On the genus of a quotient of a numerical semigroup

Adeniran, Ayomikun; Butler, Steve; Defant, Colin; Gao, Yibo; Harris, Pamela E.; Hettle, Cyrus; Liang, Qingzhong; Nam, Hayan; Volk, Adam

doi:10.1007/s00233-018-9989-3

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Cited by 3 publications

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“…Quotients of numerical semigroups have appeared in the literature over the past couple of decades [17,21] as well as recently [1,14]. See [18,Ch.…”

Section: Introductionmentioning

confidence: 99%

When Is a Numerical Semigroup a Quotient?

Bogart

O’Neill²,

Woods³

2023

Bull. Aust. Math. Soc.

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A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal {S}$ is a quotient of a numerical semigroup with k generators, we call $\mathcal {S}$ a k-quotient. We give a necessary condition for a given numerical semigroup $\mathcal {S}$ to be a k-quotient and present, for each $k \ge 3$ , the first known family of numerical semigroups that cannot be written as a k-quotient. We also examine the probability that a randomly selected numerical semigroup with k generators is a k-quotient.

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“…Quotients of numerical semigroups have appeared in the literature over the past couple of decades [17,21] as well as recently [1,14]. See [18,Ch.…”

Section: Introductionmentioning

confidence: 99%

When Is a Numerical Semigroup a Quotient?

Bogart

O’Neill²,

Woods³

2023

Bull. Aust. Math. Soc.

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show abstract

Numerical Semigroups via Projections and via Quotients

Bogart,

O’Neill,

Woods

2024

Discrete Comput Geom

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The genus of a quotient of several types of numerical semigroups

Lee,

Nam

2024

Int. J. Number Theory

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Finding the Frobenius number and the genus of any numerical semigroup [Formula: see text] is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple.

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On the genus of a quotient of a numerical semigroup

Cited by 3 publications

References 17 publications

When Is a Numerical Semigroup a Quotient?

When Is a Numerical Semigroup a Quotient?

Numerical Semigroups via Projections and via Quotients

The genus of a quotient of several types of numerical semigroups

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