On the number of L-shapes in embedding dimension four

TL;DR

This paper investigates the maximum number of L-shapes in minimum distance diagrams within embedding dimension four, extending previous results from lower dimensions and exploring their applications in semigroup factorizations and digraph properties.

Contribution

It provides new bounds on the number of L-shapes in embedding dimension four and analyzes their implications for semigroup and digraph structures.

Findings

L-shapes in dimension four can be bounded in number

L-shapes are useful for studying factorizations and catenary degree

Applications include diameter and distance in digraphs

Abstract

\textit{Minimum distance diagrams}, also known as \textit{\textsf{L}--shapes}, have been used to study some properties related to \textit{weighted Cayley digraphs} of \textit{degree} two and \textit{embedding dimension three numerical semigroups}. In this particular case, it has been shown that these discrete structures have at most two related \textsf{L}--shapes. These diagrams are proved to be a good tool for studing \textit{factorizations} and the \textit{catenary degree} for semigroups and \textit{diameter} and \textit{distance} between vertices for digraphs.