Increasingly Enumerable Submonoids of R : Music Theory as a Unifying Theme

Bras-Amorós, Maria

doi:10.1080/00029890.2020.1674073

Cited by 14 publications

(6 citation statements)

References 19 publications

(27 reference statements)

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“…Following [30], we say that a positive monoid M is strongly increasing if A pM q is the underlying set of a sequence that increases to infinity. Strongly increasing positive monoids have been considered in [8,9,30]. Example 6.1.…”

Section: The Finite Factorization Propertymentioning

confidence: 99%

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Bi-atomic classes of positive semirings

2021

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A subsemiring S of R is called a positive semiring provided that S consists of nonnegative numbers and 1 P S. Here we study factorizations in both the additive monoid pS, `q and the multiplicative monoid pSzt0u, ¨q. In particular, we investigate when, for a positive semiring S, both pS, `q and pSzt0u, ¨q have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP ñ BFP and FFP ñ BFP ñ ACCP ñ atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.

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Section: The Finite Factorization Propertymentioning

confidence: 99%

“…Submonoids of finite-rank free commutative monoids are also finite-rank positive monoids up to isomorphism, and their atomicity and arithmetic were studied in [16,17] and, more recently, in [29,32]. Positive monoids have also been considered in surprising contexts such as music theory; see the recent paper [8].…”

Section: Introductionmentioning

confidence: 99%

Bi-atomic classes of positive semirings

2021

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“…Positive monoids that are increasingly generated have been studied in [7,9,24,27]. On the other hand, positive semirings (i.e., positive monoids closed under multiplication) have been studied in [4,5], while the special case of positive monoids generated by a geometric sequence (necessarily positive semirings) have been studied in [14,16].…”

Section: Ufm Hfm Ffm Bfm Accp Monoid Atomic Monoidmentioning

confidence: 99%

Atomicity of Positive Monoids

Chapman¹,

Gotti²

2021

Preprint

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An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey many of the recent advances regarding positive monoids, and we provide numerous examples to illustrate the complexity of their atomic and arithmetic structures.

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“…This explains why Puiseux monoids have been the focus of several papers recently. They have been also studied in connection with factorization of matrices [7] and music theory [8]. Several interesting examples of Puiseux monoids have been given in [6] and [15].…”

Section: Introductionmentioning

confidence: 99%

Factorizations in reciprocal Puiseux monoids

Aguilera¹,

Gotti²,

Hamelberg³

2021

Preprint

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A Puiseux monoid is an additive submonoid of the real line consisting of rationals. We say that a Puiseux monoid is reciprocal if it can be generated by the reciprocals of the terms of a strictly increasing sequence of pairwise relatively primes positive integers. We say that a commutative and cancellative (additive) monoid is atomic if every non-invertible element x can be written as a sum of irreducibles. The number of irreducibles in this sum is called a length of x. In this paper, we identify and investigate generalized classes of reciprocal Puiseux monoids that are atomic. Moreover, for the atomic monoids in the identified classes, we study the ascending chain condition on principal ideals and also the sets of lengths of their elements.

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Increasingly Enumerable Submonoids of R : Music Theory as a Unifying Theme

Cited by 14 publications

References 19 publications

Bi-atomic classes of positive semirings

Bi-atomic classes of positive semirings

Atomicity of Positive Monoids

Factorizations in reciprocal Puiseux monoids

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