Modular numerical semigroups with embedding dimension equal to three (Q1928895)

A modular Diophantine inequality is an inequality of the form \( ax \pmod{b} \leq x \) where \(a,b\) are positive integers. The set of non-negative integer solutions to a modular Diophantine inequality is a numerical semigroup, i.e. a submonoid of \((\mathbb{N}, +)\) with finite complement in \(\mathbb{N}\). A numerical semigroup arising in this way is called modular. Every numerical semigroup has a unique minimal set of generators, and if this set consists of three elements the semigroup is called three-generated. The main result of the paper under review is the classification of three-generated numerical semigroups which are modular. Specifically, the authors provide two disjoint families of semigroups such that a three-generated numerical semigroup is modular if and only if it belongs to one of them; the families are described explicitly by listing the generators. Furthermore, symmetric modular three-generated semigroups are characterized in a similar fashion. Finally, fixed two positive integers \(n_1, n_2\) with \(\gcd(n_1, n_2)=1\), the authors characterize all integers \(n_3 > n_1, n_2\) such that the triple \(\{n_1,n_2,n_3\}\) is the minimal generating set of a modular numerical semigroup.