  In this paper, we introduce the concept of Arf special gaps of an Arf
numerical semigroup, and an algorithm for computing all Arf special gaps of a
given Arf numerical semigroup. We introduce the concept of Arf-irreducible
numerical semigroups, and draw conclusions about all these concepts. We give a
system of generators for the Frobenius variety and variety of families of Arf
numerical semigroups. Moreover, we obtain the minimal Arf system of generators
of a given Arf numerical semigroup.
