The ideals of a numerical semigroup with embedding dimension two

Abstract

Let S and Δ be numerical semigroups. We will say that S is an ideal of Δ if there exits X ⊆ Δ such that S = (X + Δ) ∪ {0}. In this work, we will study the ideals of a numerical semigroup of the form ⟨a, b⟩ with a and b positive integers such that gcd{a, b} = 1. The main results that we have obtained are the following: 1. Given a numerical semigroup S and {a, b} ⊆ N such that gcd{a, b} = 1, we present an algorithm that allows us to determine if S is an ideal of ⟨a, b⟩. 2. If S is a numerical semigroup, we show an algorithmic procedure to compute the set {{a, b} ⊆ N | gcd{a, b} = 1 and S is an ideal of ⟨a, b⟩} . 3. We obtain formulas to compute the multiplicity, Frobenius number and genus of the numerical semigroups of the form (X + ⟨a, b⟩) ∪ {0} in terms of X, a and b.