When Is a Puiseux Monoid Atomic?

Chapman, Scott T.; Gotti, Felix; Gotti, Marly

doi:10.1080/00029890.2021.1865064

Cited by 30 publications

(24 citation statements)

References 33 publications

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“…None of the implications of Diagram 1.1 are reversible in the class of positive monoids. Moreover, as is illustrated in [13], none of the implications (except UFM ⇒ HFM) is reversible in the subclass of Puiseux monoids. An example of a half-factorial positive monoid that is not a UFM is given in Example 6.6.…”

Section: A Class Of Atomic Positive Monoidsmentioning

confidence: 86%

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Atomicity of Positive Monoids

Chapman¹,

Gotti²

2021

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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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Section: A Class Of Atomic Positive Monoidsmentioning

confidence: 86%

“…Puiseux monoids are perhaps the positive monoids that have been most systematically investigated in the last five years (see [20] and references therein). A survey on the atomicity of Puiseux monoids can be found in the recent Monthly article [13].…”

Section: Ufm Hfm Ffm Bfm Accp Monoid Atomic Monoidmentioning

confidence: 99%

Atomicity of Positive Monoids

Chapman¹,

Gotti²

2021

Preprint

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“…A torsion-free rank-one monoid that is not a group is called a Puiseux monoid. It follows from [9, Theorem 3.12.1] that nontrivial submonoids of (Q ≥0 , +) account for all Puiseux monoids up to isomorphism, and their atomicity has been systematically studied recently (see [6] and references therein). Let (p n ) n∈N be the strictly increasing sequence consisting of odd primes, and consider the Puiseux monoid To formalize and generalize the last two observations, let M be a monoid, and let N be a submonoid of M .…”

Section: Generalized Grams' Constructionmentioning

confidence: 99%

Atomic semigroup rings and the ascending chain condition on principal ideals

Gotti¹,

Li²

2021

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An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals terminates. It was asserted by Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by Grams in the seventies with an explicit construction of a neat example. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams' first construction. In the first part of this paper, we generalize Grams' construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct what seems to be the first atomic semigroup ring without the ACCP in the existing literature.

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“…The key ingredient in Grams' construction is an additive submonoid of Q ≥0 , which we introduce in the next example. The atomicity of additive submonoids of Q ≥0 has been systematically investigated during the last few years (see the recent survey [13] and references therein). As we will confirm here, these monoids are effective to find counterexamples in commutative ring theory (see also [20]).…”

Section: Atomicity and The Accpmentioning

confidence: 99%

Divisibility in rings of integer-valued polynomials

Gotti¹,

Li²

2021

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In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of them have been found since the first one was constructed by A. Grams in 1974. Here we exhibit the first class of atomic rings of integer-valued polynomials without the ACCP. An integral domain is called a finite factorization domain (FFD) if it is simultaneously atomic and an idf-domain (i.e., every nonzero element is divisible by only finitely many irreducibles up to associates). We prove that a ring is an FFD if and only if its ring of integer-valued polynomials is an FFD. In addition, we show that neither being atomic nor being an idf-domain transfer, in general, from an integral domain to its ring of integer-valued polynomials. In the same class of rings of integer-valued polynomials, we consider further properties that are defined in terms of divisibility by irreducibles, including being Cohen-Kaplansky and being Furstenberg.

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When Is a Puiseux Monoid Atomic?

Cited by 30 publications

References 33 publications

Atomicity of Positive Monoids

Atomicity of Positive Monoids

Atomic semigroup rings and the ascending chain condition on principal ideals

Divisibility in rings of integer-valued polynomials

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