  Introduced is the notion of minimality for spectral representations of sum-
and max-infinitely divisible processes and it is shown that the minimal
spectral representation on a Borel space exists and is unique. This fact is
used to show that a stationary, stochastically continuous, sum- or max-i.d.
random process on $\mathbb{R}^d$ can be generated by a measure-preserving flow
on a $\sigma$-finite Borel measure space and that this flow is unique. This
development makes it possible to extend the classification program of
Rosi\'{n}ski (Ann. Probab. 23 (1995) 1163-1187) with a unified treatment of
both sum- and max-infinitely divisible processes. As a particular case, a
characterization of stationary, stochastically continuous, union-infinitely
divisible random measurable subsets of $\mathbb{R}^d$ is obtained. Introduced
and classified are several new max-i.d. random field models including fields of
Penrose type and fields associated to Poisson line processes.
