Numerical semigroups with the Arf property and Arf closures Numerical semigroups with the Arf property are a special kind of numerical semigroups with maximal embedding dimension. A numerical semigroup S is Arf if for every x,y,z in S with x\geq y\geq z, one has that x+y-z\in S.

The intersection of two Arf numerical semigroups is again Arf, and thus we can consider the Arf closure of a set of nonnegative integers with greatest common divisor equal to one. Analogously as with MED numerical semigroups, we define Arf systems of generators and minimal Arf generating system for an Arf numerical semigroup. These are also unique(see ). S is a numerical semigroup.

Returns true if S is an Arf numerical semigroup and false otherwise.
gap> IsArfNumericalSemigroup(NumericalSemigroup(3,5,7)); true gap> IsArfNumericalSemigroup(NumericalSemigroup(3,7,11)); false gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,7,11)); true
S is a numerical semigroup.

Returns the Arf closure of S.
gap> ArfNumericalSemigroupClosure(NumericalSemigroup(3,7,11)); <Numerical semigroup> gap> MinimalGeneratingSystemOfNumericalSemigroup(last); [ 3, 7, 8 ]
S is an Arf numerical semigroup.

Returns the minimal MED generating system of S.
gap> MinimalArfGeneratingSystemOfArfNumericalSemigroup( > NumericalSemigroup(3,7,8)); [ 3, 7 ]
f is an integer greater than or equal to -1. The output is the set of all Arf numerical semigroups with Frobenius number f.
gap> ArfNumericalSemigroupsWithFrobeniusNumber(10); [ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup> ] gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup); [ [ 7, 9, 11, 12, 13, 15, 17 ], [ 3, 11, 13 ], [ 6, 9, 11, 13, 14, 16 ], [ 9, 11, 12, 13, 14, 15, 16, 17, 19 ], [ 4, 11, 13, 14 ], [ 8, 11, 12, 13, 14, 15, 17, 18 ], [ 7, 11, 12, 13, 15, 16, 17 ], [ 6, 11, 13, 14, 15, 16 ], [ 11 .. 21 ] ]