Patterns on numerical semigroups

We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of the whole set of numerical semigroups. In particular, all semigroups can be arranged in an infinite non-stabilizing ascending chain whose first step consists just of the trivial semigroup and whose second step is the well-known class of Arf semigroups. We describe a procedure to compute the closure of a numerical semigroup with respect to a pattern. By using the concept of system of generators associated to a pattern, we construct recursively a directed acyclic graph with all the semigroups admitting the pattern.