Density of numerical sets associated to a numerical semigroup
| Autor: | Deepesh Singhal, Yuxin Lin |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Semigroup 010102 general mathematics Zero (complex analysis) 010103 numerical & computational mathematics 01 natural sciences Set (abstract data type) Mathematics::Category Theory Numerical semigroup FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Mathematics |
| Zdroj: | Communications in Algebra. 49:4291-4303 |
| ISSN: | 1532-4125 0092-7872 |
| DOI: | 10.1080/00927872.2021.1918136 |
| Popis: | A numerical set is a co-finite subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its complement. Each numerical set has an associated semigroup $A(T)=\{t\mid t+T\subseteq T\}$, which has the same Frobenius number as $T$. For a fixed Frobenius number $f$ there are $2^{f-1}$ numerical sets. It is known that there is a number $\gamma$ close to $0.484$ such that the ratio of these numerical sets that are mapped to $N_f=\{0\}\cup(f,\infty)$ is asymptotically $\gamma$. We identify a collection of families $N(D,f)$ of numerical semigroups such that for a fixed $D$ the ratio of the $2^{f-1}$ numerical sets that are mapped to $N(D,f)$ converges to a positive limit as $f$ goes to infinity. We denote the limit as $\gamma_D$, these constants sum up to $1$ meaning that they asymptotically account for almost all numerical sets. Comment: 13 pages |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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