Patterns on Numerical Semigroups

TL;DR

This paper introduces a new pattern-based framework for classifying numerical semigroups, generalizing Arf semigroups, and provides algorithms for computing semigroup closures and constructing a comprehensive directed graph of semigroups.

Contribution

It defines a novel pattern concept for numerical semigroups, extends the classification, and develops recursive procedures and graph structures for their analysis.

Findings

Classifies all numerical semigroups via pattern-based classes.

Provides a recursive algorithm for semigroup closure computation.

Constructs a directed acyclic graph of semigroups with the pattern property.

Abstract

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 escribe 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.