There have been month of silence in this BLOG, as I was running into serious troubles beyond the elementary Axioms for QOrders, i.e. those that equip QSpaces with a suitable topology. More over it turned out that –at least for the moment an additional axiomatic relation may be needed: while the relation Q clearly models the conformal invariant structure –the Topology of SpaceTime, it misses the projective invariant part, which only jointly as shown by Weyl long time ago fix also the metric. The work isn’t done yet. Nevertheless I would like to present the current stage with its stabilized Axioms for QOrders and the first intents to manage the additional relation.
We are looking for a combinatorial framework that, in an essential way, includes the structure of SpaceTime as a continuous model on one side and the structure of of PetriNets as a finite (countable) model on the other.
Essential means that physically different SpaceTimes and logically different PetriNets shall have different models and that different models produce different SpaceTimes and different PetriNets respectively.
For SpaceTimes a seminal contribution of S. W. Hawking^{1} introduced a unique combinatorial structure –a partialorder– attached to Lorentzian Manifolds with some additional restrictions, that up to conformal mappings defines the manifold (review^{2} on Causal Spacetimes). David Malament^{3} showed how this combinatorial structure alone, under suitable conditions, is sufficient to reconstruct Spacetime up to conformalty.
For PetriNets since 1973 there as been a systematic effort^{4} to detect the underlying combinatorial structures, specifically in the theory of Concurrency or causal structures defined by eventoccurrence systems.
The problems

the mentioned SpaceTime models in their definition make a heavy use of concepts typical for the continuous world, like Hausdorffspaces as basic modeldomain or using properties borrowed from Linear Algebra, all which as such can not be transported into the finite/countable domain.

the mentioned PetriNet models namely concurrencytheory require countable models to work and therefore as such a not suited to express all the technical concepts as used in continuous models. On structural level, there is no Linear Algebra, hence appears on first sight impossible to express concepts like convex let alone tensors or more complicated constructs.

Both models depend on Global Partial Orders even when expressing purely local concepts, a slight contradiction with the basic idea of General Relativity as something locally defined.
The ideas for solution

Both model domains use Paths respectively Curves as a basic building block, where Curves in both domains model trajectories worldlines of particles (more precisely potential trajectories see^{5}). All expressed relations and properties can be rewritten using only curves and the relations among points as defined by curves.

As Petri pointed out quite early^{6}, on partial orders there exists a generalization for the concept of Dedekindcontinuity and completeness that allows for countable models, yet if applied to fullorders produces the known results. Crucial are two types of points, closed and open, while retaining the idea that the emerging topologies should be connected.

A little bit later Petri proposed the separation relation {{a,b},{c,d}} an unordered pair of unordered pairs as the basic orderproducing relation. This relation expresses the separation of 4 points on a line, and is well defined on any JordanCurve, open or closed, i.e. there is no difference between a line, may be with suitable compactification, and a circle.

A careful analysis of the original article from Hawking, specifically analyzing the relation between local timelike cones, which form the base for the topology, the definition of regular paths in that topology and their relation to timelike curves, allowed to eliminate the reference to linear concepts like convex and to define local timelike cones and their properties using only combinatorial concepts.

This revision in turn demanded a revision of concepts in PetriNets. While in the original model the open elements are conditions and the closed elements are the events, and likewise sets border by conditions considered open while sets bordered by events closed, we need exactly the dual: events and eventbordered sets are open, conditions and conditionbordered sets are closed. It should be noted that for countable structures Petrinets are normally assumed to be countable both sets open and closed define the same dual Alexandrov^{7} Topology. However already the comparison of Dedekind continuity between total orders and halforders alas OccurrenceNets shows that the common type of elements in both the nonbranching conditions must be closed.

In a Hawkingspace all points are closed. Therefore it was necessary to overcome the initial interpretation of Einstein as if world points would correspond to physical events. They do not! If the Hawkingspace models the loci the geometry, then a physical event can not have an exact place as QuantumMechanics tells us. A similar observation made decades ago Pauli^{8}. Curiously enough, in this interpretation nothing ever happens in HawkingSpace as there are no events. To have events we must coarse grain first.

Likewise a too naive interpretation by NetTheory of GRT had to be abandoned, as if each worldpoint branches into infinite many worldlines. Actually a worldpoint summarizes the whole timelike pre respectively postcones as such and not individual lines. This is the essence of the construction of regular paths by Hawking and the distinguishing conditions from Malament.
Based on the above I obtained the Axiomsets shown in the picture below.
Some models for QOrders are:
(1) OccurrenceNets (with the above change and some additional requirements) as subclass of PetriNets
(2) The Real Numbers (but not Rationals nor Integers) (Qorder is derived from classical order)
(3) The UnitCircle S1 (and the Circle Group) (but not ncyclic Groups) and the Real Line (Qorder is derived from the relation among four points)
(4) The MinkowskiSpace and the Quaterions (Qorder is derived from QTopology)
(5) The Causal structure of a Lorentzian manifold as defined by Hawking and others (Qorder is derived from the relation among four points on a timelike curve)
1 S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved spacetime which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
2Alfonso GarcíaParrado, José M. M. Senovilla, Causal structures and causal boundaries, arXiv:grqc/0501069v2
3 D. Malament, The class of continuous timelike curves determines the topology of spacetime Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 13991404
4 Olaf Kummer, MarkOliver Stehr: Petri's Axioms of Concurrency  A Selection of Recent Results , Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 2327, 1997, Lecture Notes in Computer Science 1248, © SpringerVerlag , 1997
5 David B. Malament, Classical Relativity Theory, arxiv.org/abs/grqc/0506065v2
6 Petri, C.A., Concurrency. Lecture Notes in Computer Science Vol. 84: Net Theory and Applications, Proc. of the Advanced Course on General Net Theory of Processes and Systems, Hamburg, 1979 / Brauer, W. (ed.)  Berlin, Heidelberg, New York: SpringerVerlag, 1980, Pages: 251260
7Not to be confused with the Alexandrov Topology used by Hawking
8Pauli, Vorlesungen in Turin über nichtlokale Feldtheorien in GoogleBooks http://books.google.com/books?id=NU9OUjf8cYC&hl=es Page 34 ff.
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