Notes

Chapter 9: Fundamental Physics

Section 13: Space, Time and Relativity


Posets

The way I set things up, collections of events can be thought of as partially ordered sets (posets). If all events occurred in a definite sequence in time, this would define a total linear ordering for them. But with the setup I use, there is only a partial ordering of events, defined by causal connections. The causal networks I draw are so-called Hasse or order diagrams of the posets of events. If a connection goes directly from x to y in this network then x is said to cover y. And in general if there is a path from x to y then one writes x \[Succeeds] y. The collection of all events that will lead to a given set of events (the union of their past light cones) is known as the filter of that set. Within a poset, there can be sequences of elements that are totally ordered, and these are called chains. (The maximum length of any chain is sometimes called the dimension of a poset, but this is unrelated to the notions of dimension I consider.) There can also be sets of elements between which no ordering relations at all are defined, and these are called antichains.

Standard examples of posets include subsets of a set ordered by the subset relation, complex numbers ordered by magnitude, and integers ordered by divisibility. Posets first arose as general concepts in the late 1800s in connection with the development of mathematical logic, and to some extent abstract algebra. They became somewhat popular in the mid-1900s, both as formal generalizations in lattice theory, and as structures in various combinatorics applications. It was already noted in the 1920s that events in relativity theory formed posets.

The pictures below show the first few distinct possible Hasse diagrams for posets. For successive numbers of elements the total numbers of these are 1, 2, 5, 16, 63, 318, 2045, 16999, ...

From Stephen Wolfram: A New Kind of Science [citation]