Radical Numerical Semigroups

Abstract

This work contributes to the study of radical numerical semigroups. If 𝑛� ∈ℤ where 𝑛� ≥2, then the product of all its prime positive divisors is called the radical of n. It is denoted by r⁡(𝑛�). A radical numerical semigroup is a numerical semigroup S such that 𝑠� +r⁡(𝑠�) ∈𝑆� for every 𝑠� ∈𝑆�\{0}. We present three algorithms that will help us understand the structure of radical semigroups. These algorithms allow us to calculate all radical numerical semigroups with a fixed genus, with a fixed Frobenius number, as well as with a fixed multiplicity. Furthermore, given X, a set of positive integers such that gcd⁡(𝑋�) =1, we will prove the existence of the smallest radical semigroup containing X. We will also present an algorithm to obtain it.