Transit Functions and Clustering Systems

TL;DR

This paper explores the mathematical properties of transit functions related to clustering systems, providing new characterizations and classifications inspired by hypergraph structures like pyramids and weak hierarchies.

Contribution

It introduces alternative characterizations of weak hierarchies and classifies union-closed and weakly pyramidal clustering systems within the framework of transit functions.

Findings

Characterization of weak hierarchies

Union-closed clustering systems as pyramids

Weakly pyramidal systems as generalizations

Abstract

Transit functions serve not only as abstractions of betweenness and convexity but are also closely connected with clustering systems. Here, we investigate the canonical transit functions of binary clustering systems inspired by pyramids, i.e., interval hypergraphs. We provide alternative characterizations of weak hierarchies, and describe union-closed binary clustering systems as a subclass of pyramids and weakly pyramidal clustering systems as an interesting generalization.