Lines, Cuts and Dedekind

Reformulated without changing its essential content.
Axiom 7 finally ties together classically continuous and classically countable models. As shown in a later post, it defines a new type of Topology valid for both.

Axiom 7 Continuous 

We will walk through the construction step by step.

Step 1: Lines 
Here the second definition introduces sets, where all of its elements are q-related. Then the first line defines as Lines maximal sets of this type. If one accepts the Hausdorff-Maximal Principle for chains of partially ordered sets (or equivalently the Axiom of Choice), then Lines is actually a non-empty set and for any totally q-related set there is at least on containing line.

Step 2: Generalized Dedekind cuts
Traditionally a Dedekind cut splits an ordered set into two halves, one below the other. This notion doesn’t make sense on a circle.  Now consider {a,b} as an interval on a line or a circle, then a set is defined as Generalized Dedekind cut, if with endpoints it contains either the interior of the interval or its complement, which on a circle is again an interval; likewise if we would work with the 1-point compactified Real numbers, yet there –without compactification- also the traditional cuts would do.

Step 3: Borders of a Generalized Dedekind cut and what a cut cuts
As border of a Generalized Dedekind cut we define all elements, that are not properly contained in some interval in D, properly means that there should be at least one complementary element outside. A cut cuts a set, if there are elements inside and outside the cut.

Step 4: Line-Limits of a Generalized Dedekind cut
Step 4 is crucial in the whole construction. Actually we are only interested in limiting points on lines, yet for these we will accept only those border-elements as true or objective limits, that do not depend on a specific cut, i.e. when the piece cut-out by two cuts are the same, then there should be –if any- the same limit.

Step 5: The Symmetry-Axiom

In the traditional definition of a Dedekind-cut, either the lower set has a maximal or the upper set has a minimal element, whence crossing the border a maximal element becomes minimal vice versa. 

Step 6: The Cut-Axiom

After these lengthy preparations, the final axioms is short: all Lines should have well defined limits for their pieces, when cut by a Generalized Dedekind cut .

Some Remarks:

1. The basic idea for this type of Generalized Dedekind-cut for partial orders appears already in early papers of Carl Adam Petri. While keeping the basic idea, our definition of a cut itself is different and not only limited to combinatorial i.e. countable partial orders. Yet Petri was the first to note that Dedekind-complete for partial orders might be something quite different than rather requiring classical continuity on all lines (respectively all completely ordered subsets).
2. Nonetheless, for sets consisting only out of a single line or that are totally ordered sets, our definition is equivalent to the one introduced by Dedekind. Hence the Real Numbers are Dedekind-complete respectively –continuous, while the Long Line and the Long Ray are not. And –as intended- the Real Circle is Dedekind–continuous.
3. In earlier attempts –see older parts of this BLOG- I tried to avoid the use of the Hausdorff principle respectively the Axiom of Choice, due to their non-constructive character. These attempts failed so far. So to escape from being stuck in an non-essential detail, I finally accepted the Definition for Lines as a building block.

This ends the presentation of the 7 Axiom-sets for Q-Spaces.

Revised: 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  and 6 define then how the world appears to us.

Axiom 6 Coherent

Axiom 6 is still quite easy to understand: it claims that the relation Q is connected by q-sets. Whatever it’s cut into two pieces, there will be always 2 q-sets that differ only in 1 element to connect the pieces. This claim generalizes Axiom 4 Ordered 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 Axiom 5, which is intrinsically related to one of the most complicated problems in Algebraic Topology: the problem 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.

Axiom 5 Oriented

For our current purpose of motivation only, some picturesque  considerations and some simple supporting lemma hopefully are 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 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 on one circle, 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 desired rule.
Lemma A 11 
At this stage we know therefore also that to be consistently clockwise oriented on any circle all quintuples of 5 points on this circle must fit into the above definition and that if the rule expressed as lemma for one circle holds.

Axiom 5.2 converts hence the 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 of any circle. We define then that Q allows a consistent orientation.

Due to Axiom 6, 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. (Question for the interested reader: why there can’t be a counter-experiment to axiom 5).

With these considerations, we end the presentation of the first 6 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

Revised: Going in Circles – Part II

In Going in Circles – Part I we presented the basic reasoning underlying Q-Orders by analyzing the configurations of elements on 1 Circle and the combination of 2 circles. Already in the discussion of Axiom 4 we said informally that the combination of circles should create new ones and specified which would be permitted based on pictures.

Yet it would be a fruitless and hence useless task, to continue that way, since very rapidly we would be defeated by combinatorial explosion, as 4 elements on 1 circle already allow 3 configurations according to Axiom 1 and Lemma A 1. We need urgently rules of interference! For more than 4 elements on 1 Circle, Lemma A 2 solved the problem, as shown by Lemma A 5.

Yet for the combination of more than 2 circles –at least 3^3 = 27 possibilities- we have no rules yet and analyzing even in this case the configurations one by one more would be very clumsy, and literally impossible for only countable many elements.

A part of the problem is solved by Axiom 3.

Axiom 3  Regular

It claims that

1. whenever all pairs of 3 points can be found on some circle –regardless their configurations there-, there should be a 4th element to give a complete circle, where all 4 may be positioned.
2. whenever all triplets of 4 points have their circle, the 4 points themselves may be positioned on 1 circle.

As this is a global claim, Axiom 3 is truly an Axiom i.e. a Thinking – Hypothesis that may be false in the real world (as the Parallel Axiom in non-Euclidian Geometries). Actually I’ve done some investigation on the consequences, if it were false in the frame work of Q-Orders. Due to its hypothetical character we will flag further on any proof that uses it.

Sorry, no pictures for Axiom 3! (it would have been 91 for 3.1 and 271 for 3.2) …. but whoever has the time and resources to draw them up all, including the configuration for the resulting circle, is kindly invited to provide them, or even better a small animation applet …).

Axiom 3 and 4 combined express a second Thinking Hypothesis as a rule of interference to allow later on for consistent orientation.

Lemma A.6

Whenever a pair {c,d} is separated by another pair {a,b}, and this pair appears on the same side of some circle , then {a,b} separate {c,d} also on this circle. Or more loosely, if {a,b} separates {c,d}, then it does so anywhere or even more general: {{a,b},{c,d}} is a universal relation.

This Lemma has far reaching consequences as it’s implied by the concept of transitivity of partial orders and likewise by the concept of continuous functions in Real Analysis.

Then Lemma 7 shows that a pair {u,v} not only separates all circles that run trough {u,v} but cuts them into two equivalence classes of points- say inner-outer, left-right, upper-lower - without defining which is which.

Lemma A 7 

Q-Orders sometimes have adjacent points:
Lemma A 8 Adjacent Points 

Adjacent are those points which can not be separated by Q. It’s a local property, i.e. something that can’t be separated by one interval, can’t be separated by any other.

With the above, we are done with all local Axioms, remaining only the Axioms 5 and 6  that will generalize our findings about orientation.

Revised: Going in Circles – Part I

Strange as is may sound, to me Axioms are not just formal statements –obviously if we like to do Mathematics later on, they have to be also formal statements in some formal language-, but rather the intent to express as precisely as possible a concept found in reality, Plato would have said an idea.

In this post we’re going to show for 3 of the first 5 Axioms how they have been found and what was the essential input for the remaining 2 of them. Our basic idea are the circle and relations among 4 distinct points on that circle, expressed by an unordered pair of 2 unordered pairs.

Axiom I Q-Relation

Why this relation? Let’s have a look on a circle and just any 4 distinct points and their possible configurations.

To avoid any premature symbolic interpretation, we use 4 small hearts in 4 different colors to mark the 4 distinct points. Then the simplest way to express the only invariant of these configurations is by stating that the set {red heart, yellow heart} separates the set {blue heart, green heart}, a statement that remains true from which ever side we look on the circle; from before, from behind, from top, from bottom, from left, from right … and anywhere in-between.

The remaining definitions just introduce short notations for the relation itself and to denote that 4 points occur actually jointly on one Circle.

As easily seen, any 4 distinct points may be grouped into exactly into 1 set of 2 sets. As candidate for an axiom we note therefore (it will be a consequence of our final Q-Order Axiom 4):
Lemma A.1 

Any additional statement would need additional information from either some external frame of reference –like the blue heart is on the left- or an additional convention to describe a way how one heart after another may be visited, i.e. an inner orientation. We note that apparently there are just 2 inner orientations.

Axiom 2 Reduced
 
Axiom 2 states then that all shall be expressed in terms of points and circles, such that 2 points which appear anywhere –at any circle- in identical configurations shall be considered identical. And for the sake of completeness, we eliminate the all-isolated and the all-connect points.

What happens if a 5th heart enters the game? Let’s see: The 4 purple hearts with 5 show just the 4 possible positions of the 5th point. Fortunately we may describe its position exactly using only the already defined relation Q, as the next axiom candidate show (again as Lemma a consequence of the later Axiom 4)

Lemma A.2 

In words: If 5 points are on a circle and for 4 of them their configuration is known, then the 5th falls in exactly 1 of 4 alternatives. As a consequence, the configuration of any set of distinct points on a circle may be defined using only the relation Q.

In the finite case, there are (n-1)! circle-configurations for n elements. As stated already elsewhere, the Sumerian used 5, 6 and 7 elements and their configurations as the base for their mathematics and geometry.

Looking on the above configurations, we find a triplet of relations of 5 points that never occurs:

Lemma A.3 

It just expresses that {a,b} splits the circle in 2 halves.

What if we have two circles and not only one? Axiom 4 deals with these cases and the above at the same time.

Axiom 4 Ordered

Let’s picture the first case. Obviously there should be some new circles formed by elements of both and the new circles will share the elements a and b. Yet not all combinations of 4 new circles would be mutually compatible (lemma 3). Axiom 4.1 offers the only two choices possible: (an inner {{a,b},{d,e}} and an outer circle {{a,b},{c,f}}) or (a left {{a,b},{c,e}} and a right circle {{a,b},{d,f}}).

Now the second case:

Again not all possible combinations would be compatible. Axiom 4.2 offers the only two choices possible: Either an inner circle with 5 elements {a,b,d,e,f} or an outer circle with again 5 elements {a,b,c,e,f}.

Please note that the terms like inner, outer, left and right have no intrinsic meaning yet. They just may help in seeing the circles.

Finally the third possible case:  Here the original circles are {{a,c},{d,b}} and {{a,f}},{{e,b}} and almost any additional circle is possible, as that there is none at all. Hence Axiom 4 makes no claims for this situation.

Lemma 1 and 2 were initial working hypotheses (candidates for axioms). We leave it to the reader that now - as intended -they are consequences of Axioms 1, 2, 4.1 and 4.2.

We will go back for a moment to a single circle and show how implicitly there is an order defined as soon as one decides, which points should be the beginning and the end respectively.

With Lemma 2 already holds the following:
Lemma A.4  
This may be used, once fixed a and b to define a partial order.

Lemma A.5 

Have a look on the following picture

and you easily verify that {{b,n},{m,e}} as defined for the circle, will order the whole set {1,2,3,4,5,6} as one would expect. Note that inverting the order of (b,e) to (e,b) inverts the order of the set, again as one should expected.

The dual orientation of circles is the underlying principle. The remaining parts of the first 6 Axioms will insure that there is a consistent orientation –up to duality- for the whole Q and in all of its admitted circles, by virtue of which in any Interval we may operate as usual with partial orders.

Revised Axioms for Q-Spaces: Q-Orders

There have been month of silence in this BLOG, as I was running into serious troubles beyond the elementary Axioms for Q-Orders, i.e. those that equip Q-Spaces 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 Space-Time, 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 Q-Orders 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 Space-Time as a continuous model on one side and the structure of of Petri-Nets as a finite (countable) model on the other.

Essential means that physically different Space-Times and logically different Petri-Nets shall have different models and that different models produce different Space-Times and different Petri-Nets respectively.

For Space-Times a seminal contribution of S. W. Hawking¹ introduced a unique combinatorial structure –a partial-order– attached to Lorentzian Manifolds with some additional restrictions, that up to conformal mappings defines the manifold (review² on Causal Space-times). David Malament³ showed how this combinatorial structure alone, under suitable conditions, is sufficient to reconstruct Space-time up to conformalty.

For Petri-Nets since 1973 there as been a systematic effort⁴ to detect the underlying combinatorial structures, specifically in the theory of Concurrency or causal structures defined by event-occurrence systems.

The problems

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

2. the mentioned Petri-Net models -namely concurrency-theory- 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.

3. 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

1. Both model domains use Paths respectively Curves as a basic building block, where Curves in both domains model trajectories -world-lines- of particles (more precisely potential trajectories see⁵). All expressed relations and properties can be re-written using only curves and the relations among points as defined by curves.

2. As Petri pointed out quite early⁶, on partial orders there exists a generalization for the concept of Dedekind-continuity and -completeness that allows for countable models, yet if applied to full-orders produces the known results. Crucial are two types of points, closed and open, while retaining the idea that the emerging topologies should be connected.

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

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

5. This revision in turn demanded a revision of concepts in Petri-Nets. 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 event-bordered sets are open, conditions and condition-bordered sets are closed. It should be noted that for countable structures -Petri-nets are normally assumed to be countable- both sets -open and closed- define the same dual Alexandrov⁷ Topology. However already the comparison of Dedekind continuity between total orders and half-orders alas Occurrence-Nets shows that the common type of elements in both -the non-branching conditions- must be closed.

6. In a Hawking-space 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 Hawking-space models the loci -the geometry-, then a physical event can not have an exact place as Quantum-Mechanics tells us. A similar observation made decades ago Pauli⁸. Curiously enough, in this interpretation nothing ever happens in Hawking-Space as there are no events. To have events we must coarse grain first.
   

7. Likewise a too naive interpretation by Net-Theory of GRT had to be abandoned, as if each world-point branches into infinite many world-lines. Actually a world-point summarizes the whole time-like pre- respectively post-cones 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 Axiom-sets shown in the picture below.

Some models for Q-Orders are:

(1) Occurrence-Nets (with the above change and some additional requirements) as subclass of Petri-Nets
(2) The Real Numbers (but not Rationals nor Integers) (Q-order is derived from classical order)
(3) The Unit-Circle S1 (and the Circle Group) (but not n-cyclic Groups) and the Real Line (Q-order is derived from the relation among four points)
(4) The Minkowski-Space and the Quaterions (Q-order is derived from Q-Topology)
(5) The Causal structure of a Lorentzian manifold as defined by Hawking and others (Q-order is derived from the relation among four points on a time-like curve)

1 S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved space-time which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
2Alfonso García-Parrado, José M. M. Senovilla, Causal structures and causal boundaries, arXiv:gr-qc/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. 1399-1404
4 Olaf Kummer, Mark-Oliver 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 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997
5 David B. Malament, Classical Relativity Theory, arxiv.org/abs/gr-qc/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: Springer-Verlag, 1980, Pages: 251-260
7Not to be confused with the Alexandrov Topology used by Hawking
8Pauli, Vorlesungen in Turin über nichtlokale Feldtheorien in Google-Books http://books.google.com/books?id=NU9OUj-f8cYC&hl=es Page 34 ff.