The Frobenius problem for a class of quotients of numerical semigroups

TL;DR

This paper characterizes the Apéry set and derives explicit formulas for the Frobenius number and genus of quotients of numerical semigroups, expanding understanding of their algebraic structure.

Contribution

It provides new characterizations and explicit formulas for Frobenius numbers of certain quotients of numerical semigroups, a novel extension in the field.

Findings

Characterized the Apéry set for a class of quotients of numerical semigroups.

Derived formulas for Frobenius number and genus under specific conditions.

Obtained explicit Frobenius number formulas for particular quotient cases.

Abstract

Given a numerical semigroup $S$ and a positive integer $p$, the quotient $\frac{S}{p}=\{x\in \mathbb{N} \mid px\in S\}$ also forms a numerical semigroup. In this paper, we first characterize the Ap\'ery set for a class of quotients of numerical semigroups. Under certain conditions, we then derive half-closed form formulas for their Frobenius number and genus. Furthermore, for specific values of part parameters, we obtain explicit formulas for the Frobenius number of certain quotients of numerical semigroups.