Unions of sets of lengths

Freeze, Michael; Geroldinger, Alfred

doi:10.7169/facm/1229696561

Cited by 21 publications

(17 citation statements)

References 14 publications

(13 reference statements)

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“…(e) ⇒ (a): Let n ∈ N + such that nρ ∈ N + and n = λ nρ . Then λ nρ < ∞ and, similarly to the previous analysis, there exists L ∈ L with {λ nρ , nρ} ⊆ L. So nρ ≤ sup L and inf L ≤ λ nρ = n, which, again by Lemma 2.10(iv), yields ρ = ρ(L).The next two propositions are the key (technical) results of this paper: In particular, the first of them is a substantial improvement of[7, Lemma 3.4] (see also Claim 3 in the proof of [4, Theorem 2.2(2)]).…”

supporting

confidence: 57%

“…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: 99%

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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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confidence: 57%

Section: Introductionmentioning

confidence: 99%

Structural properties of subadditive families with applications to factorization theory

Tringali

2019

Isr. J. Math.

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“…Unless U = g(−g), this contradicts the fact that U ∈ A(G 0 ). Theorem 4.3 implies that ρ 2k (G P ) = kD(G P ) = 5k and that 5k + 1 ≤ ρ 2k+1 (G P ) ≤ 5k + 2 for all k ∈ N. In order to prove that U k (H) is an interval for each k ∈ N, it suffices to show that U k (G P ) ∩ N ≥k is an interval for each k ∈ N. Indeed, this follows from a simple symmetry argument (see [20,Lemma 3.5]). We proceed conclude the proof by proving the following three claims A1, A2, and A3 which clearly imply the assertion.…”

Section: Monoids Of Modules Over Commutative Noetherian Local Ringsmentioning

confidence: 99%

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

et al. 2014

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Abstract. Let R be a ring and let C be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let V(C) denote a set of representatives of isomorphism classes in C and, for any module M in C, let [M ] denote the unique element in V(C) isomorphic to M . Then V(C) is a reduced commutative semigroup with operation defined by [M ] , and this semigroup carries all information about direct-sum decompositions of modules in C. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if End R (M ) is semilocal for all M ∈ C, then V(C) is a Krull monoid. Suppose that the monoid V(C) is Krull with a finitely generated class group (for example, when C is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case we study the arithmetic of V(C) using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid V(C) for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

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“…Suppose S is a Krull monoid such that every class contains a prime divisor. Then it was shown only recently that, for all k ∈ N, the unions V k (S) are arithmetical progressions with difference 1 (see [17,Theorem 4.1], [21] for a simpler proof, and also [19]). In [34], unions of sets of lengths are studied for non-principal order in number fields, and in [9], for domains of the form V + XB[X], where V is a discrete valuation domain and B the ring of integers in a finite extension field over the quotient field of V .…”

mentioning

confidence: 99%

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

Blanco¹,

García-Sánchez²,

Geroldinger³

2011

Actes Des Rencontres Du CIRM

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Unions of sets of lengths

Cited by 21 publications

References 14 publications

Structural properties of subadditive families with applications to factorization theory

Structural properties of subadditive families with applications to factorization theory

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

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