The Great Simplification – Part II

After slashing the original definition of Axiom 6, here comes a similar reduction of Axiom 7, originally introduced in The trapped Arrow of Time – Part III.

The new
Axiom 7 Q-Loops
    
Axiom 7.1 are technical definitions:  completely Q-ordered sets and the closed hull of a set. 

Axiom 7.2-4 introduce a substitute for Jordan-Curves. 7.2 defines the property of being connect for a set in terms of topology. 7.3 defines the relation of being topologically separated for two points.  Please note that we don’t require that the whole space is T1. 7.4 defines J-Curves as sets that contain for each point at least one separated partner and fall apart exactly if a separated pair is removed. This definition requires implicitly the Axiom of Choice, therefore it’s flagged.

For the moment let’s assume that J-Curves exist -later on we will claim the existence of rather specific ones- and see whether the definition meets our expectations. Axiom 7.5 introduces the set of connected subsets into which a separated pair splits the J-Curve. By definition for J-Curves, there must be at least 2 of them.    
Lemma S 2 Segments  
 
The lemma shows that –as intended- a separated pair splits a J-Curve in just 2 segments. And these 2 segments contain at least each a point for a separated pair, that is we’ve got two pairs that mutually separate each other.

Sounds  familiar? Well, be aware that the term J-Curve was introduced and some of it’s properties shown without reference to the initial Q-Order, yet –as intended- on a J-Curve there exists a natural Q-Order, as defined by Axiom 7.6.
Lemma S 3 J-Order 

Be aware that by no means all J-Curves correspond to q-ordered sets, as shown below. 

The above yellow Rhombus  is a  J-Curve, yet it’s not a Q-ordered set but rather built by of 2 Q-ordered sets, the left and right side.

It appears as if we might start out just with some topology with some nice properties … and [re-]construct the Q-Order. For the moment we ask only –consistent with our whole approach- that every totally q-ordered set shall be consistently embeddable into some J-Curve, where consistency means that original Q-Order and derived Q-order of the curve are the same.

Axiom A 7.7 Consistent Embedding 

Axiom A 7.7 has backward consequences for the Q-topology.
Lemma S 4 Connected Space 
  
As immediate consequence of Axiom 7.6, J-Curves finally do exist. J-Curves connect the whole space (S 4.2), which is hence a connected topological space (S 4.3).

Finally
Axiom 7.8 Local Orientation

This Axiom establishes an intrinsic relation between J-Curves –remember they are closed- and the underlying Q-Topology. For each point –respectively its closed hull- there shall exist at least one neighborhood, sufficiently large that it can be split into two subsets –sometimes called local future and local past, or local input and local output- such that  each of these subsets is J-convex – i.e. any two points can be connected by a J-Curves, but sufficiently small that any J-Curve that connects between the sets contains at least one external point.

The picture below illustrates the concept.

The two sets are
{ {(0,0),(-1,0)}, (-1,0), {(-1,0),(-1,1)}, (-1,1), {(-1,1),(0,0)}, {(-1,1),(0,1)}, (0,1),{(0,1),(0,0)} }
{ {(0,0),(0,-1)}, (0,-1), {(0,-1),(1,-1)}, (1,-1), {(1,-1),(0,0)} {(1,-1),(1,0)}, (1,0),{(1,0),(0,0)} }

Axiom 7.8 requires at least two dimensions (or two J-Curves). As to be shown, it captures the  underlying  essence of the Hawking-construction  for regular curves, which in the original text is scattered between local properties of the manifold –existence of local convex neighborhoods in terms of the Manifold-Topology, global causality-conditions –strong causality-, all needed to effectively define Regular Curves, and finally the properties  defined by the construction as such.

This ends our preliminary presentation of the new Axiom 7 Q-Loops.

Going in Circles – Part III

There is an old saying When the only tool at hand is a hammer, the world appears to be a bunch of nails. Our hammer are the sets {{a,b},{c,d}}, the nails are the points, Axiom 5 defines then how the world appears to us.

Axiom 5 ORI Orientation

Axiom 5.3 is still quite easy to understand: it claims that the relation Q is connected by q-sets. Whatever cut into two pieces, there will be always 2 q-sets that differ only in 1 element to connect the pieces. This claim generalizes Q-Order Axiom 4 ORD as a careful inspection shows: the implications of Axiom 4 establish exactly this type of connection between their left and right sides.

Lemma A 9 shows that any two q-sets can be connected in a finite number of steps.

Lemma A 10 shows how this connectivity extends to Q: any two points are connected through a finite number of q-sets.

Hence our world is, as intended, connected only by hammers and nails. In a later post about Q-Topology we will see the equivalence to path-connected.

Now the harder parts of Axiom 5, which are intrinsically related to one of the most complicated problems in Algebraic Topology: the problems whether a space has a systematic orientation. In our case there is also a close, independent and direct relation to Group Theory. Both relations will be touched, maybe, in some later posts.

It should be mentioned that there have been related, relevant investigation also in the realm of Petri-Nets, e.g. the Orientation of Concurrency Structures by Olaf Kummer and Mark-Oliver Stehr(1), the Construction of globally Cyclic Orders by Stehr(2) and the proposal of cycloids as basic building blocks by Carl Adam Petri. Again these subjects will be touched, maybe, in a later post.

For our current purpose of motivation only, some picturesque considerations and some simple supporting lemma appear sufficient.

We know already that the points on any circle may be visited one after one in exactly two fashions or orientations: clockwise or counter-clockwise. And –as the following picture shows once more- interchanging (a,b) to (b,a) in {{a,b},{c,d}} reverses the direction.

Axiom 5.1 constructs first a set, a double-cover, that for each {{a,b},{c,d}} contains exactly two sets of sequences to run the circle, either clock- or counter-clockwise. We do not define which is which (same column = same orientation).

The idea of double-covers in Topology is to split the double-cover later into 2 subsets such that each original element has exactly 1 orientation, applying consistency rules to guarantee a orientation defined as uniform for all elements. From our experience with 5 points, we know it should be sufficient to have a consistency rule just for 5 points and then extend this rule. Let’s see.

Lemma A 2 tells us which are the q-sets for this configuration, which enables us to construct manually the double-cover and the desired single-covers .. to find the rule.
Lemma A 11
 

At this stage we know that to be consistently clockwise oriented all quintuples of 5 points must fit into the above definitions and that then the rule expressed as lemma holds.

Axiom 5.2 converts these findings for 5 points into a claim. We claim: there shall exist a split of the double-cover into two single-covers, such that the found rule holds for all quintuples in each of the single-covers. We define then that these Q allow a consistent orientation. Due to Axiom 5.3, there are exactly 2 possible global orientations. Therefore it’s sufficient to define the orientation for any {{a,b}},{{c,d}} to have it defined it for all Q or: there is only one direction of the arrow of time -even though it might be actually cyclic- and its orientation extends from anywhere to everywhere in finitely many steps.

Though somewhat surprising, it’s almost mathematical routine to convert a property found in a specific case into an axiomatic claim: all new model shall have this same property. And it’s relatively risk-less w.r.t.o inconsistencies as the specific case serves as a model in the sense of formal logic consistency.

However and again this is an axiomatic claim, not a fact: the world should be as we think it were, something that can not be proven, only disproven by a found counter-experiment. Similar as in the preceding cases of global claims, we will raise a red flag whenever Axiom 5.2 is actually used. (Question for the interested reader: why there can’t be a counter-experiment to axiom 5.3?)

With these considerations, we end the presentation of the first 5 Axiom-Sets for Q-Orders.

(1) Olaf Kummer and Mark-Oliver StehrPetri's Axioms of Concurrency - A Selection of Recent Results , Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997
(2) Mark-Oliver Stehr Cyclic Orders: A Foundation for Concurrent Synchronization Schemes. University Paris 7, Group Preuves, Programmes et Systemes, December 16, 2003. Part I: Thinking in Cycles