Let N denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid S ⊆ N. For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid S ⊆ N^d . The cardinality of N^d 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 ⊆ 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 Ng,d and with suggestions for further avenues of research.

Algorithms and basic asymptotics for generalized numerical semigroups in N^d

UTANO, Rosanna
Ultimo
2016

Abstract

Let N denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid S ⊆ N. For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid S ⊆ N^d . The cardinality of N^d 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 ⊆ 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 Ng,d and with suggestions for further avenues of research.
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[doi 10.1007%2Fs00233-015-9690-8] G. Failla; C. Peterson; R. Utano -- Algorithms and basic asymptotics for generalized numerical semigroups in $${mathbb {N}}^d$$ N d.pdf

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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11570/3035172
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