Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups
Dumitru I. Stamate, Francesco Strazzanti
TL;DR
The paper studies shifted numerical semigroups $M_n=\langle n,n+r_1,\dots,n+r_k\rangle$ and proves that, for sufficiently large $n$, key properties like nearly Gorenstein and almost symmetric are preserved under the shift to $M_{n+r_k}$, establishing a form of periodicity with period $r_k$. A central advance is a corrected, explicit bijection $\varphi_n$ on the pseudo-Frobenius sets, enabling formulas for Frobenius and pseudo-Frobenius numbers of $M_{n+\lambda r_k}$ and a stable ordering of these numbers under iteration. The authors also explore invariants such as the residue and reduced type, showing that residue can fail to be periodic, while canonical reductions and reduced type are periodic along the shifted family beyond explicit thresholds. Collectively, the results provide explicit, computable descriptions of how Frobenius-type data and tight invariants behave under large shifts, deepening understanding of asymptotic semigroup behavior and guiding computations in this family.
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 $M_{n+r_k}$.