Radical Numerical Semigroups

This work contributes to the study of radical numerical semigroups. If $n\in \mathbb{Z}$ where $n\ge 2$, then the product of all its prime positive divisors is called the radical of n. It is denoted by $\mathrm{r}\left(n\right)$. A radical numerical semigroup is a numerical semigroup S such that $s+\mathrm{r}\left(s\right)\in S$ for every $s\in S\setminus \left\{0\right\}$. 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 $gcd\left(X\right)=1$, we will prove the existence of the smallest radical semigroup containing X. We will also present an algorithm to obtain it.