Foliations and automorphisms of causal sets

Abstract

We consider foliations of causal sets and their behavior under causal automorphisms. For a connected, countably infinite causal set containing an infinite antichain, and such that every antichain has finite intersection with the past and future of any point, we prove that each automorphism admits a foliation whose slices it either preserves or translates. In the latter case, the slices admit a rational coordinate which the automorphism changes by one unit, with a common choice of direction. The existence of foliations for arbitrary partially ordered sets follows from a theorem of Milner and Pouzet. Here the slices are ordered by finite paths of links between slices. We distinguish this condition from direct linkage and from the additional condition defining a temporal foliation. Counterexamples show that linked foliations need not exist, and that passage to relation space does not ensure temporal existence.

17 pages, 3 figures. Major revision: revised foliation definition, corrected proofs, and amended theorem statements; previous temporal-existence claims withdrawn. Relation to the Milner-Pouzet theorem clarified

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