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

Abstract

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.