Numerical semigroups with embedding dimension three and minimal catenary degree

TL;DR

This paper characterizes certain numerical semigroups with embedding dimension three that achieve a specific equality involving their Delta set and catenary degree, advancing understanding of their algebraic structure.

Contribution

It provides a complete characterization of numerical semigroups with embedding dimension three that attain equality in a key inequality relating Delta set and catenary degree.

Findings

Identifies conditions for equality in the inequality involving Delta set and catenary degree

Characterizes numerical semigroups with embedding dimension three meeting these conditions

Enhances understanding of the algebraic structure of these semigroups

Abstract

We characterize numerical semigroups $S$ with embedding dimension three attaining equality in the inequality $\max\Delta(S)+2\leq \operatorname{cat}(S)$, where $\Delta(S)$ denotes the Delta set of $S$ and $\operatorname{cat}(S)$ denotes the catenary degree of $S$.