The concept of a causal complement is suitable to establish a causal disjointness relation on index sets where the indices are subsets of a given set. This generalizes the concept of causal complement of subsets of the Minkowski spacetime, see for example Haag-Kastler vacuum representation.
Let be an arbitrary set and . An assignment
is called a causal complement if the following conditions hold:
A set is causally closed iff .
The set is the causal closure of .
Causal complements are always causally closed. The intersection of two causally closed sets is again a causally closed set. The causal complement of a set may be empty.
A causal disjointness relation on an index set of subsets of a given set can be defined via
if all sets have a causal complement and if
(iv) there is a sequence of mutually different subsets with and .