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

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.