THE NUMBER OF ADDENDS IN THE DECOMPOSITION OF AN ELEMENT OF A NUMERICAL MONOID INTO ATOMS

2019 ◽  
Vol 44 (1) ◽  
pp. 63-80
Author(s):  
Hamid Kulosman
Keyword(s):  
Numerical Monoid

scholarly journals Higher Nash blowups

2007 ◽  
Vol 143 (6) ◽  
pp. 1493-1510 ◽  
Author(s):  
Takehiko Yasuda
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Negative Integer ◽  
Curve Singularity ◽  
Irreducible Curve ◽  
Numerical Monoid

AbstractFor each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.

THE CATENARY AND TAME DEGREES ON A NUMERICAL MONOID ARE EVENTUALLY PERIODIC

2014 ◽  
Vol 97 (3) ◽  
pp. 289-300 ◽  
Author(s):  
SCOTT T. CHAPMAN ◽  
MARLY CORRALES ◽  
ANDREW MILLER ◽  
CHRIS MILLER ◽  
DHIR PATEL
Keyword(s):  
Periodicity Condition ◽  
Numerical Monoid ◽  
Cancellative Monoid ◽  
Krull Monoids ◽  
Catenary Degree

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.

ON DELTA SETS OF NUMERICAL MONOIDS

2006 ◽  
Vol 05 (05) ◽  
pp. 695-718 ◽  
Author(s):  
CRAIG BOWLES ◽  
SCOTT T. CHAPMAN ◽  
NATHAN KAPLAN ◽  
DANIEL REISER
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Finitely Generated ◽  
Generating Set ◽  
Numerical Monoid ◽  
Delta Set ◽  
Minimal Generating Set

Let S be a numerical monoid (i.e. an additive submonoid of ℕ0) with minimal generating set 〈n1,…,nt〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by [Formula: see text] (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {mi+1 - mi|1 ≤ i < k} and the Delta set of S by Δ(S) = ∪m∈SΔ(m). In this paper, we address some basic questions concerning the structure of the set Δ(S). In Sec. 2, we find upper and lower bounds on Δ(S) by finding such bounds on the Delta set of any monoid S where the associated reduced monoid S red is finitely generated. We prove in Sec. 3 that if S = 〈n, n + k, n + 2k,…,n + bk〉, then Δ(S) = {k}. In Sec. 4 we offer some specific constructions which yield for any k and v in ℕ a numerical monoid S with Δ(S) = {k, 2k,…,vk}. Moreover, we show that Delta sets of numerical monoids may contain natural "gaps" by arguing that Δ(〈n, n + 1, n2 - n - 1〉) = {1,2,…,n - 2, 2n - 5}.

Shifts of generators and delta sets of numerical monoids

2014 ◽  
Vol 24 (05) ◽  
pp. 655-669 ◽  
Author(s):  
S. T. Chapman ◽  
Nathan Kaplan ◽  
Tyler Lemburg ◽  
Andrew Niles ◽  
Christina Zlogar
Keyword(s):  
Positive Integer ◽  
Generating Set ◽  
Numerical Monoid ◽  
Delta Set ◽  
A Value ◽  
Positive Integers ◽  
Minimal Generating Set

Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by ℒ(m) = {m1, …, mk} (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {mi+1 - mi | 1 ≤ i < k} and the Delta set of S by Δ(S) = ⋃m∈SΔ(m). In this paper, we expand on the study of Δ(S) begun in [C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006) 1–24] in the following manner. Let r1, r2, …, rt be an increasing sequence of positive integers and Mn = 〈n, n + r1, n + r2, …, n + rt〉 a numerical monoid where n is some positive integer. We prove that there exists a positive integer N such that if n > N then |Δ(Mn)| = 1. If t = 2 and r1 and r2 are relatively prime, then we determine a value for N which is sharp.

Elements in a Numerical Semigroup with Factorizations of the Same Length

2011 ◽  
Vol 54 (1) ◽  
pp. 39-43 ◽  
Author(s):  
S. T. Chapman ◽  
P. A. García-Sánchez ◽  
D. Llena ◽  
J. Marshall
Keyword(s):  
Recent Literature ◽  
Numerical Semigroup ◽  
Embedding Dimension ◽  
Numerical Monoid ◽  
Irreducible Elements

AbstractQuestions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

scholarly journals REALISABLE SETS OF CATENARY DEGREES OF NUMERICAL MONOIDS

2017 ◽  
Vol 97 (2) ◽  
pp. 240-245 ◽  
Author(s):  
CHRISTOPHER O’NEILL ◽  
ROBERTO PELAYO
Keyword(s):  
Finite Subset ◽  
Numerical Monoid ◽  
Catenary Degree

The catenary degree is an invariant that measures the distance between factorisations of elements within an atomic monoid. In this paper, we classify which finite subsets of$\mathbb{Z}_{\geq 0}$occur as the set of catenary degrees of a numerical monoid (that is, a co-finite, additive submonoid of$\mathbb{Z}_{\geq 0}$). In particular, we show that, with one exception, every finite subset of$\mathbb{Z}_{\geq 0}$that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid.

An algorithm to compute ω-primality in a numerical monoid

2010 ◽  
Vol 82 (1) ◽  
pp. 96-108 ◽  
Author(s):  
David F. Anderson ◽  
Scott T. Chapman ◽  
Nathan Kaplan ◽  
Desmond Torkornoo
Keyword(s):  
Numerical Monoid

scholarly journals A realization theorem for sets of lengths in numerical monoids

2018 ◽  
Vol 30 (5) ◽  
pp. 1111-1118 ◽  
Author(s):  
Alfred Geroldinger ◽  
Wolfgang Alexander Schmid
Keyword(s):  
Nonempty Subset ◽  
Numerical Monoid ◽  
Sets Of Lengths

AbstractWe show that for every finite nonempty subset of {\mathbb{N}_{\geq 2}} there are a numerical monoid H and a squarefree element {a\in H} whose set of lengths {\mathsf{L}(a)} is equal to L.

scholarly journals Minimal presentations of shifted numerical monoids

2018 ◽  
Vol 28 (01) ◽  
pp. 53-68 ◽  
Author(s):  
Rebecca Conaway ◽  
Felix Gotti ◽  
Jesse Horton ◽  
Christopher O’Neill ◽  
Roberto Pelayo ◽  
...  
Keyword(s):  
Recent Result ◽  
Commutative Algebra ◽  
Shift Parameter ◽  
Numerical Monoid ◽  
Factorization Theory ◽  
The Family ◽  
Combinatorial Commutative Algebra ◽  
Minimal Presentations

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

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