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

Chapman, Scott T.; Corrales, Marly; Miller, Andrew; Miller, Chris; Patel, Dhir

doi:10.1080/00927872.2017.1310878

Cited by 9 publications

(5 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is in line with the earlier work done for numerical monoids. Indeed, several of the papers we have already cited are dedicated to showing that while general numerical monoids have complicated factorization properties, those that are generated by an arithmetic sequence have very predictable factorization invariants (see [1,6,8,10]). After fixing a positive rational r > 0, we will study the additive submonoid of Q ≥0 generated by the set {r n | n ∈ N 0 }.…”

Section: Prologuementioning

confidence: 99%

“…In addition, we call is called the set of positive catenary degrees. Recent studies of the catenary degree of numerical monoids can be found in [8] and [41].…”

Section: Factorization Invariantsmentioning

confidence: 99%

“…This changed with the appearance of [6] and [11], both of which view factorization problems in additive submonoids of the natural numbers known as numerical monoids. These two papers generated a flood of work in this area, from both the pure [1,8,9,10,12,13,14,41] and the computational [5,2,16,17] points of view. Over the past three years, similar studies have emerged for additive submonoids of the nonnegative rational numbers, also known as Puiseux monoids [26,28,30,31,32,33].…”

Section: Prologuementioning

confidence: 99%

See 2 more Smart Citations

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

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 S r := r n | n ∈ N 0 , where r is a positive rational. As the atomic monoids S r are nicely generated, we are able to give detailed descriptions of many of their factorization invariants. One distinguishing characteristic of S r is that all its sets of lengths are arithmetic sequences of the same distance, namely |a − b|, where a, b ∈ N are such that r = a/b and gcd(a, b) = 1. We prove this, and then use it to study the elasticity and tameness of S r .

show abstract

Section: Prologuementioning

confidence: 99%

“…In addition, we call is called the set of positive catenary degrees. Recent studies of the catenary degree of numerical monoids can be found in [8] and [41].…”

Section: Factorization Invariantsmentioning

confidence: 99%

Section: Prologuementioning

confidence: 99%

See 1 more Smart Citation

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…A classical example is the Frobenius number F (S) (taking its name from the aforementioned coin-exchange Date: September 24, 2018. problem), defined as the largest integer that lies outside of S [11]. Closed formulas are also known for the genus, Apéry set, delta set, and catenary degree of an arithmetical numerical monoid [3,9,12]. Arithmetical numerical monoids are also one of the only families of numerical monoids (or monoids in general) whose set of length sets is completely parametrized [1].…”

Section: Introductionmentioning

confidence: 99%

On arithmetical numerical monoids with some generators omitted

Lee¹,

O’Neill²,

Over³

2017

Preprint

View full text Add to dashboard Cite

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: (i) we characterize which individual generators can be omitted from S without changing the set of length sets or Frobenius number; and (ii) we prove that under certain conditions, nearly every generator of S can be omitted without changing its set of length sets or Frobenius number.

show abstract

“…This paper concerns the catenary degree (Definition 2.4), a factorization invariant that has been the subject of much recent work [2,6,17]. The catenary degree c(n) of a monoid element n ∈ S is a nonnegative integer derived from combinatorial properties of the set of factorizations of n. Although much of the literature on the catenary degree focuses on the maximum catenary degree attained within S, some recent papers [4,5] examine the catenary degees of individual monoid elements. In this paper, we investigate the set C(S) of catenary degrees occurring within S as a factorization invariant, focusing on the setting where S is finitely generated.…”

Section: Introductionmentioning

confidence: 99%

On the set of catenary degrees of finitely generated cancellative commutative monoids

O’Neill

Ponomarenko

Tate

et al. 2016

Int. J. Algebra Comput.

View full text Add to dashboard Cite

The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integer measuring the distance between the irreducible factorizations of n. The catenary degree of the monoid S, defined as the supremum over all catenary degrees occurring in S, has been studied as an invariant of nonunique factorization. In this paper, we investigate the set C(S) of catenary degrees achieved by elements of S, focusing on the case where S in finitely generated (where C(S) is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of C(S) that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for C(S), and present some open questions for further study.

show abstract

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

Cited by 9 publications

References 11 publications

Factorization invariants of Puiseux monoids generated by geometric sequences

Factorization invariants of Puiseux monoids generated by geometric sequences

On arithmetical numerical monoids with some generators omitted

On the set of catenary degrees of finitely generated cancellative commutative monoids

Contact Info

Product

Resources

About