The numerical duplication of a numerical semigroup

TL;DR

This paper introduces and analyzes the numerical duplication of a numerical semigroup, a construction that generates new semigroups with specific symmetry properties, expanding understanding of their algebraic structure.

Contribution

It defines the numerical duplication of a semigroup, characterizes when the resulting semigroup is almost symmetric, and determines its type.

Findings

Characterization of ideals leading to almost symmetric duplicated semigroups

Determination of the type of the duplicated semigroup

Conditions under which the duplication preserves certain properties

Abstract

In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup $S$ and a semigroup ideal $E\subseteq S$, produces a new numerical semigroup, denoted by $S\Join^b\E$ (where $b$ is any odd integer belonging to $S$), such that $S=(S\Join^b\E)/2$. In particular, we characterize the ideals $E$ such that $S\Join^bE$ is almost symmetric and we determine its type.