$p$-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.