# Minimal presentations of shifted numerical monoids

@article{Conaway2018MinimalPO, title={Minimal presentations of shifted numerical monoids}, author={Rebecca Conaway and Felix Gotti and Jesse Horton and Christopher O'Neill and Roberto Pelayo and Mesa Pracht and Brian Wissman}, journal={Int. J. Algebra Comput.}, year={2018}, volume={28}, pages={53-68} }

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of “shifted” monoids Mn obtained by adding n to each generator of S. In this paper, we examine minimal relations among the generators of Mn when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We explore several applications to computation and factorization theory, and improve a recent result of Thanh Vu from combinatorial…

## 11 Citations

On arithmetical numerical monoids with some generators omitted

- MathematicsSemigroup Forum
- 2018

Numerical monoids (cofinite, additive submonoids of the non-negative integers) arise frequently in additive combinatorics, and have recently been studied in the context of factorization theory.…

PARAMETRIZED AND SHIFTED NUMERICAL SEMIGROUPS

- Mathematics
- 2018

A numerical semigroup S is an additive subgroup of the non-negative integers. Previous works have developed the shifted numerical semigroup family Mn which comes from adding n to each generator of S…

On parametrized families of numerical semigroups

- Mathematics
- 2019

Abstract A numerical semigroup is an additive subsemigroup of the non-negative integers. In this article, we consider parametrized families of numerical semigroups of the form for polynomial…

On minimal presentations of shifted affine semigroups with few generators

- MathematicsInvolve, a Journal of Mathematics
- 2021

An affine semigroup is a finitely generated subsemigroup of $(\mathbb Z_{\ge 0}^d, +)$, and a numerical semigroup is an affine semigroup with $d = 1$. A growing body of recent work examines shifted…

Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

- MathematicsAlgebra Colloquium
- 2019

Let C(n) be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve C(n + wv), where w > 0 is an…

Canonical trace ideal and residue for numerical semigroup rings

- MathematicsSemigroup Forum
- 2021

For a numerical semigroup ring $K[H]$ we study the trace of its canonical ideal. The colength of this ideal is called the residue of $H$. This invariant measures how far is $H$ from being symmetric,…

On length densities

- MathematicsForum Mathematicum
- 2021

Abstract For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x∈M{x\in M}, denoted LD(x){\operatorname{LD}(x)}, and the entire monoid M, denoted…

Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic

- Mathematics
- 2019

Let $f_1(n), \ldots, f_k(n)$ be polynomial functions of $n$. For fixed $n\in\mathbb{N}$, let $S_n\subseteq \mathbb{N}$ be the numerical semigroup generated by $f_1(n),\ldots,f_k(n)$. As $n$ varies,…

Betti numbers for numerical semigroup rings

- Mathematics
- 2016

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.

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