# Contents

## 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

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

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.

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

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

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

(iv) there is a sequence $(Y_n)_{n=1}^{\infty}$ of mutually different subsets with $Y_n^{\perp} \neq \emptyset$ 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.

Revised on June 11, 2010 09:02:48 by Toby Bartels (173.190.150.217)