A realization theorem for sets of distances

Geroldinger, Alfred; Schmid, Wolfgang

doi:10.1016/j.jalgebra.2017.03.003

Cited by 19 publications

(11 citation statements)

References 12 publications

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“…Then sets of lengths have a well-described structure ( [16,Chapter 4]) and the given description is known to be best possible ( [28]). The set of distances ∆(H) is an interval with min ∆(H) = 1 ([21]) whose maximum is unknown in general ( [22]) (this is in contrast to the fact that in finitely generated Krull monoids any finite set ∆ with min ∆ = gcd ∆ may occur as set of distances [18]). The standing conjecture is that the system of sets of lengths is characteristic for the group (see [15] for a survey, and [17,23,32,31] for recent progress).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Which sets are sets of lengths in all numerical monoids?

Geroldinger

Schmid

2019

Banach Center Publ.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Which sets are sets of lengths in all numerical monoids?

Geroldinger

Schmid

2019

Banach Center Publ.

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“…Along the same lines of thought, the present paper is aimed to establish a kind of periodicity of directed families that applies primarily to unions of sets of lengths: Nothing similar had been known so far, modulo the fact that, for important but rather special categories of monoids and domains, the sets U k are arithmetic progressions, if not even intervals as in the case of the ring of integers of a number field or, more in general, of a commutative Krull monoid with finite class group such that each class contains a prime, see [7,Theorem 4.1]. Moreover, some of the achievements of this work will probably help with one of the long term goals in all studies on unions of sets of lengths: To prove a realization theorem in the same spirit of what has already been done with sets of lengths [19] and sets of distances [11].…”

Section: Introductionmentioning

confidence: 87%

Structural properties of subadditive families with applications to factorization theory

Tringali

2019

Isr. J. Math.

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Let H be a multiplicatively written monoid. Given k ∈ N + , we denote by U k the set of all ℓ ∈ N + such that a 1 · · · a k = b 1 · · · b ℓ for some atoms (or irreducible elements) a 1 , . . . , a k , b 1 , . . . , b ℓ ∈ H. The sets 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, which is usually expressed by saying that H satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem.More precisely, we will show that, under mild assumptions on H, not only does the Structure Theorem for Unions hold, but there also exists µ ∈ N + such that, for every M ∈ N, the sequencesare µ-periodic from some point on. The result applies, for instance, 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" (where no explicit reference to monoids or atoms is ever made), by inquiring into the properties of what we call a subadditive family, i.e., a collection L of subsets of N such that, for all L 1 , L 2 ∈ L , there is L ∈ L with L 1 + L 2 ⊆ L.

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“…Besides the sets of lengths, there are many factorization invariants, including the set of distances and the catenary degree (definitions can be found in [10]), for which the realization problem restricted to several classes of atomic monoids have been studied lately. Indeed, theorems in this direction have been established in [4,12,22,23].…”

Section: A Bf-puiseux Monoid With Full System Of Sets Of Lengthsmentioning

confidence: 99%

Systems of sets of lengths of Puiseux monoids

Gotti

2019

Journal of Pure and Applied Algebra

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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 | p is prime with the Goldbach's conjecture; in particular, we show that L(2) is precisely the set of Goldbach's numbers.

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A realization theorem for sets of distances

Abstract: Let H be an atomic monoid.

Cited by 19 publications

References 12 publications

Which sets are sets of lengths in all numerical monoids?

Which sets are sets of lengths in all numerical monoids?

Structural properties of subadditive families with applications to factorization theory

Systems of sets of lengths of Puiseux monoids

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