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 set*s {{a,b},{c,d}}, the nails *are the points, Axiom 5 and 6 define then how the *world* appears to us.

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.

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

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