Which sets are sets of lengths in all numerical monoids?

Geroldinger, Alfred; Schmid, Wolfgang

doi:10.4064/bc118-12

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…Before finding sets of lengths which are characteristic for a given group, we determine those sets of non-negative integers which are sets of lengths over all finite groups. It turns out that this is a simple consequence of the associated result in the abelian setting (sets which are sets of lengths in all numerical monoids are determined in [22]).…”

Section: Arithmetic Of the Monoid Of Product-one Sequencesmentioning

confidence: 87%

On the algebraic and arithmetic structure of the monoid of product-one sequences II

2018

Period Math Hung

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Let G be a finite group and G ′ its commutator subgroup. By a sequence over G, we mean a finite unordered sequence of terms from G, where repetition is allowed, and we say that it is a productone sequence if its terms can be ordered such that their product equals the identity element of G. The monoid B(G) of all product-one sequences over G is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if G is abelian (equivalently, B(G) is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if |G ′ | ≤ 2 if and only if B(G) is seminormal, and we study sets of lengths in B(G).

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Section: Arithmetic Of the Monoid Of Product-one Sequencesmentioning

confidence: 87%

On the algebraic and arithmetic structure of the monoid of product-one sequences II

2018

Period Math Hung

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“…Geroldinger and Schmid [31] investigated the intersection of systems of sets of lengths of numerical monoids. In particular, they proved ∩ L(M) = {{0}, {1}, {2}}, where the intersection is taken over all numerical monoids M = N. Gotti [36,Corollary 5.7] showed that if we take the previous intersection over all nontrivial atomic Puiseux monoids then we obtain ∩ L(M) = {{0}, {1}}.…”

Section: Sets Of Lengths and Their Unionsmentioning

confidence: 99%

On the factorization invariants of the additive structure of exponential Puiseux semirings

Polo¹

2021

Preprint

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Exponential Puiseux semirings are additive submonoids of Q ≥0 generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. Additionally, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

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Factorization under local finiteness conditions

Cossu

Tringali

2023

Journal of Algebra

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Which sets are sets of lengths in all numerical monoids?

Abstract: We explicitly determine those sets of nonnegative integers which occur as sets of lengths in all numerical monoids.Dedicated to Jerzy Kaczorowski on the occasion of his 60th birthday.

Cited by 4 publications

References 21 publications

On the algebraic and arithmetic structure of the monoid of product-one sequences II

On the algebraic and arithmetic structure of the monoid of product-one sequences II

On the factorization invariants of the additive structure of exponential Puiseux semirings

Factorization under local finiteness conditions

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