Abstract
Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote \({\rm R}(S,a,m) = \{ s-as \bmod m \mid s \in S \}\) (where asmodm is the remainder of the division of as by m). In this paper we characterize the pairs (a,m) such that \({\rm R}(S,a,m)\) is a numerical semigroup. In this way, if we have a pair (a,m) with such characteristics, then we can reduce the problem of computing the genus of S=〈n 1,…,n p 〉 to computing the genus of a “smaller” numerical semigroup 〈n 1−an 1modm,…,n p −an p modm〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.
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Communicated by Fernando Torres.
Both of the authors are supported by FQM-343 (Junta de Andalucía), MTM2010-15595 (MICINN, Spain), and FEDER funds. The second author is also partially supported by Junta de Andalucía/Feder grant number FQM-5849.
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Robles-Pérez, A.M., Rosales, J.C. Modular retractions of numerical semigroups. Semigroup Forum 87, 553–568 (2013). https://doi.org/10.1007/s00233-013-9481-z
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DOI: https://doi.org/10.1007/s00233-013-9481-z