Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups

TL;DR

This paper investigates the properties of shifted numerical semigroups, proving that certain algebraic properties are preserved under shifts and providing explicit formulas for key numerical invariants.

Contribution

It establishes the preservation of nearly Gorenstein and almost symmetric properties under shifts in numerical semigroups and corrects previous misconceptions about pseudo-Frobenius elements.

Findings

Preservation of properties under shifts for large n

Explicit formulas for Frobenius and pseudo-Frobenius numbers

Correction of a previous incorrect claim in the literature

Abstract

Given the integers $0<r_1<\dots<r_k$, we consider the shifted family of semigroups $M_n=\langle n, n+r_1,\dots, n+r_k\rangle$, where $n>0$. For sufficiently large $n$, we prove that if $M_n$ is nearly Gorenstein or almost symmetric, then so is $M_{n+r_k}$. A key ingredient is to relate the pseudo-Frobenius elements of $M_n$ and $M_{n+r_k}$, correcting a wrong claim in the literature. Moreover, we derive explicit formulas for the Frobenius and pseudo-Frobenius numbers of $M_{n+r_k}$.