Decomposition and classification of length functions

We study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Pr\"ufer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.