p-numerical semigroups with p-symmetric properties, II

Recently, the concept of the p-numerical semigroup with p-symmetric properties has been introduced. When $p=0$, the classical numerical semigroup with symmetric properties is recovered. In this paper, we further study the p-numerical semigroup with p-almost symmetric properties. We also give p-generalized formulas of Watanabe and Johnson, and introduce p-Arf numerical semigroup and study its properties.