All these are explained in detail in my paper with Rafael Sorkin on the subject, or in my Ph. Please write me at the above email address to inquire on details.
This `time' in not some physical external time, but is `purely gauge'.
One might regard physical time as being `embodied in the growth', rather than the growth `occuring in time'.
For example, each `node' of a Cartesian lattice in three dimensions has six nearest neighbors.
Random spatial lattices, such as a Voronoi complex, will similarly have valences of order 1 (or perhaps more properly of order of the spatial dimension).
A familiar example of a causal set is a family tree. It is plotted using the convention of spacetime diagrams in which the oldest ancestors appear at the bottom.
My uncle James Lynn Rideout has compiled an extensive database of 8000 of our relations, which is depicted here. Note that the only parent-child relationship is considered.The bottom left image shows the same points as viewed by an observer moving at v=-4/5.The same region (by the v=0 observer's coordinates) is blown up on the right.Two Topics in Elementary Particle Physics: (1) Quark Graphs and Angular Distributions in the Decays of the Axial-Vector Mesons.(2) Universal Current-Current Theories and the Non-Leptonic Hyperon Decays.Causal sets are discrete partially ordered sets, which are postulated to be a discrete substratum to continuum spacetime.The order gives rise to macroscopic causal order, while the discreteness or 'counting' gives rise to macroscopic spacetime volume.The green or blue arrows indicate gregarious children (those which arise when the new element appears unrelated to any others), and red arrows timid children (the child which occurs when the new element appears above the entire parent causet).The numbers attached to the edges give the "number of ways to transition from the parent to the child". [These diagrams are available in a variety of file formats, colors schemes, and licenses.The colors of the links do not play an essential role.(The purple links connect elements on neighboring 'layers', where the layer of an element is the length of the longest past directed chain which ends at that element.