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… 

Figures and Tables from this paper

On arithmetical numerical monoids with some generators omitted
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
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
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
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
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
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
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
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
We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.
...
...

References

SHOWING 1-10 OF 16 REFERENCES
The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids
Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then
Finitely generated commutative monoids
There is a lack of effective methods for studying properties of finitely generated commutative monoids. This was one of the main reasons for developing a self-contained book on finitely generated
Shifts of generators and delta sets of numerical monoids
TLDR
If t = 2 and r1 and r2 are relatively prime, then the value for N which is sharp is determined.
ON DELTA SETS OF NUMERICAL MONOIDS
Let S be a numerical monoid (i.e. an additive submonoid of ℕ0) with minimal generating set 〈n1,…,nt〉. For m ∈ S, if , then is called a factorization length of m. We denote by (where mi < mi+1 for
On the delta set and the Betti elements of a BF-monoid
AbstractWe examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection
The catenary and tame degree of numerical monoids
Abstract We construct an algorithm which computes the catenary and tame degree of a numerical monoid. As an example we explicitly calculate the catenary and tame degree of numerical monoids generated
On the set of elasticities in numerical monoids
In an atomic, cancellative, commutative monoid S, the elasticity of an element provides a coarse measure of its non-unique factorizations by comparing the largest and smallest values in its set of
Factorization Invariants in half-Factorial Affine Semigroups
TLDR
It is proved that for half-factorial monoids, the tame degree and the ω-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.
THE CATENARY AND TAME DEGREE OF NUMERICAL SEMIGROUPS
We construct an algorithm which computes the catenary and tame degree of a numerical semigroup. As an example we explicitly calculate the catenary and tame degree of numerical semigroups generated by
Periodicity of betti numbers of monomial curves
...
...