Sets of arithmetical invariants in transfer Krull monoids

Geroldinger, Alfred; Zhong, Qinghai

doi:10.1016/j.jpaa.2018.12.011

Cited by 25 publications

(10 citation statements)

References 41 publications

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“…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.…”

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confidence: 99%

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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confidence: 99%

Factorization invariants of Puiseux monoids generated by geometric sequences

Chapman

Gotti

2019

Communications in Algebra

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“…There has been a great deal of research devoted to the arithmetic of transfer Krull monoids (cf. [31]) and in Section 3 we show that the monoid of isomorphism classes of lattices over a Bass ring is transfer Krull of finite type. Consequently, we can measure the degree to which direct-sum decompositions over a Bass ring are not unique by instead studying the arithmetic of certain finitely generated Krull monoids and, in particular, certain Diophantine monoids introduced in Section 3 as well as monoids of graph agglomerations in Section 4.…”

Section: Transfer Homomorphisms and Krull Monoidsmentioning

confidence: 98%

Lattices over Bass Rings and Graph Agglomerations

Baeth

Smertnig

2021

Algebr Represent Theor

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We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of T(R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of T(R). As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull–Remak–Schmidt–Azumaya, is characterized by a theorem of Levy–Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

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“…The fact that

O

is a transfer Krull monoid of finite type, implies that many questions on factorizations in

O

, in particular all the ones on sets of lengths, can be reduced to questions in combinatorial and additive number theory over finite abelian groups, specifically the stable class group