Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups

Abstract

Given the integers $0<r_1<\cdots <r_k$, we consider the shifted family of semigroups $M_n=⟨n,n+r_1,\dots ,n+r_k⟩$, 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 $M_{n+r_k}$.

References in corpus (8)

• The trace of the canonical module
• On the defining equations of the tangent cone of a numerical semigroup ring
• On the set of elasticities in numerical monoids
• Canonical trace ideal and residue for numerical semigroup rings
• Minimal presentations of shifted numerical monoids
• Homogeneous numerical semigroups
• Apéry sets of shifted numerical monoids
• Torsion in Differentials and Berger's Conjecture