Defining Rank-Based Taxa Mathematically

Let U be the set of all individuals.

Let ranks be represented by a contiguous series of natural numbers (ℕ). 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 ⊇ … T_n.

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 p_r be a function mapping each member, t, of T_r to a taxon (set of individuals): p_r(t) := {x ∈ U | for all s ∈ T_r, d(x, t) ≤ d(x, s)}. Let P_r be the image of p_r. Then P_r 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 P_r 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.