# Almost symmetric numerical semigroups with odd generators

## TL;DR

This paper investigates the structure of almost symmetric numerical semigroups generated by odd integers, providing characterizations and construction methods specifically for embedding dimension four and type three.

## Contribution

It offers a complete characterization and construction approach for almost symmetric semigroups with embedding dimension four generated by odd integers, focusing on type three cases.

## Key findings

- Characterization of when certain symmetric semigroups are generated by odd integers
- A method to construct all almost symmetric semigroups with embedding dimension four and type three
- Proof that all pseudo-Frobenius numbers are multiples of one generator

## Abstract

We study almost symmetric semigroups generated by odd integers. If the embedding dimension is four, we characterize when a symmetric semigroup that is not complete intersection or a pseudo-symmetric semigroup is generated by odd integers. Moreover, we give a way to construct all the almost symmetric semigroups with embedding dimension four and type three generated by odd elements. In this case we also prove that all the pseudo-Frobenius numbers are multiple of one of them and this gives many consequences on the semigroup and its defining ideal.