When Is a Numerical Semigroup a Quotient?

Abstract

A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal {S}$ is a quotient of a numerical semigroup with k generators, we call $\mathcal {S}$ a k-quotient. We give a necessary condition for a given numerical semigroup $\mathcal {S}$ to be a k-quotient and present, for each $k \ge 3$ , the first known family of numerical semigroups that cannot be written as a k-quotient. We also examine the probability that a randomly selected numerical semigroup with k generators is a k-quotient.

Arithmetic Varieties of Numerical Semigroups

“…(4) Return A (F). Therefore, by Algorithm 12, we conclude that For example, if F = { 2, 5 , 3, 5, 7 } we write [3,5,7]]; F:=List(F,i->NumericalSemigroup(i)); SmallestArithmeticVariety(F); provided that the package NumericalSgps and the function Arithmetic Extensions have already been loaded into GAP.…”

“…In [10], it is proved its existence; however no example is given. Recently, in [3], some examples are exhibited. Have an algorithm to decide whether a numerical semigroup belongs to S d | S ∈ ED(3) and d ∈ N\{0} is still an open problem.…”