The AxiomSets for QSpaces
There has been another round of silence in this BLOG, partially to attend some breadandbutter business, partially to solidify the inclusion of a second basic relation besides order, a relation that reflects locally geometry: while the relation Q clearly models the conformal invariant structure –the Topology of SpaceTime, it misses the projective invariant part, which as shown by Hermann Weyl^{1} long time ago is needed to define the complete EinsteinWeyl Causality.
There as been a second change: while all earlier versions of the Axioms constructed the Topology using the Qorder relation, this version puts the definition of a class of Topologies at the beginning and then ties the Qorder relation to that topology. There are two reasons for this change: first it comes closer to the classical definition of the basic realm of classical General Relativity: a topological space with additional properties i.e. as differential manifold (which ties in geometric concepts) with an additional Lorentzmetric (which introduces order).
The second reason: the class of topologies singled out includes the topology of all PetriNets without directions and the topology of all locally pathconnected spaces, which appears as the common topological ground for both: coarse graining a locally pathconnected space such that grains are either open or closed one arrives at a PetriNet and reverse: blowing up a PetriNet i.e substituting its elements by convex open respectively closed sets one gets a locally pathconnected space.
Introduction
We are searching 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^{2} introduced a unique combinatorial structure –a partialorder– attached to Lorentzian Manifolds with some additional restrictions, that up to conformal mappings defines the manifold (review^{3} on Causal Spacetimes). David Malament^{4} showed how this combinatorial structure alone, under suitable conditions, is sufficient to reconstruct Spacetime up to conformal factor.
There has been another approach somewhat close to ours. HansJürgen Borchers and Rathindra Nath Sen reconstruct the complete EinsteinWeyl Causality^{5} starting from the total order on light rays. Light rays in a certain sense connect the conformal and the projective structure by their inherent order and being locally geodesic. Yet the authors still assume a global partial order and that light rays are orderdense, which precludes finite structures i.e. Petrinets.
For PetriNets since 1973 there as been a systematic effort^{6} 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 are 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.

Geometry without additional constraints can not be derived from order alone. It must be introduced as an additional concept. It's long known that linegeometry i.e. Geometry based on Points, Lines and Incidences has finite, countable and continuous models. Yet the concept of a geodesic line is neither present in Causal Structures nor as far as I know in Petri NetTheory.
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^{7}). 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^{8}, 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^{9}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 a 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^{10}. 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.

W.r.t. Geometry, we will start at the most elementary level: locally a line shall be uniquely defined by 2 points, locally any 2 points shall be connected by a line, finally the geometry shall be nontrivial i.e the local space shall be connected by lines with at least 3 points. Obviously, once defined, lines shall split into the three known classes: timelike, lightrays, spacelike.
1 Hermann Weil, ZeitRaumMaterie, III. Edition, Julius Springer Verlag, Berlin, 1919
2 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
3 Alfonso GarcíaParrado, José M. M. Senovilla,
Causal structures and causal boundaries,
arXiv:grqc/0501069v2 4 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
5 HansJürgen Borchers, Rathindra Nath Sen, Mathematical Implications of EinsteinWeyl Causality, Lect. Notes Phys. 709 (Springer, Berlin Heidelberg 2006)
6 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
7 David B. Malament,
Classical Relativity Theory,
arxiv.org/abs/grqc/0506065v2 8 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
9 Not to be confused with the Alexandrov Topology used by Hawking
10 Pauli,
Vorlesungen in Turin über nichtlokale Feldtheorien in GoogleBooks
http://books.google.com/books?id=NU9OUjf8cYC&hl=es Page 34 ff.
Based on the above I obtained the Axiomsets shown below. In the second part of this post we will comment on each set.
The Axioms
The Topology and QOrder Axioms
Geometry and Compatibility Axioms
The Axiomsets group by group
Topology Axioms
Axiom 1 TOP Topological Space
Axiom TOP defines a Topology with 2 additional properties, connected and distinguishing or T0 i.e. different points have different open neighborhoods.
Axiom 2 PAT Path connected
Normally locally pathconnected is defined by the existence of a continuous mapping of the [0,1] interval into the space or subsspaces. As this mapping carries some of the topological characteristics of R, we first define onedimensional connected subspaces J, and then claim the existence of neighborhoods where all points are connected by some of these subspace in J (the letter reflects JordanCurves, actually subspaces in in J are the traces of continuous, monotone mappings of S^{1})
Local QOrder Axioms
Axiom 3 REL QRelation
Axiom 4 LIN QLinear
Axiom 5 ORD QOrdered
The above QOrder Axioms remain unchanged compared to earlier versions. The model the situation of points on closed curves. For details and the underlying heuristics please consult: Revised: Going in Circles – Part I and Revised: Going in Circles – Part II.
Global Orientation of QOrders
Axiom 6 ORI QOriented
Axiom ORI combines into a single Axiomset the necessary and sufficient conditions to extend an orientation once it’s define on a single circle. For details and the underlying heuristics please consult: Revised: Going in Circles – Part III. It should be noted that partial orders without identical points satisfy all the above QOrder Axioms. The formal proof that a Lorentzian Causal Order is orientable iff it’s orientable in the sense of Qorders is still incomplete.
Generalized Dedekind Completeness of QOrders
Axiom 7 COM QComplete
Axiom COM is a localized version of the Generalized Dedekind Completeness for partial orders as introduced first by Carl Adam Petri. Whereas the original version by Petri and our own introduces and uses Lines as maximal totally ordered sets, this version uses only totally ordered Intervals. The essence that any dcut that cuts an interval defines a unique limit point for that interval, remains the same. Crucial is the distinction between border points bdr and limit points lim. is For details and the underlying heuristics please consult: Lines, Cuts and Dedekind.
When thinking in Lorentzian Causal Order, remember that QOrders correspond to timelike curves, hence the tip of a timelike cone (i.e. a lightcone without its border) is a limit to the whole cone, not only to each of its lines. If –as in nondistinguishing causal structures see David Malament^{4}  there are more but one tippoint, these are not timelike to each other, hence they remain unique for each timelike curve. In the case of QOrders and PetriNets, limit points are the singleentry – single exit elements. A Qcomplete PetriNet resembles an occurrencenet with conditions as limits.
Compatibility between QOrder and Topology
Axiom 8 CON QContinuous
Axiom CON claims the compatibility between limitpoints and closed points in the underlying topology, and sets which contain an interval for each of its closed points and open sets of the underlying topology. In QTopology – I we showed that the Qorder Axioms themselves are sufficient to construct a topology. In a yet to be published post we will show that this topology satisfies the new Topology Axioms. Moreover topology + incomplete QOrder for the countable case still define a globally orientable PetriNet. Finally there is a procedure to complete Qorders, such that the completed QOrder is compatible.
Geometry Axioms
Axiom 9 GEO Geometry
The GeometryAxioms try to capture the most elementary of a local geometry. The second line defines a new threeelement unordered relation g. The third line defines normal sets as sets which (1) are connected by the geometryrelation, where (2) different elements can be separated and (3) any three points define a unique set of grelated elements. Before looking into more details, let’s translate these conditions in the usual terminology for Incidence Geometry.
Definition 1 Linear Space
The first line defines the set of lines belonging to a normal set. The following Lemmata are straight forward almost by definition: (1) any two points lie on at least 1 line, any line has at least 2 different points on it. (2) any point lies on at least 2 different lines and finally (3) 2 different lines cut each other at at most 1 point. The later implies that 2 points define a unique line.
These structures are well known as Linear Spaces. The structure arises naturally if ones selects or finds a set a set of points out of a projective or affine geometry or similar, with the additional constraint that the set shall be distinguishing: different points differ in at least one line they lie on. Or from a different perspective: it’s what one obtains by selecting points out of a normal neighborhood of a Lorentzian Manifold, taking the geodesics as lines.
Going back to Axiom GEO any point shall be contained in some normal set and the class of normal sets connected w.r.t. to g.
Compatibility between QOrder and Geometry
Axiom 10 COH QCoherent
The axiom claims that lightrays are geodesic and that any geodesic is either timelike, spacelike or lightlike. The explanation and heuristics for the crucial second definition of lightray related will be provided soon.
Problems remaining
We would like to embed Linear Structures into Vectorspaces as an initial step towards the construction of Tangentbundles, which will be needed if we would like to have something like Fieldequations