The Characterization of Finite Elasticities

Grynkiewicz, David J.

doi:10.1007/978-3-031-14869-9

Cited by 3 publications

(1 citation statement)

References 70 publications

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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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Atoms of root-closed submonoids of $$\mathbb {Z}^2$$

Lettl

2022

Semigroup Forum

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We describe how one can explicitly obtain all atoms of an arbitrary root-closed monoid, whose quotient group is isomorphic to $$\mathbb {Z}^2$$ Z 2 . For this purpose, we solve this task for three special types of such monoids in Theorems 5 and 6, and then transfer these results to the general case. It turns out that all atoms can be obtained from the (regular) continued fraction expansion of the slopes of the bounding rays of the cone, which is spanned by the monoid.

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On Dedekind domains whose class groups are direct sums of cyclic groups

Chang

Geroldinger

2024

Journal of Pure and Applied Algebra

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The Characterization of Finite Elasticities

Cited by 3 publications

References 70 publications

On the finiteness of certain factorization invariants

On the finiteness of certain factorization invariants

Atoms of root-closed submonoids of $$\mathbb {Z}^2$$

On Dedekind domains whose class groups are direct sums of cyclic groups

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