Abstract
Let \({\mathbb {N}}\) denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid \(S\subseteq {\mathbb {N}}\). For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid \(S\subseteq {\mathbb {N}}^d\). The cardinality of \({\mathbb {N}}^d \setminus S\) is called the genus. We describe a family of algorithms, parameterized by (relaxed) monomial orders, that can be used to generate trees of semigroups with each GNS appearing exactly once. Let \(N_{g,d}\) denote the number of generalized numerical semigroups \(S\subseteq {\mathbb {N}}^d\) of genus \(g\). We compute \(N_{g,d}\) for small values of \(g,d\) and provide coarse asymptotic bounds on \(N_{g,d}\) for large values of \(g,d\). For a fixed \(g\), we show that \(F_g(d)=N_{g,d}\) is a polynomial function of degree \(g\). We close with several open problems/conjectures related to the asymptotic growth of \(N_{g,d}\) and with suggestions for further avenues of research.
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Acknowledgments
The authors would like to thank INDAM for support. The second author would like to thank the kind hospitality of the Mediterranean University of Reggio Calabria and the University of Messina. Motivation for this paper was inspired by algorithms and open problems presented in a talk by Shalom Eliahou (to whom we express our thanks). We also wish to thank an anonymous referee who provided several valuable comments which led to improved exposition.
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Communicated by Fernando Torres.
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Failla, G., Peterson, C. & Utano, R. Algorithms and basic asymptotics for generalized numerical semigroups in \({\mathbb {N}}^d\) . Semigroup Forum 92, 460–473 (2016). https://doi.org/10.1007/s00233-015-9690-8
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DOI: https://doi.org/10.1007/s00233-015-9690-8