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

TL;DR

This paper introduces and explores $p$-numerical semigroups, generalizing classical concepts like gaps and symmetry by considering the number of solutions to linear Diophantine equations, extending the theory of numerical semigroups.

Contribution

It develops a new framework for $p$-numerical semigroups, generalizing classical semigroup properties based on the $p$-Frobenius number and solution counts.

Findings

Defined $p$-gaps, $p$-symmetric, and $p$-pseudo-symmetric semigroups

Established properties of these $p$-semigroups

Connected $p$-semigroups to classical cases when $p=0$

Abstract

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation $a_1 x_1+\cdots+a_k x_k=n$ ($a_1,\dots,a_k$ are given positive integers with $\gcd(a_1,\dots,a_k)=1$) does not have a non-negative integer solution $(x_1,\dots,x_k)$. The generalized Frobenius number (called the $p$-Frobenius number) is the largest integer such that this linear equation has at most $p$ solutions. That is, when $p=0$, the $0$-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss $p$-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer $p$, $p$-gaps, $p$-symmetric semigroups, $p$-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When $p=0$, they correspond to the original gaps, symmetric semigroups, and pseudo-symmetric semigroups, respectively.