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Published in: Applicable Algebra in Engineering, Communication and Computing 6/2021

18-02-2020 | Original Paper

Numerical semigroups closed under addition of their divisors

Authors: J. C. Rosales, M. B. Branco

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 6/2021

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Abstract

A numerical semigroup S is closed under addition of its divisors (\({\mathcal {C}}\)-semigroup) if the following condition holds: if \(s\in S\setminus \{0\}\) and d is a non trivial divisor of s then \(s+d\in S\). In this paper we prove that the set of all \({\mathcal {C}}\)-semigroups is a Frobenius variety. As a consequence, we give algorithms to compute the set \({\mathcal {C}}\)-semigroups with fixed multiplicity, genus and Frobenius number.

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Literature
1.
go back to reference Alfonsín, J.L.R.: The Diophantine Frobenius Problem. Oxford University Press, London (2005) CrossRef Alfonsín, J.L.R.: The Diophantine Frobenius Problem. Oxford University Press, London (2005) CrossRef
2.
go back to reference Robles-Pérez, A.M., Rosales, J.C.: Frobenius pseudo-varieties in numerical semigroups. Annali di Matematica 194, 275–287 (2015) MathSciNetCrossRef Robles-Pérez, A.M., Rosales, J.C.: Frobenius pseudo-varieties in numerical semigroups. Annali di Matematica 194, 275–287 (2015) MathSciNetCrossRef
4.
go back to reference Rosales, J.C.: Families of numerical semigroups closed under finite intersections and for the Frobenius number. Houston J. Math. 34, 339–348 (2008) MathSciNetMATH Rosales, J.C.: Families of numerical semigroups closed under finite intersections and for the Frobenius number. Houston J. Math. 34, 339–348 (2008) MathSciNetMATH
5.
go back to reference Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups. Developments in Mathematics, vol. 20. Springer, New York (2009) CrossRef Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups. Developments in Mathematics, vol. 20. Springer, New York (2009) CrossRef
6.
go back to reference Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Jimenez-Madrid, J.A.: The oversemigroups of a numerical semigroup. Semigroup Forum 67, 145–158 (2003) MathSciNetCrossRef Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Jimenez-Madrid, J.A.: The oversemigroups of a numerical semigroup. Semigroup Forum 67, 145–158 (2003) MathSciNetCrossRef
Metadata
Title
Numerical semigroups closed under addition of their divisors
Authors
J. C. Rosales
M. B. Branco
Publication date
18-02-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 6/2021
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00420-4

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