$p$-numerical semigroups with $p$-symmetric properties, II

TL;DR

This paper explores advanced properties of $p$-numerical semigroups, generalizing classical symmetric and almost symmetric cases, and introduces $p$-Arf semigroups with new formulas and properties.

Contribution

It extends the theory of $p$-numerical semigroups by defining $p$-almost symmetric and $p$-Arf semigroups, providing generalized formulas and analyzing their properties.

Findings

Derived $p$-generalized formulas of Watanabe and Johnson.

Introduced and studied properties of $p$-Arf numerical semigroups.

Extended classical semigroup concepts to the $p$-setting.

Abstract

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.