While between partial orders it is usual to consider order-preserving functions, the most important type of functions between prefix orders are so-called history preserving functions. Given a prefix ordered set "P", a history of a point "p∈P" is the (by definition totally ordered) set "p- ≜ {q | q ≤ p}". A function "f : P → Q" between prefix orders P and Q is then history preserving if and only if for every "p∈P" we find "f(p-) = f(p)-". Similarly, a future of a point "p∈P" is the (prefix ordered) set "p+ ≜ {q | p ≤ q}" and "f" is future preserving if for all "p∈P" we find "f(p+) = f(p)+".