Proportionally modular numerical semigroups with embedding dimension three. (Q2875437)

A numerical semigroup is a subset \(S\) of \(\mathbb N\) that is closed under addition, \(0\in S\) and \(\mathbb N\setminus S\) is finite. A numerical semigroup is called a proportionally modular numerical semigroup (\(\mathbf{PM}\)-semigroup) if it is the set of integer solutions of a proportionally modular Diophantine inequality which is an expression of the form: \(ax\bmod b\leq cx\), where \(a,b,c\) are positive integers. If in the mentioned equality \(c=1\), the corresponding numerical semigroup is called a modular numerical semigroup (\(\mathbf M\)-semigroup). A numerical semigroup is an irreducible numerical semigroup if it can not be expressed as the intersection of two numerical semigroups containing it properly.NEWLINENEWLINE Some results concerning these notions are obtained. One of the results says that irreducible \(\mathbf{PM}\)-semigroups generated by three positive integers are \(\mathbf M\)-semigroups.