Products of k atoms in Krull monoids

Fan, Yushuang; Zhong, Qinghai

doi:10.1007/s00605-016-0942-9

Cited by 7 publications

(6 citation statements)

References 17 publications

(18 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. Let G be a cyclic group of order |G| = n ∈ [4,12]. By Theorem 3.14, we infer that n − 2 ∈ ∆ * ρ (G) = ∆ ρ (G).…”

Section: This Implies Thatmentioning

confidence: 96%

Long Sets of Lengths With Maximal Elasticity

Geroldinger¹,

Zhong²

2018

Can. j. math.

Self Cite

View full text Add to dashboard Cite

Abstract. We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid H, let ∆ ρ (H) be the set of all positive integers d which occur as di erences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity ρ(H). We study ∆ ρ (H) for transfer Krull monoids of nite type (including commutative Krull domains with nite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number elds) for which we use ideal theoretic methods.

show abstract

“…Proof. Let G be a cyclic group of order |G| = n ∈ [4,12]. By Theorem 3.14, we infer that n − 2 ∈ ∆ * ρ (G) = ∆ ρ (G).…”

Section: This Implies Thatmentioning

confidence: 96%

Long Sets of Lengths With Maximal Elasticity

Geroldinger¹,

Zhong²

2018

Can. j. math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…If G 0 = G, then all sets U k (H) are intervals (see [21,Theorem 3.1.3] or [18,Theorem 4.1]). Recent progress on the invariants ρ k B(G) can be found in [17].…”

Section: Cancellative Semigroupsmentioning

confidence: 99%

“…This holds true, among others, for Krull monoids with the property that every class contains a prime divisor ([21, Theorem 3.1.3]). In this case the study of the elasticities ρ k (H) has received much attention in recent literature ( [22,17]).…”

mentioning

confidence: 99%

Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

et al. 2017

Self Cite

View full text Add to dashboard Cite

Abstract. Let H be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every k ∈ N, let U k (H) denote the set of all ℓ ∈ N with the property that there are atomsis the union of all sets of lengths containing k).The Structure Theorem for Unions states that, for all sufficiently large k, the sets U k (H) are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds.This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

show abstract

“…. , b ℓ ∈ H (see also Example 2.2), where an element of H is an atom if it is neither a unit nor the product of two non-units: The sets U k (H) are called unions of sets of lengths and have been studied in factorization theory since decades, see [6] for recent progress and [9,20] for surveys. In particular, we say that H satisfies the Structure Theorem for Unions if there exist d ∈ N + and M ∈ N such that, for all but finitely many k ∈ N,…”

Section: Introductionmentioning

confidence: 99%

Structural properties of subadditive families with applications to factorization theory

Tringali

2019

Isr. J. Math.

View full text Add to dashboard Cite

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.

show abstract

Products of k atoms in Krull monoids

Cited by 7 publications

References 17 publications

Long Sets of Lengths With Maximal Elasticity

Long Sets of Lengths With Maximal Elasticity

Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

Structural properties of subadditive families with applications to factorization theory

Contact Info

Product

Resources

About