Algorithms and 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 numericalsemigroup (GNS) is a cofinite submonoid S ⊆ N^d. The cardinality of N^dS is calledthe 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 Ng,d for large values of g, d. For a fixed g, we show that F_g(d) = N_g,d isa polynomial function of degree g. We close with several open problems/conjecturesrelated to the asymptotic growth of N_g,d and with suggestions for further avenues of research.