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.