On the atomic structure of exponential Puiseux monoids and semirings

Albizu-Campos, Sofía; Bringas, Juliet; Polo, Harold

doi:10.1080/00927872.2020.1820514

Cited by 11 publications

(12 citation statements)

References 16 publications

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“…Specifically, we prove that if S r,N is an atomic exponential Puiseux semiring then there exists B ∈ N such that every L ∈ L(S r,N ) is an AAP with difference |n(r) − d(r)| and bound B. But first let us collect some technical lemmas; the first one was borrowed from [1] (Lemma 3.8), and the second one is its counterpart for the case where r > 1.…”

Section: Sets Of Lengths and Their Unionsmentioning

confidence: 99%

“…In [10], it was proved that, for every bounded factorization monoid M, the minimum and maximum of ∆(M) are both attained at Betti elements. But notice that atomic exponential Puiseux semirings are not, in general, BFMs (see [1,Example 4.7]).…”

Section: Betti Elements Catenary Degrees and Sets Of Distancesmentioning

confidence: 99%

“…The class of atomic rational cyclic semirings is one of the rare classes of non-finitely generated monoids for which we can provide detailed descriptions for many of their factorization invariants, and various generalizations of this class (e.g., [1,16,43]) have been scrutinized to gain insight about the atomic structure and factorization invariants of positive monoids (i.e., additive submonoids of the nonnegative cone of the real line). In [1], Albizu-Campos et al studied the atomic properties of exponential Puiseux semirings, that is, Puiseux monoids generated by almost all of the nonnegative powers of a positive rational number. The purpose of the present article is to show that atomic exponential Puiseux semirings retain some of the nice factorization properties of atomic rational cyclic semirings.…”

Section: Introductionmentioning

confidence: 99%

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On the factorization invariants of the additive structure of exponential Puiseux semirings

Polo¹

2021

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Exponential Puiseux semirings are additive submonoids of Q ≥0 generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. Additionally, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

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Section: Sets Of Lengths and Their Unionsmentioning

confidence: 99%

Section: Betti Elements Catenary Degrees and Sets Of Distancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the factorization invariants of the additive structure of exponential Puiseux semirings

Polo¹

2021

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“…Generalizations of N 0 [α] (with α rational) were also studied in [2] from the factorizationtheoretical point of view. More recently, a deeper and more systematic study of the atomic structure of N 0 [α] was carried out in [16] for any positive real α.…”

Section: Introductionmentioning

confidence: 99%

On the primality and elasticity of algebraic valuations of cyclic free semirings

Jiang¹,

Li²,

Zhu³

2022

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A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number α, the additive monoid M α of the evaluation semiring N 0 [α] is atomic. The atomic structure of both the additive and the multiplicative monoids of N 0 [α] has been the subject of several recent papers. Here we focus on the monoids M α , and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when α is less than 1, the atoms of N 0 [α] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of N 0 [α], including that when α is rational, the elasticity of M α is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti).

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“…In addition, Baeth et al [4] recently studied the atomic structure of both the additive and the multiplicative monoids of subsemirings of R ≥0 . Finally, factorizations in certain subsemirings of Q ≥0 have also been considered in [1] by Albizu-Campos et al and in [5] by Baeth and Gotti. We begin by introducing the main terminology in Section 2.1 and outlining the main known results we use later. Then, in Section 3, we discuss the atomicity of the monoids M α .…”

Section: Introductionmentioning

confidence: 99%

Factorizations in evaluation monoids of Laurent semirings

Zhu¹

2021

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where N 0 is the set of nonnegative integers and N 0 [x, x −1 ] is the semiring of Laurent polynomials with coefficients in N 0 . In this paper, we study various factorization properties of the additive structure of N 0 [α, α −1 ]. We characterize when N 0 [α, α −1 ] is atomic. Then we characterize when N 0 [α, α −1 ] satisfies the ascending chain condition on principal ideals in terms of certain well-studied factorization properties. Finally, we characterize when N 0 [α, α −1 ] satisfies the unique factorization property and show that, when this is not the case, N 0 [α, α −1 ] has infinite elasticity.

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On the atomic structure of exponential Puiseux monoids and semirings

Cited by 11 publications

References 16 publications

On the factorization invariants of the additive structure of exponential Puiseux semirings

On the factorization invariants of the additive structure of exponential Puiseux semirings

On the primality and elasticity of algebraic valuations of cyclic free semirings

Factorizations in evaluation monoids of Laurent semirings

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