Remarks on cluster systems

CREATED: 200910030211

Presented by Andreas Dress at GCM2009

• $X$ be a finite set (eg. set of species)
• A cluster system, $\mathcal{C}$ is a subset of $P(X)$ (eg. clades)
• Special kinds of $\mathcal{C}$: partition, hierarchy
• Charles Ray, a Catholic, coined the term //species// in 1600
• Lineals? proposed a hierarchical system for all species
• and also one for minerals but without not much success
• Freiberg (Germany) has the oldest mineral museum
• Hierachy: $A, B \in \mathcal{C} \rightarrow A \cap B \in {\emptyset, A, B}$, size is $O(n)$
• Weak hierachy: $A, B, C \in \mathcal{C} \rightarrow A \cap B \cap C \in {\phi, A \cap B, A \cap C, B \cap C}$, size is $O(n^2)$
• Neighbourhoods: $B(x|r) = {y | D(x,y) \le r}$, size is $O(n^2)$
• Relative distances, instead of $r$, $D(x,y) \le D(a,b)$ but may not have much sense
• Compare relative to a fix element, $D(x,y) \le D(x,z)$, leads to ranking
• Ranking functions are more robust: $rk_x(x) = 0$ $rk_x(y) \le k$
• $H(\mathcal{C}) = {C \in \mathcal{C} | C \cap C’ \in {\emptyset, C, C’}, \forall C’ \in \mathcal{C} }$