Numerical Monoids Research Articles

Approximating length-based invariants in atomic Puiseux monoids

A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. Using that a Puiseux monoid is an increasing union of copies of numerical monoids, we prove that some of the factorization invariants of these two classes of monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux monoids.

On strongly primary monoids, with a focus on Puiseux monoids

Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is Noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.

Factorization theory in commutative monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

Structural properties of subadditive families with applications to factorization theory

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets $\mathscr U_k$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large $k$, namely, $H$ satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we show that, under mild assumptions on $H$, not only does the Structure Theorem for Unions hold, but there also exists $\mu\in{\bf N}^+$ such that, for every $M\in{\bf N}$, the sequences $$ \bigl((\mathscr U_k-\inf\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} \quad\text{and}\quad \bigl((\sup\mathscr U_k-\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} $$ are $\mu$-periodic from some point on. The result applies, e.g., to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a "purely additive model", by inquiring into the properties of what we call a subadditive family, i.e., a collection $\mathscr L$ of subsets of $\bf N$ such that, for all $L_1,L_2\in\mathscr L$, there is $L\in\mathscr L$ with $L_1+L_2\subseteq L$.

Factorization invariants of Puiseux monoids generated by geometric sequences

We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study here consists of all atomic monoids of the form , where r is a positive rational. As the atomic monoids Sr are nicely generated, we are able to give detailed descriptions of many of their factorization invariants. One distinguishing characteristic of Sr is that all its sets of lengths are arithmetic sequences of the same distance, namely , where are such that and . We prove this, and then use it to study the elasticity and tameness of Sr.

Which sets are sets of lengths in all numerical monoids?

We explicitly determine those sets of nonnegative integers which occur as sets of lengths in all numerical monoids.

Systems of sets of lengths of Puiseux monoids

In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of Q≥0). We begin by presenting a BF-monoid M with full system of sets of lengths, which means that for each subset S of Z≥2 there exists an element x∈M whose set of lengths L(x) is S. It is well known that systems of sets of lengths do not characterize numerical monoids. Here, we prove that systems of sets of lengths do not characterize non-finitely generated atomic Puiseux monoids. In a recent paper, Geroldinger and Schmid found the intersection of systems of sets of lengths of numerical monoids. Motivated by this, we extend their result to the setting of atomic Puiseux monoids. Finally, we relate the sets of lengths of the Puiseux monoid P=〈1/p|pis prime〉 with the Goldbach's conjecture; in particular, we show that L(2) is precisely the set of Goldbach's numbers.

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. Arithmetical numerical monoids, which are minimally generated by arithmetic sequences, are particularly well-behaved, admitting closed forms for many invariants that are difficult to compute in the general case. In this paper, we answer the question “when does omitting generators from an arithmetical numerical monoid S preserve its (well-understood) set of length sets and/or Frobenius number?” in two extremal cases: (1) we characterize which individual generators can be omitted from S without changing the set of length sets or Frobenius number; and (2) we prove that under certain conditions, nearly every generator of S can be omitted without changing its set of length sets or Frobenius number.

A realization theorem for sets of lengths in numerical monoids

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

Minimal presentations of shifted numerical monoids

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.

The catenary degrees of elements in numerical monoids generated by arithmetic sequences

ABSTRACTWe compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.

On dynamic algorithms for factorization invariants in numerical monoids

Studying the factorization theory of numerical monoids relies on understanding several important factorization invariants, including length sets, delta sets, and $\omega$-primality. While progress in this field has been accelerated by the use of computer algebra systems, many existing algorithms are computationally infeasible for numerical monoids with several irreducible elements. In this paper, we present dynamic algorithms for the factorization set, length set, delta set, and $\omega$-primality in numerical monoids and demonstrate that these algorithms give significant improvements in runtime and memory usage. In describing our dynamic approach to computing $\omega$-primality, we extend the usual definition of this invariant to the quotient group of the monoid and show that several useful results naturally extend to this broader setting.

Digital semigroups

The well-known expansion of rational integers in an arbitrary integer base different from 0,1, − 1 is exploited to study relations between numerical monoids and certain subsemigroups of the multiplicative semigroup of nonzero integers.

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 lengths (called its length set). In this paper, we show that the set of length sets $\mathcal L(S)$ for any arithmetical numerical monoid $S$ can be completely recovered from its set of elasticities $R(S)$; therefore, $R(S)$ is as strong a factorization invariant as $\mathcal L(S)$ in this setting. For general numerical monoids, we describe the set of elasticities as a specific collection of monotone increasing sequences with a common limit point of $\max R(S)$.

Computation of Delta sets of numerical monoids

Let $$\{a_1,\ldots ,a_p\}$$ be the minimal generating set of a numerical monoid S. For any $$s\in S$$ , its Delta set is defined by $$\Delta (s)=\{l_{i}-l_{i-1}\mid i=2,\ldots ,k\}$$ where $$\{l_1<\dots <l_k\}$$ is the set $$\{\sum _{i=1}^px_i\mid s=\sum _{i=1}^px_ia_i \text { and } x_i\in \mathbb {N}\text { for all }i\}.$$ The Delta set of a numerical monoid S, denoted by $$\Delta (S)$$ , is the union of all the sets $$\Delta (s)$$ with $$s\in S.$$ As proved in Chapman et al. (Aequationes Math. 77(3):273–279, 2009), there exists a bound N such that $$\Delta (S)$$ is the union of the sets $$\Delta (s)$$ with $$s\in S$$ and $$s<N$$ . In this work, we obtain a sharpened bound and we present an algorithm for the computation of $$\Delta (S)$$ that requires only the factorizations of $$a_1$$ elements.

On the computation of the Apéry set of numerical monoids and affine semigroups

A simple way of computing the Ap\'ery set of a numerical semigroup (or monoid) with respect to a generator, using Groebner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.

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

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.

How Do You Measure Primality?

In commutative monoids, the ω-value measures how far an element is from being prime. This invariant, which is important in understanding the factorization theory of monoids, has been the focus of much recent study. This paper provides detailed examples and an overview of known results on ω-primality, including several recent and surprising contributions in the setting of numerical monoids. As many questions related to ω-primality remain, we provide a list of open problems accessible to advanced undergraduate students and beginning graduate students.

Shifts of generators and delta sets of numerical monoids

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 &lt; mi+1 for each 1 ≤ i &lt; 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 &lt; 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 &gt; 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.

Factorization properties of Leamer monoids

The Huneke-Wiegand conjecture has prompted much recent research in Commutative Algebra. In studying this conjecture for certain classes of rings, Garc\'ia-S\'anchez and Leamer construct a monoid S_\Gamma^s whose elements correspond to arithmetic sequences in a numerical monoid \Gamma of step size s. These monoids, which we call Leamer monoids, possess a very interesting factorization theory that is significantly different from the numerical monoids from which they are derived. In this paper, we offer much of the foundational theory of Leamer monoids, including an analysis of their atomic structure, and investigate certain factorization invariants. Furthermore, when S_\Gamma^s is an arithmetical Leamer monoid, we give an exact description of its atoms and use this to provide explicit formulae for its Delta set and catenary degree.