Symmetries on almost symmetric numerical semigroups

TL;DR

This paper characterizes almost symmetric numerical semigroups through pseudo-Frobenius numbers, provides criteria for duals to be almost symmetric, and explores the properties of gluings of such semigroups.

Contribution

It introduces a new characterization of almost symmetric numerical semigroups and analyzes their behavior under duality and gluing operations.

Findings

Almost symmetric semigroups are characterized by the symmetry of pseudo-Frobenius numbers.

A criterion is provided for the dual of a semigroup to be almost symmetric.

Gluing of non-symmetric semigroups results in a non-almost symmetric semigroup.

Abstract

The notion of almost symmetric numerical semigroup was given by V. Barucci and R. Fr\"oberg. We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for $H^*$ (the dual of $M$) to be almost symmetric numerical semigroup. Using these results we give a formula for multiplicity of an opened modular numerical semigroups. Finally, we show that if $H_1$ or $H_2$ is not symmetric, then the gluing of $H_1$ and $H_2$ is not almost symmetric.