A Class of Numerical Semigroups Defined by Kunz and Waldi

TL;DR

This paper characterizes a specific class of numerical semigroups called KW semigroups, provides criteria for their principal matrices, describes their minimal resolutions for small dimensions, and explores a three-dimensional generalization using lattice paths.

Contribution

It introduces KW semigroups, characterizes them via principal matrices, and extends the concept to three dimensions with initial results.

Findings

Characterization of KW semigroups by principal matrices

Necessary and sufficient criteria for principal matrices of KW semigroups

Explicit minimal resolutions for dimensions 3 and 4

Abstract

In this paper, we explore a class of numerical semigroups initiated by Kunz and Waldi containing two coprime numbers $p < q$, which we call KW semigroups. We characterize KW numerical semigroups by their principal matrices. We present a necessary and sufficient criterion for a matrix to be the principal matrix of a KW semigroup. An explicit description of the minimal resolutions of numerical semigroups in the same class with small embedding dimensions 3 and 4 is given. We give a generalization of this notion to three dimensions using lattice paths under a plane and present some preliminary results and questions.