nLab
causal complement

Idea

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.

Definition

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

$$M\mapsto M^{\perp }\subset X$$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) $M\bigcap M^{\perp }=\varnothing$

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.

A causal disjointness relation on an index set of subsets of a given set $X$ can be defined via

$$M_1\perp M_2\text{iff}{M}_1\subseteq (M_2{)}^{\perp }$$M_1 \perp M_2 \; \text{iff} \; M_1 \subseteq (M_2)^{\perp}

if all sets $M$ have a causal complement and if

(iv) there is a sequence $(Y_n{)}_{n=1}^{\mathrm{\infty }}$ of mutually different subsets with $Y_n^{\perp }\ne \varnothing$ and $\bigcup Y_n=X$.

The latter condition is needed to get a $\sigma$-bounded poset; the $\sigma$-boundedness is part of the definition of a causal index set.