Algorithms and basic asymptotics for generalized numerical semigroups in N

Let denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid . For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid . The cardinality of 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 denote the number of generalized numerical semigroups of genus . We compute for small values of and provide coarse asymptotic bounds on for large values of . For a fixed , we show that is a polynomial function of degree . We close with several open problems/conjectures related to the asymptotic growth of and with suggestions for further avenues of research.