Random Event Structures
Manfred Droste, Guo‐Qiang Zhang · Publikation Server (Leipzig University) · 2008
In a line of recent development, probabilistic constructions of universal, homo- geneous objects have been provided in various categories of ordered structures, such as causal sets (12) , biflnite domains (13) , and countable partial orders (10) . These constructions have been shown to produce objects with the desired properties with probability 1 in an appropriately deflned measure space. A common strategy for these constructions is successive point-wise extension of an existing flnite structure, with decisions on the relationships between the newly added point and the existing structure made according to well-specifled probabilistic choices. This strategy is a departure from (and understandably so due to the increased com- plexity) the original one for random graphs (16) where a universal homogeneous countable graph is constructed with probability 1 in a single step (i.e., a single round of countably many probabilistic choices made independently). It would be interesting to see which of the categories studied more recently may admit such \one-step constructions. The main focus of this paper is a new strategy, consisting of a single round of countably many probabilistic choices made independently, for the construction of a universal, homogeneous prime event structure. The intuition that the one-round construction is desirable has a similar ∞avor to a more general setting in e.g. Calculus/Real Analysis. When taking limits, iterative step- by-step processes are usually given, but a set of machineries was invented to determine the limit, i.e., achieving a \one-round direct and explicit description of the limit.