Numerical semigroups of coated odd elements

TL;DR

This paper investigates a special class of numerical semigroups called coated odd element semigroups, focusing on their Frobenius number, gender, and embedding dimension to understand their structural properties.

Contribution

It introduces and studies coated odd element numerical semigroups, analyzing their key invariants and expanding the understanding of their algebraic structure.

Findings

Characterization of Frobenius number for coated odd semigroups

Analysis of gender and embedding dimension in these semigroups

Identification of structural properties unique to coated odd semigroups

Abstract

A numerical semigroup $S$ is coated with odd elements (Coe-semigroup), if $\left\{x-1, x+1\right\}\subseteq S$ for all odd element $x$ in $S$. In this note, we will study this kind of numerical semigroups. In particular, we are interested in the study of the Frobenius number, gender and embedding dimension of a numerical semigroup of this type.