Transit Functions and Clustering Systems

TL;DR

The paper analyzes canonical transit functions for binary clustering systems, focusing on interval-hypergraph (pyramidal) structures. It provides axiomatic characterizations of weak hierarchies via transit-function properties, and introduces union-closure and weakly pyramidal clustering to situate between hierarchies and pyramids. By establishing equivalences and covering/union properties, the work clarifies how betweenness-based clustering can be axiomatized and how various generalizations relate. It also presents open questions about finite first-order axioms for pyramidal transit sets and explores the boundaries of these clustering families. Overall, the results advance the formal understanding of how transit functions encode interval-like clustering and betweenness in discrete structures.

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.

Figures (6)

• Figure 1: The set system $\mathcal{C}$ on $X$ comprising the singletons, $X$, the edge $A=\{x,z\}$, and the two triples $B=\{x,y,p\}$ and $C=\{y,z,q\}$ is not a weak hierarchy since $A$, $B$, and $C$ intersect pair-wisely but $A\cap B\cap C=\emptyset$. On the other hand, it is a binary clustering system and satisfies (MM).
• Figure 2: Top: the set system $\mathcal{C}$ on $X$ comprising the singletons, $X$, the three edges $\{a,b\}$, $\{a,c\}$, $\{b,c\}$, and the set $\{a,b,c,d\}$ satisfies (KS), (KR), (KC). It satisfies (K3) since the three pairs of overlapping edges are contained in the unique inclusion-minimal cover $\{a,b,c,d\}$. It does not satisfy (MM) since the inclusion minimal set containing $p,q\in\{a,b\}\cup\{b,c\}$ is always one of the three edges. Bottom: the set system $\mathcal{C}$ on $X$ comprising the singletons, $X$, the three edges $\{a,b\}$, $\{a,c\}$, $\{b,c\}$, and the two sets $B_1=\{a,b,c,d\}$ and $B_2=\{a,b,c,e\}$ satisfies (KS), (KR), (KC), and (K1). It violates (K3) since both $B_1$ and $B_2$ are inclusion-minimal covering sets for $\{a,b\}\cup\{b,c\}$.
• Figure 3: The set system $\mathcal{C}$ on $X$ comprising the singletons, $X$, and the three intervals $[x,v]$, $[u,y]$, and $[u,v]$ is clearly pyramidal with the linear order $<$ shown from left to right. For its canonical transit function $R$ we have $R(x,z)=[x,v]$, $R(z,y)=[u,y]$ but $R(x,y)=X\ne [x,v]\cup[u,y]=[x,y]\notin\mathcal{C}$. Therefore $R$ is pyramidal but satisfies neither (uc) nor (u).
• Figure 4: The set system $\mathcal{C}$ on $X$ comprising the singletons, $X$, and the four edges $C_1=\{a,b\}$, $C_2=\{b,c\}$, $C_3=\{c,d\}$ and $C_4=\{d,a\}$ is a weak hierarchy (since any triple of sets that intersect pairwise contains either $X$ or a singleton), and satisfies axiom (WP). Since the four edges form a 4-cycle $(C_1,C_2,C_3,C_4)$ in $(X,\mathcal{C})$ there is no linear ordering on $X$ compatible with $\mathcal{C}$. Thus $(X,\mathcal{C})$ is weakly (pre-)pyramidal but not (pre-)pyramidal.
• Figure 5: Top: The set system $\mathcal{C}$ consists of the singletons, the three edges $\{x,z\}$, $\{y,z\}$, $\{u,z\}$, and $X$. One easily checks that it is a weak hierarchy. Its canonical transit function $R$ satisfies $R(p,q)=X$ if and only if $p\ne q$ and $p,q\in\{x,y,u\}$. Obviously $z\in R(p,q)=X$. However, $R(p,z)\cup R(z,q)=\{p,q,z\}\ne R(p,q)$; thus $R$ does not satisfy (o). Below: The set system comprises the singletons, $X$, and the three pairs $\{p,q\}$ with $p,q\in\{x,y,z\}$. These three pairs intersect pair-wisely, but $\{x,y\}\cap\{x,z\}\cap \{y,z\}=\emptyset$, thus $\mathcal{C}$ is not a weak hierarchy. The canonical transit function $R$ satisfies $R(p,q)=\{p,q\}$ for $p,q\in\{x,y,z\}$ and $R(p,u)=X$ for $p\in\{x,y,z\}$. For pairs, the statement of (o) is trivially true, and for $z\in R(p,u)=X$ we have $R(p,u)=R(p,z)\cup R(z,u)$ since $R(z,u)=X$ for $z\ne u$ and $R(p,z)=X$ for $z=u$. Thus $R$ satisfies (o) but not (w).

Theorems & Definitions (47)

• Proposition 1: Changat:19a
• Definition 1
• Proposition 2: Nebesky:83
• Theorem 1
• proof
• Example 1
• Lemma 1
• proof
• Example 2
• Theorem 2