On the Molecules of Numerical Semigroups, Puiseux Monoids, and Puiseux Algebras

Gotti, Felix; Gotti, Marly

doi:10.1007/978-3-030-40822-0_10

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…A didactic exposition of the factorization-theoretical aspects of Z[ √ −5] can be found in [9]. Following [21], we say that a non-invertible element x ∈ M is a molecule if x has exactly one factorization in M, and we let M (M) denote the set consisting of all molecules of M.…”

Section: Introductionmentioning

confidence: 99%

“…This is hardly a surprise given that factorization theory has its origin in algebraic number theory, one of the pioneering works being [7]. More recently, the molecules of additive monoids such as numerical monoids and some generalizations of them have been studied in [21,.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we study the sizes of the sets of molecules that are not atoms in additive submonoids of Q ≥0 . One of the initial motivations of this project was the following realizability question posed by F. Gotti and the first author in [21].…”

Section: Introductionmentioning

confidence: 99%

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On the set of molecules of numerical and Puiseux monoids

Gotti¹,

Tirador²

2020

Preprint

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Additive submonoids of Q ≥0 , also known as Puiseux monoids, are not unique factorization monoids (UFMs) in general. Indeed, the only unique factorization Puiseux monoids are those generated by one element. However, even if a Puiseux monoid is not a UFM, it may contain nonzero elements having exactly one factorization. We call such elements molecules. Molecules were first investigated by W. Narkiewicz in the context of algebraic number theory. More recently, F. Gotti and the first author studied molecules in the context of Puiseux monoids. Here we address some aspects related to the size of the sets of molecules of various subclasses of Puiseux monoids with different atomic behaviors. In particular, we positively answer the following recent realization conjecture: for each m ∈ N ≥2 there exists a numerical monoid whose set of molecules that are not atoms has cardinality m.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the set of molecules of numerical and Puiseux monoids

Gotti¹,

Tirador²

2020

Preprint

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On the Set of Molecules of Numerical and Puiseux Monoids

Gotti

Tirador

2021

Springer Proceedings in Mathematics &Amp; Statistics

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On the Set of Betti Elements of a Puiseux Monoid

CHAPMAN,

JANG,

MAO

et al. 2024

Bull. Aust. Math. Soc.

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Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of $b \in M$ is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.

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On the Molecules of Numerical Semigroups, Puiseux Monoids, and Puiseux Algebras

Cited by 5 publications

References 15 publications

On the set of molecules of numerical and Puiseux monoids

On the set of molecules of numerical and Puiseux monoids

On the Set of Molecules of Numerical and Puiseux Monoids

On the Set of Betti Elements of a Puiseux Monoid

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