Realisable Sets of Catenary Degrees of Numerical Monoids

O’Neill, Christopher; Pelayo, Roberto

doi:10.1017/s0004972717000995

Cited by 14 publications

(12 citation statements)

References 13 publications

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“…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%

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Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

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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 .

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“…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%

“…Note that Corollary 3.4(4) contrasts with[41, Theorem 4.2] and[25, Proposition 4.3.1], where it is proved that most subsets of N 0 can be realized as the set of catenary degrees of a numerical monoid and a Krull monoid (finitely generated with finite class group), respectively. The Elasticity.…”

mentioning

confidence: 99%

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

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“…Indeed, as shown in Example 5.4.2, for every n 2 N there is an order R in a quadratic number field whose localization R p at a maximal ideal p is finitely primary of exponent greater than or equal to the given n but the catenary degree cðR p Þ is bounded by 5 [9, Theorem 1.1]. Let H & F ¼ F Â Â FðfpgÞ be finitely primary of rank one, suppose that its value monoid v p ðHÞ ¼ fv p ðaÞ j a 2 Hg ¼ hd 1 , :::, d s i, with s 2 N, 1 < d 1 < ::: < d s , and gcdðd 1 , :::, d s Þ ¼ 1: The catenary degree of numerical monoids has been studied a lot in recent literature (see [17,[49][50][51][52], for a sample). By (5.1), we have 2 þ maxDðHÞ cðHÞ: There are also results for minDðHÞ: Indeed, by [27,Lemma 4.1], we have…”

Section: Arithmetic Of Stable Orders In Dedekind Domainsmentioning

confidence: 99%

On the arithmetic of stable domains

Bashir

Geroldinger

Reinhart

2021

Communications in Algebra

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A commutative ring R is stable if every non-zero ideal I of R is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the 2-generator property. In the present paper, we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains.

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“…We mention some striking recent results. O'Neill and Pelayo showed that for every finite nonempty subset C ⊂ N ≥2 there is a numerical monoid H such that Ca(H) = C ( [77]).…”

Section: Weakly Krull Monoidsmentioning

confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

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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.

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Realisable Sets of Catenary Degrees of Numerical Monoids

Cited by 14 publications

References 13 publications

Factorization invariants of Puiseux monoids generated by geometric sequences

Factorization invariants of Puiseux monoids generated by geometric sequences

On the arithmetic of stable domains

Factorization theory in commutative monoids

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