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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).
The growth process is depicted by the poset of finite causal sets, sometimes called poscau, in which two causets are related (in the poset's order) if one can reach the other by a sequence of sequential growth.
(Equivalently, a causet A comes before, or precedes, B, if B contains a past-closed subcauset which is isomorphic to A.) I have generated diagrams of poscau for all causets up to four elements, and five.
It is this `hyper-connectivity' that allows them to maintain Lorentz invariance in the presence of discreteness.
(In the case of causal sets, it is the mean valence of the Hasse diagram which is important, not of the causal relation itself.) Below is a demonstration of the Lorentz invariant character of causal sets.
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.
I grant permission to use the images in presentation slides, provided the attribution and copyright notice remain intact. ] Below are some Hasse diagrams of random causal sets generated by the transitive percolation dynamics.
Such discrete structures cannot hope to capture the noncompact Lorentz symmetry of spacetime.
Causal sets, however, have a `mean valence' which grows with some finite power of the number of elements in the causal set.
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.