Abstract
Let \(S,T\) be two numerical semigroups. We study when \(S\) is one half of \(T\), with \(T\) almost symmetric. If we assume that the type of \(T\), \(t(T)\), is odd, then for any \(S\) there exist infinitely many such \(T\) and we prove that \(1 \le t(T) \le 2t(S)+1\). On the other hand, if \(t(T)\) is even, there exists such \(T\) if and only if \(S\) is almost symmetric and different from \({\mathbb {N}}\); in this case the type of \(S\) is the number of even pseudo-Frobenius numbers of \(T\). Moreover, we construct these families of semigroups using the numerical duplication with respect to a relative ideal.
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Acknowledgments
The author would like to thank Marco D’Anna for his help and support during the drafting of the paper and Pedro García-Sánchez for his useful suggestions.
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Communicated by Fernando Torres.
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Strazzanti, F. One half of almost symmetric numerical semigroups. Semigroup Forum 91, 463–475 (2015). https://doi.org/10.1007/s00233-014-9641-9
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DOI: https://doi.org/10.1007/s00233-014-9641-9