On the Arithmetic of Mori Monoids and Domains

Zhong, Qinghai

doi:10.1017/s0017089519000132

Cited by 9 publications

(3 citation statements)

References 17 publications

(14 reference statements)

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“…Since the late 1980s, elasticity has received wide attention in the literature; see Anderson's survey [3] for an overview of results prior to 1997, [17, Theorems 6.2 and 7.2] for some of the strongest finiteness criteria so far available in the cancellative commutative setting, and [5], [7], [14], [19], [24], and [25] for a non-exhaustive list of recent contributions. Introduced by Valenza [22] in his study of factorization in number rings, the notion was made popular by Zaks [23] who used it as a measure of the deviation of an atomic monoid from the condition of half-factoriality (a monoid is half-factorial if it is atomic and any two atomic factorizations of the same element have the same length, i.e., the same number of factors).…”

Section: Introductionmentioning

confidence: 99%

On the finiteness of certain factorization invariants

Cossu,

Tringali

2024

Arkiv För Matematik

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Let H be a monoid and π H be the unique extension of the identity map on H to a monoid homomorphism F (H)→H, where we denote by F (X) the free monoid on a set X. Given A⊆H, an A-word z (i.e., an element of F (A)) is minimal if π H (z) =π H (z ) for every permutation z of a proper subword of z. The minimal A-elasticity of H is then the supremum of all rational numbers m/n with m, n∈N + such that there exist minimal A-words a and b of length m and n, resp., with π H (a)=π H (b).Among other things, we show that if H is commutative and A is finite, then the minimal A-elasticity of H is finite. This provides a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where H is cancellative, commutative, and finitely generated modulo units, and A is the set A (H) of atoms of H. We also demonstrate that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, finitely generated monoid with trivial group of units whose minimal A (H)-elasticity is infinite.

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Section: Introductionmentioning

confidence: 99%

On the finiteness of certain factorization invariants

Cossu,

Tringali

2024

Arkiv För Matematik

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“…Since the late 1980s, elasticity and its finiteness have received wide attention in the literature (see Anderson's survey [3] for a thorough overview of results prior to 1997, [16, Theorems 6.2 and 7.2] for some of the strongest finiteness criteria so far available in cancellative commutative settings, and [5,7,13,18,22,23] for a selection of recent contributions). Introduced by Valenza [20] in connection with the study of factorization in number rings, the notion was made popular by Zaks [21] who used it as a measure of the departure of an atomic monoid from the condition of half-factoriality (a monoid is half-factorial if it is atomic and any two atomic factorizations of the same element have the same length, i.e., the same number of factors).…”

Section: Introductionmentioning

confidence: 99%

On the finiteness of certain factorization invariants

Cossu¹,

Tringali²

2023

Preprint

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Let H be a monoid, F (X) be the free monoid on a set X, and π H be the unique extension of the identity map on H to a monoid homomorphism F (H) → H. Given A ⊆ H, an A-word z (i.e., an element of F (A)) is minimal if π H (z) = π H (z ′ ) for every permutation z ′ of a proper subword of z. The minimal A-elasticity of H is then the supremum of all rational numbers m/n with m, n ∈ N + such that there exist minimal A-words a and b of length m and n, resp., with π H (a) = π H (b).Most notably, we show that if H is commutative and A is finite, then the minimal A-elasticity of H is finite. As with previous results in the same vein, the proof relies on Dickson's lemma, but other than that, the key ideas are fundamentally new. This yields a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where H is cancellative, commutative, and finitely generated modulo units and A is its set of atoms.

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“…A domain R is weakly Krull if and only if R • is a weakly Krull monoid. The arithmetic of weakly Krull monoids is studied via transfer homomorphisms to T -block monoids (see [42,Sections 3.4 and 4.5] for T -block monoids and the structure of sets of lengths and [91,90] for the structure of their unions). We cannot develop these concepts here whence we restrict to the monoid of their divisorial ideals whose arithmetic can be deduced easily from the local case.…”

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confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

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This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

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On the Arithmetic of Mori Monoids and Domains

Cited by 9 publications

References 17 publications

On the finiteness of certain factorization invariants

On the finiteness of certain factorization invariants

On the finiteness of certain factorization invariants

Factorization theory in commutative monoids

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