The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids

Chapman, Scott T.; García-Sánchez, Pedro A.; Llena, D.; Ponomarenko, Vadim; Rosales, J. C.

doi:10.1007/s00229-006-0008-8

Cited by 62 publications

(94 citation statements)

References 10 publications

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“…The present paper will focus on the three closely related invariants, namely the set of distances, the (H) The study of these arithmetical invariants (in settings ranging from numerical monoids to Mori rings with zero-divisors) has attracted a lot of attention in the recent literature (for a sample see [10,9,24,14,25,7,11,8]). Our main focus here will be on Krull monoids with finite class group G such that each class contains a prime divisor.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Catenary Degree of Krull Monoids Ii

Geroldinger

Zhong

2014

J. Aust. Math. Soc.

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Abstract. Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property:for each a ∈ H and each two factorizations z, z ′ of a, there exist factorizations z = z 0 , . . . , z k = z ′ of a such that, for each i ∈ [1, k], z i arises from z i−1 by replacing at most N atoms from z i−1 by at most N new atoms. To exclude trivial cases, suppose that |G| ≥ 3. Then the catenary degree depends only on the class group G and we have c(H) ∈

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The Catenary Degree of Krull Monoids Ii

Geroldinger

Zhong

2014

J. Aust. Math. Soc.

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“…An algorithm which computes the catenary degree of a finitely generated monoid can be found in [6], and a more specific version for numerical monoids in [5]. We shift from considering particular factorizations to analyzing their lengths.…”

Section: Definitions and Backgroundmentioning

confidence: 99%

On delta sets and their realizable subsets in Krull monoids with cyclic class groups

2014

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Abstract. Let M be a commutative cancellative monoid. The set ∆(M ), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M , is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that ∆(M ) ⊆ {1, . . . , n − 2}. Moreover, equality holds for this containment when each class contains a prime divisor from M . In this note, we consider the question of determining which subsets of {1, . . . , n−2} occur as the delta set of an individual element from M . We first prove for x ∈ M that if n − 2 ∈ ∆(x), then ∆(x) = {n − 2} (i.e., not all subsets of {1, . . . , n − 2} can be realized as delta sets of individual elements). We close by proving an Archimedean-type property for delta sets from Krull monoids with finite cyclic class group: for every natural number m, there exist a Krull monoid M with finite cyclic class group such that M has an element x with |∆(x)| ≥ m.

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“…The second step contains the main new idea behind the present progress. In a series of recent papers (see for example [1,3,4]) and a talk at this conference [10], arithmetical invariants of a monoid have been characterized in abstract semigroup theoretical terms, such as the monoid of relations and presentations. Of course, these semigroup theoretical invariants are far beyond reach in the case of non-principal orders.…”

Section: Our Approachmentioning

confidence: 99%

Arithmetic of non-principal orders in algebraic number fields

Philipp¹

2011

Actes Des Rencontres Du CIRM

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Arithmetic of non-principal orders in algebraic number fieldsAndreas Philipp Abstract Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

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The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids

Cited by 62 publications

References 10 publications

The Catenary Degree of Krull Monoids Ii

The Catenary Degree of Krull Monoids Ii

On delta sets and their realizable subsets in Krull monoids with cyclic class groups

Arithmetic of non-principal orders in algebraic number fields

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