Irreducible numerical semigroups with multiplicity three and four. (Q382929)

A numerical semigroup is a submonoid of \(\mathbb N\) (the set of nonnegative integers) with finite complement in \(\mathbb N\). The multiplicity of a numerical semigroup is the smallest positive integer belonging to it. A numerical semigroup is irreducible if it cannot be expressed as the intersection of two numerical semigroups properly containing it. If we fix a multiplicity \(m\), then a numerical semigroup with multiplicity \(m\) is said to be \(m\)-irreducible if it cannot be written as the intersection of two numerical semigroups properly containing it and with multiplicity \(m\). Every numerical semigroup can be expressed as a finite intersection of irreducible numerical semigroups, and the same holds with \(m\)-irreducible if we fix the multiplicity. This paper characterizes \(3\)- and \(4\)-irreducible numerical semigroups, and decompositions of numerical semigroups with multiplicity three and four into \(3\)- and \(4\)-irreducible numerical semigroups, respectively.