Let $X$ be an arbitrary set and $M \subset X$. An assignment

$M \mapsto M^{\perp} \subset X$

is called a causal complement if the following conditions hold:

(i) $M \subseteq M^{\perp\perp}$

(ii) $(\bigcup_j M_j)^{\perp} = \bigcap_j (M_j)^{\perp}$

(iii) $M \bigcap M^{\perp} = \emptyset$

A set $M$ is causally closed iff $M = M^{\perp\perp}$.

The set $M^{\perp\perp}$ is the causal closure of $M$.

Properties

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.