Frobenius Allowable Gaps of Generalized Numerical Semigroups Article Swipe
A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ for which the complement $\mathbb{N}^d\setminus S$ is finite. The points in the complement $\mathbb{N}^d\setminus S$ are called gaps. A gap $F$ is considered Frobenius allowable if there is some relaxed monomial ordering on $\mathbb{N}^d$ with respect to which $F$ is the largest gap. We characterize the Frobenius allowable gaps of a generalized numerical semigroup. A generalized numerical semigroup that has only one maximal gap under the natural partial ordering of $\mathbb{N}^d$ is called a Frobenius generalized numerical semigroup. We show that Frobenius generalized numerical semigroups are precisely those whose Frobenius gap does not depend on the relaxed monomial ordering. We estimate the number of Frobenius generalized numerical semigroup with a given Frobenius gap $F=(F^{(1)},\dots,F^{(d)})\in\mathbb{N}^d$ and show that it is close to $\sqrt{3}^{(F^{(1)}+1)\cdots (F^{(d)}+1)}$ for large $d$. We define notions of quasi-irreducibility and quasi-symmetry for generalized numerical semigroups. While in the case of $d=1$ these notions coincide with irreducibility and symmetry, they are distinct in higher dimensions.
Related Topics
Concepts
Numerical semigroup
Mathematics
Irreducibility
Complement (music)
Monomial
Semigroup
Combinatorics
Discrete mathematics
Pure mathematics
Chemistry
Phenotype
Gene
Biochemistry
Complementation
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.37236/10748
- https://www.combinatorics.org/ojs/index.php/eljc/article/download/v29i4p12/pdf
- OA Status
- diamond
- Cited By
- 6
- References
- 18
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W3151354368
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W3151354368Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.37236/10748Digital Object Identifier
- Title
-
Frobenius Allowable Gaps of Generalized Numerical SemigroupsWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2022Year of publication
- Publication date
-
2022-10-21Full publication date if available
- Authors
-
Deepesh Singhal, Yuxin LinList of authors in order
- Landing page
-
https://doi.org/10.37236/10748Publisher landing page
- PDF URL
-
https://www.combinatorics.org/ojs/index.php/eljc/article/download/v29i4p12/pdfDirect link to full text PDF
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YesWhether a free full text is available
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diamondOpen access status per OpenAlex
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https://www.combinatorics.org/ojs/index.php/eljc/article/download/v29i4p12/pdfDirect OA link when available
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Numerical semigroup, Mathematics, Irreducibility, Complement (music), Monomial, Semigroup, Combinatorics, Discrete mathematics, Pure mathematics, Chemistry, Phenotype, Gene, Biochemistry, ComplementationTop concepts (fields/topics) attached by OpenAlex
- Cited by
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6Total citation count in OpenAlex
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-
2025: 1, 2024: 5Per-year citation counts (last 5 years)
- References (count)
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18Number of works referenced by this work
- Related works (count)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.ordering. | 108 |
| abstract_inverted_index.precisely | 96 |
| abstract_inverted_index.semigroup | 3, 67, 117 |
| abstract_inverted_index.submonoid | 6 |
| abstract_inverted_index.symmetry, | 159 |
| abstract_inverted_index.complement | 13, 22 |
| abstract_inverted_index.considered | 32 |
| abstract_inverted_index.semigroup. | 63, 87 |
| abstract_inverted_index.semigroups | 94 |
| abstract_inverted_index.dimensions. | 165 |
| abstract_inverted_index.generalized | 1, 61, 65, 85, 92, 115, 144 |
| abstract_inverted_index.semigroups. | 146 |
| abstract_inverted_index.characterize | 54 |
| abstract_inverted_index.(F^{(d)}+1)}$ | 132 |
| abstract_inverted_index.$\mathbb{N}^d$ | 9, 43, 80 |
| abstract_inverted_index.irreducibility | 157 |
| abstract_inverted_index.quasi-symmetry | 142 |
| abstract_inverted_index.quasi-irreducibility | 140 |
| abstract_inverted_index.$\mathbb{N}^d\setminus | 14, 23 |
| abstract_inverted_index.$\sqrt{3}^{(F^{(1)}+1)\cdots | 131 |
| abstract_inverted_index.$F=(F^{(1)},\dots,F^{(d)})\in\mathbb{N}^d$ | 123 |
| cited_by_percentile_year.max | 98 |
| cited_by_percentile_year.min | 91 |
| countries_distinct_count | 1 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile.value | 0.93249636 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |