  Abstract approaches to maximum and anti-maximum principles for differential
operators typically rely on the condition that all vectors in the domain of the
operator are dominated by the leading eigenfunction of the operator. We study
the necessity of this condition. In particular, we show that under a number of
natural assumptions, so-called individual versions of both the maximum and the
anti-maximum principle simultaneously hold if and only if the aforementioned
domination condition is satisfied.
  Consequently, we are able to show that a variety of concrete differential
operators do not satisfy an anti-maximum principle.
