2009/01/02 by Graham Brightwell, Brightwell, Graham, Malwina Luczak +1
Computer Science · Mathematics · #06A07 #60C05 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Probability (math.PR) #math.CO #math.PR #msc:06A07 #msc:60C05
paper · pdf · doi:10.48550/arxiv.0901.0242
25 pages; to appear in Combinatorics, Probability and Computing
openalex publication_date 2009/01/02 · arxiv created 2012/01/29 · arxiv updated 2012/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers -- we call such a linear extension a \em natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of \em order-invariance: if we condition on the set of the bottom k elements of the natural extension, each possible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.