Dilatations of numerical semigroups

TL;DR

This paper introduces the concept of dilatation for numerical semigroups, analyzing how key invariants and properties like Gorenstein-related conditions are preserved across infinite families generated from a base semigroup.

Contribution

It defines dilatation as a new construction for numerical semigroups and characterizes how invariants and properties are maintained within these families.

Findings

Dilatation preserves several semigroup invariants.

Properties like almost Gorenstein, 2-AGL, nearly Gorenstein are preserved.

Provides explicit descriptions of minimal generators and invariants for dilatations.

Abstract

This paper is focused on numerical semigroups and presents a simple construction, that we call dilatation, which, from a starting semigroup $S$, permits to get an infinite family of semigroups which share several properties with $S$. The invariants of each semigroup $T$ of this family are given in terms of the corresponding invariants of $S$ and the Ap\'ery set and the minimal generators of $T$ are also described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that $S$ satisfies one of these properties if and only if each dilatation of $S$ satisfies the corresponding one.