The right-generators descendant of a numerical semigroup

TL;DR

This paper introduces an efficient algorithm for exploring the descendants of a numerical semigroup within its tree structure, focusing on primitive elements and leveraging specific properties to optimize the process.

Contribution

It presents a novel method to quickly generate descendant sets of numerical semigroups, improving the exploration of the semigroup tree up to a certain genus.

Findings

Efficient algorithm for semigroup tree exploration

Utilizes properties of primitive elements and the second nonzero element

Applicable to pseudo-ordinary cases with conductor

Abstract

For a numerical semigroup, we encode the set of primitive elements that are larger than its Frobenius number and show how to produce in a fast way the corresponding sets for its children in the semigroup tree. This allows us to present an efficient algorithm for exploring the tree up to a given genus. The algorithm exploits the second nonzero element of a numerical semigroup and the particular pseudo-ordinary case in which this element is the conductor.