1, k+2}. Where k is an integer
greater than or equal to 5 and k is congruent to 1 (modulo 4). We prove that all these
semigroups are both numerical semigroups with Arf property and saturated numerical
semigroup.
There is not any formulas to calculate invariants as Frobenius number, gaps,
n(S) and genus of S even for numerical semigroup with multiplicity four. But this
invariants have been calculated by imposing some conditions on elements of the
numerical semigroup S. We calculate the Frobenius number, the genus and the set
of gaps of each of these numerical semigroups.
Additionally, we give a relation between the set of pseudo- Frobenius numbers
and the set of all fundamental gaps of these numerical semigroups.