On almost-symmetry in generalized numerical semigroups

TL;DR

This paper introduces the concept of almost-symmetry in generalized numerical semigroups, explores its properties, and provides methods to classify and organize these semigroups based on Frobenius elements.

Contribution

It defines almost-symmetry for generalized numerical semigroups and develops a framework for their characterization, including a tree structure for classification.

Findings

Almost-symmetric generalized numerical semigroups form a new Frobenius family.

The class extends irreducible generalized numerical semigroups.

A method to compute and organize these semigroups by Frobenius element.

Abstract

In this work we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that this class yields a new family of Frobenius generalized numerical semigroups and extends the class of irreducible generalized numerical semigroups. This investigation allows us to provide a method of computing all almost symmetric generalized numerical semigroup having a fixed Frobenius element and organizing them in a rooted tree depending on a chosen monomial order.