Causal Sets - Definition

Definition

A causal set (or causet) is a set with a partial order relation that is

  • Reflexive: For all, we have .
  • Antisymmetric: For all, we have .
  • Transitive: For all, we have implies .
  • Locally finite: For all, we have card.

Here card denotes the cardinality of a set . We'll write if and .

The set represents the set of spacetime events and the order relation represents the causal relationship between events (see causal structure for the analogous idea in a Lorentzian manifold).

Although this definition uses the reflexive convention we could have chosen the irreflexive convention in which the order relation is irreflexive. The causal relation of a Lorentzian manifold (without closed causal curves) satisfies the first three conditions. It is the local finiteness condition that introduces spacetime discreteness.

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