  Inspired by Schwartz, Jang-Lewis and Victory, who study in particular
generalizations of triangularizations of matrices to operators, we shall give
for positive operators on Lebesgue spaces equivalent definitions of atoms
(maximal irreducible sets). We also characterize positive power compact
operators having a unique non-zero atom which appears as a natural
generalization of irreducible operators and are also considered in
epidemiological models. Using the different characterizations of atoms, we also
provide a short proof for the representation of the ascent of a positive power
compact operator as the maximal length in the graph of critical atoms.
