A semilattice structure for the set of numerical semigroups with fixed Frobenius number

TL;DR

This paper introduces a structured approach to enumerate all numerical semigroups with a fixed Frobenius number by partitioning the set into classes and reconstructing the entire set from key elements, using a novel congruence relation and Kunz-coordinates vectors.

Contribution

It presents a new method for enumerating numerical semigroups with a given Frobenius number using a semilattice structure and an efficient reconstruction technique.

Findings

Successfully enumerates all semigroups with fixed Frobenius number

Identifies unique irreducible and homogeneous semigroups in each class

Proposes an efficient algorithm using Kunz-coordinates vectors

Abstract

We present a procedure to enumerate the whole set of numerical semigroups with a given Frobenius number F, S(F). The methodology is based on the construction of a partition of S(F) by a congruence relation. We identify exactly one irreducible and one homogeneous numerical semigroup at each class in the relation, and from those two elements we reconstruct the whole class. An alternative more efficient method is proposed based on the use of the Kunz-coordinates vectors of the elements in S(F).