The Set of Numerical Semigroups with Frobenius Number Belonging to a Fixed Interval

Let a and b be positive integers such that a<b and [a,b]={x∈N∣a≤x≤b}. In this work, we will show that A([a,b])={S∣S is a numerical semigroup whose Frobenius number belongs to [a,b]} and is a covariety. This fact allows us to present an algorithm which computes all the elements from A([a,b]). We will prove that A([a,b],m)={S∈A([a,b])∣S has multiplicity m} and is a ratio-covariety. As a consequence, we will show an algorithm which calculates all the elements belonging to A([a,b],m). Based on the above results, we will develop an interesting algorithm that calculates all numerical semigroups with a given multiplicity and complexity.