  The extremal coefficient function (ECF) of a max-stable process $X$ on some
index set $T$ assigns to each finite subset $A\subset T$ the effective number
of independent random variables among the collection $\{X_t\}_{t\in A}$. We
introduce the class of Tawn-Molchanov processes that is in a 1:1 correspondence
with the class of ECFs, thus also proving a complete characterization of the
ECF in terms of negative definiteness. The corresponding Tawn-Molchanov process
turns out to be exceptional among all max-stable processes sharing the same ECF
in that its dependency set is maximal w.r.t. inclusion. This entails sharp
lower bounds for the finite dimensional distributions of arbitrary max-stable
processes in terms of its ECF. A spectral representation of the Tawn-Molchanov
process and stochastic continuity are discussed. We also show how to build new
valid ECFs from given ECFs by means of Bernstein functions.
