  A condition of geometric modular action is proposed as a selection principle
for physically interesting states on general space-times. This condition is
naturally associated with transformation groups of partially ordered sets and
provides these groups with projective representations. Under suitable
additional conditions, these groups induce groups of point transformations on
these space-times, which may be interpreted as symmetry groups. The
consequences of this condition are studied in detail in application to two
concrete space-times -- four-dimensional Minkowski and three-dimensional de
Sitter spaces -- for which it is shown how this condition characterizes the
states invariant under the respective isometry group. An intriguing new
algebraic characterization of vacuum states is given. In addition, the logical
relations between the condition proposed in this paper and the condition of
modular covariance, widely used in the literature, are completely illuminated.
