On the genus of a quotient of a numerical semigroup

TL;DR

This paper establishes formulas linking the genus of a quotient of a numerical semigroup to the genus of the original semigroup, with specific computations for semigroups of embedding dimension two and those generated by arithmetic progressions.

Contribution

It introduces new identities connecting the genus and Frobenius number of quotients of numerical semigroups, expanding understanding of their algebraic structure.

Findings

Derived a relation between the genus of a quotient and the original semigroup

Computed the genus for semigroups with embedding dimension 2

Established identities for semigroups generated by arithmetic progressions

Abstract

We find a relation between the genus of a quotient of a numerical semigroup $S$ and the genus of $S$ itself. We use this identity to compute the genus of a quotient of $S$ when $S$ has embedding dimension $2$. We also exhibit identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions.