On the cardinality of minimal presentations of numerical semigroups

In this paper, we consider the following question: "given the multiplicity m and embedding dimension e of a numerical semigroup S, what can be said about the cardinality η of a minimal presentation of S?" We approach this question from a combinatorial (poset-theoretic) perspective, utilizing the recently-introduced notion of a Kunz nilsemigroup. In addition to making significant headway on this question beyond what was previously known, in the form of both explicit constructions and general bounds, we provide a self-contained introduction to Kunz nilsemigroups that avoids the polyhedral geometry necessary for much of their source material.