## 27 February 2010

### Defining Rank-Based Taxa Mathematically

Let U be the set of all individuals.

Let ranks be represented by a contiguous series of natural numbers (). Let 1 represent the lowest (finest) rank and let some natural number n represent the highest (coarsest) rank.

Let T be a sequence of n sets of type individuals (i.e., individuals represented by type specimens). Let each set in the sequence (other than the last set) be a superset of the next set, i.e., T1 ⊇ T2 ⊇ … Tn.

Let d be a metric function measuring some distance between any two individuals: d(x, y) ∈ ℝ0+ (the set of nonnegative real numbers). Note that, because it is a metric, d(x, x) = 0 and d(x, y) = d(y, x).

For each rank level r, let pr be a function mapping each member, t, of Tr to a taxon (set of individuals): pr(t) := {x ∈ U | for all s ∈ Tr, d(x, t) ≤ d(x, s)}. Let Pr be the image of pr. Then Pr is the taxonomy of rank level r.

Note that some individuals may be placed in multiple taxa of the same rank if they are equidistant between type individuals. These individuals may be considered unclassifiable for that rank. Let U′ be the set of all individuals except for those which are unclassifiable for some rank. Similarly, let P′r be Pr but with all unclassifiable individuals removed from each member taxon. P′r is a partition on U′. For any two rank levels q and r, if q < r, then P′q is a refinement of (or equal to) P′r.

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