Let {a1, . . . , ap} be the minimal generating set of a numerical monoid S. For any s ∈ S, its Delta set is defined by ∆xiai and xi ∈ N for all i}. The Delta set of a numerical monoid S, denoted by ∆(S), is the union of all the sets ∆(s) with s ∈ S. As proved in [5], there exists a bound N such that ∆(S) is the union of the sets ∆(s) with s ∈ S and s < N . In this work, we obtain a sharpened bound and we present an algorithm for the computation of ∆(S) that requires only the factorizations of a1 elements.

We give an algorithm to compute the ω-primality of finitely generated atomic monoids. Asymptotic ω-primality is also studied and a formula to obtain it in finitely generated quasi-Archimedean monoids is proven. The formulation is applied to numerical semigroups, obtaining an expression of this invariant in terms of its system of generators.

In this paper we present a new kind of semigroups called convex body semigroups which are generated by convex bodies of R^k. They generalize to arbitrary dimension the concept of proportionally modular numerical semigroup of [7]. Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of R^2 are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them

A numerical semigroup is perfect if it does not have isolated gaps. In this paper we will order the perfect numerical semigroups with a fixed multiplicity. This ordering allows us to give an algorithm procedure to obtain them. We also study the perfect monoid, which is a subset of N that can be expressed as an intersection of perfect numerical semigroups, and we present the perfect monoid generated by a subset of N. We give an algorithm to calculate it. We study the perfect closure of a numerical semigroup, as well as the perfect numerical semigroup with maximal embedding dimension, in particular Arf and saturated numerical semigroups.

A numerical semigroup is perfect if it has no isolated gaps. In this paper, we will characterize the perfect numerical semigroups with embedding dimension three, and we show how to obtain them all. Also, we obtain formulas for each of the genus and the pseudo-Frobenius numbers of these semigroups.